In the present article, we study a vector optimization problem involving convexificator-based locally Lipschitz approximately convex functions and give some ideas for approximate efficient solutions. In terms of the convexificator, we approximate Stampacchia-Minty type vector variational inequalities and use them to describe an approximately efficient solution to the nonsmooth vector optimization problem. Moreover, we give a numerical example that attests to the credibility of our results.
Citation: Faizan A. Khan, Rohit K. Bhardwaj, Tirth Ram, Mohammed A. S. Tom. On approximate vector variational inequalities and vector optimization problem using convexificator[J]. AIMS Mathematics, 2022, 7(10): 18870-18882. doi: 10.3934/math.20221039
[1] | Ruifang Yang, Shilin Yang . Representations of a non-pointed Hopf algebra. AIMS Mathematics, 2021, 6(10): 10523-10539. doi: 10.3934/math.2021611 |
[2] | Haijun Cao, Fang Xiao . The category of affine algebraic regular monoids. AIMS Mathematics, 2022, 7(2): 2666-2679. doi: 10.3934/math.2022150 |
[3] | Mingze Zhao, Huilan Li . A pair of dual Hopf algebras on permutations. AIMS Mathematics, 2021, 6(5): 5106-5123. doi: 10.3934/math.2021302 |
[4] | Qining Li . Double Ore extensions of anti-angle type for Hopf algebras. AIMS Mathematics, 2022, 7(7): 12566-12586. doi: 10.3934/math.2022696 |
[5] | Yaguo Guo, Shilin Yang . Projective class rings of a kind of category of Yetter-Drinfeld modules. AIMS Mathematics, 2023, 8(5): 10997-11014. doi: 10.3934/math.2023557 |
[6] | Yang Zhang, Jizhu Nan . A note on the degree bounds of the invariant ring. AIMS Mathematics, 2024, 9(5): 10869-10881. doi: 10.3934/math.2024530 |
[7] | Panpan Jia, Jizhu Nan, Yongsheng Ma . Separating invariants for certain representations of the elementary Abelian p-groups of rank two. AIMS Mathematics, 2024, 9(9): 25603-25618. doi: 10.3934/math.20241250 |
[8] | Piyapat Dangpat, Teerapong Suksumran . Regularity of extended conjugate graphs of finite groups. AIMS Mathematics, 2022, 7(4): 5480-5498. doi: 10.3934/math.2022304 |
[9] | Fengxia Gao, Jialei Chen . Conjugacy classes of left ideals of Sweedler's four-dimensional algebra H4. AIMS Mathematics, 2022, 7(5): 7720-7727. doi: 10.3934/math.2022433 |
[10] | Hashem Bordbar, Sanja Jančič-Rašovič, Irina Cristea . Regular local hyperrings and hyperdomains. AIMS Mathematics, 2022, 7(12): 20767-20780. doi: 10.3934/math.20221138 |
In the present article, we study a vector optimization problem involving convexificator-based locally Lipschitz approximately convex functions and give some ideas for approximate efficient solutions. In terms of the convexificator, we approximate Stampacchia-Minty type vector variational inequalities and use them to describe an approximately efficient solution to the nonsmooth vector optimization problem. Moreover, we give a numerical example that attests to the credibility of our results.
Throughout this paper, 𝕜 is assumed to be an algebraically closed field of characteristic 0, and all matrices, vector spaces, algebras and Hopf algebras are assumed to be over 𝕜.
Roughly speaking, the concept of Hopf algebras is still valid under taking the linear duals. This elementary point of view suggests to us that structures of the dual are effective when we deal with certain problems on a Hopf algebra. The most basic definition of the dual for a Hopf algebra H would be the finite dual H∘ described by Heyneman and Sweedler [11], which is also a Hopf algebra over the same base field. Of course, there are a number of relationships between H and H∘, and this motivates the authors to consider the structure of duals for varieties of Hopf algebras.
Of course it is known that when H is a finite-dimensional Hopf algebra, its dual Hopf algebra H∗ has the same dimension as well as similar or dual properties with H. Indeed, the structures of H are determined conversely by those of H∗. However for an infinite-dimensional Hopf algebra H, its finite dual H∘ might be much too large (or small) to be studied easily. This motivates us to consider some kinds of dual Hopf algebras H∙ for certain (infinite-dimensional) Hopf algebras H, where the original evaluation is replaced by Hopf pairings H∙⊗H→𝕜. Our primary goal is to formulate such Hopf pairings for noetherian affine Hopf algebras H of finite GK-dimensions, such that:
(HP1) The pairing is non-degenerate;
(HP2) H∙ and H are both affine and noetherian;
(HP3) GKdim(H∙)=GKdim(H).
Meanwhile, we wish that some properties of H∙ and H would be dual to each other (such as (HP4) in Section 6.2), which should be noted in different situations. Finally, H∙ would be better to exist uniquely for any given H under our assumptions. In this paper, we try to establish the desired Hopf pairings for affine prime regular Hopf algebras H of GK-dimension one. For this purpose, the finite dual H∘ of each H is determined, and afterwards H∙ would be chosen as a suitable Hopf subalgebra of H∘.
However, it is not easy to determine the structure of H∘ for a Hopf algebra H in general, especially for infinite-dimensional ones. Of course, the most direct way to get H∘ is by definition. For this, recall that H∘ is the Hopf algebra generated by f∈H∗ which vanish on an ideal I⊆H of finite codimension. This means that we need a description of all finite codimensional ideals which is impossible in general. To the authors' knowledge, there are two other ways to get H∘ if H is good enough. One is applying the well-known Cartier-Konstant-Milnor-Moore's theorem [20] if H happens to be commutative. The related idea and method were generalized further (see[7] and [21,Chapter 9]). Another one is applying representation theoretical way if the representation category Rep-H of finite dimensional modules happens to be very nice (see [27]).
In the literature, significant progress has been made in classifying infinite dimensional noetherian Hopf algebras of low Gelfand-Kirillov dimensions (GK-dimension for short). The program of classifying Hopf algebras of GK-dimension one was initiated by Lu, Wu and Zhang [18]. Then the second author found a new class of examples about prime regular Hopf algebras of GK-dimension one [16]. Brown and Zhang [5] made further efforts in this direction and classified all prime regular Hopf algebras of GK-dimension one under an extra hypothesis. In 2016, Wu, Ding and the second author [32] removed this hypothesis and gave a complete classification of prime regular Hopf algebras of GK-dimension one, in which some non-pointed Hopf algebras were found. Recently, the second author classified prime Hopf algebras of GK-dimension one satisfying two certain conditions [17]. For Hopf algebras H of GK-dimension two, Goodearl and Zhang [10] gave a classification of the case when H is a domain satisfying Ext1H(𝕜,𝕜)≠0. For those with vanishing Ext-groups, some interesting examples were constructed by Wang, Zhang and Zhuang [29]. The progress can also be found in a survey by Brown and Zhang [6]. As for Hopf algebras of GK-dimensions three and four, all connected ones were classified by Zhuang [33] and Wang, Zhang and Zhuang [30] respectively.
Our main interest in this paper is to deal with the affine prime Hopf algebras of GK-dimension one, whose detailed structures are recalled in Section 2.1. An interesting fact is that all affine prime regular Hopf algebras are commutative-by-finite [3], that is, a finite module over a normal commutative Hopf subalgebra. This suggests that we have a chance to get the finite duals of affine prime regular Hopf algebras of GK-dimension one explicitly. Actually, Ge and the second author use this observation together with some other techniques to determine the finite dual of the infinite dihedral group algebra 𝕜D∞ recently [9]. In addition, after finishing this paper we realized that A. Jahn and M. Couto already considered this question along this line in their theses [8,12].
This paper is to develop the idea and techniques in [9] further and as the main result we get the finite duals of all the affine prime regular Hopf algebras of GK-dimension one in terms of generators and relations. Now let us recall in our notations that there are five kinds of such Hopf algebras listed in [32]:
𝕜[x],𝕜[x±1],𝕜D∞,T∞(n,v,ξ),B(n,ω,γ),D(m,d,ξ). |
Since the finite duals of 𝕜[x],𝕜[x,x−1] and 𝕜D∞ are known, it remains for us to determine the latter three kinds of Hopf algebras. Our main result is:
Theorem 1.1. (1) As a Hopf algebra, T∞(n,v,ξ)∘ is isomorphic to the Hopf algebra T∞∘(n,v,ξ) constructed in Subsection 3.1.
(2) As a Hopf algebra, B(n,ω,γ)∘ is isomorphic to the Hopf algebra B∘(n,ω,γ) constructed in Subsection 4.1.
(3) As a Hopf algebra, D(m,d,ξ)∘ is isomorphic to the Hopf algebra D∘(m,d,ξ) constructed in Subsection 5.1.
These three claims are presented respectively as Theorems 3.4, 4.4, and 5.7. In brief, our computation for finite dual H∘ starts with properties of certain polynomial or Laurent polynomial subalgebra P, by which we figure cofinite left ideals of H. Then we construct some key elements in H∘ and determine their relations under Hopf operations. Finally it is proved that these elements generate H∘ with desired relations.
Using our descriptions of finite duals, we find that for an affine prime regular Hopf algebra H one can always get a Hopf algebra H∙ such that there is a non-degenerate Hopf pairing (see Majid [19] for the definition) between H and H∙. The definition can also be found as Definition 6.1 in this paper. Some properties of H∙ are listed in the following.
Proposition 1.2. (1) All Hopf algebras (𝕜D∞)∙, T∞(n,v,ξ)∙, B(n,ω,γ)∙ and D(m,d,ξ)∙ have GK-dimension one and are minimal under inclusion relation;
(2) As algebras, (𝕜D∞)∙ is regular while T∞(n,v,ξ)∙, B(n,ω,γ)∙ and D(m,d,ξ)∙ are not for n,m≥2.
Proposition 1.2 is a direct consequence of Propositions 6.4 and 6.5. Through the pairing we constructed and this proposition, we get two observations: 1) For each affine prime regular Hopf algebra H of GK-dimension one, we get a quantum group in the sense of Takeuchi [28] naturally (see Remark 6.8 for details); 2) We give a counterexample to an infinite-dimensional analogous version of the semisimplicity result by Larson and Radford [14] which states that for a finite-dimensional Hopf algebra H, H is semisimple if and only if H∗ is so.
After we finished the present paper, we found the nice preprint [4] written by K.A. Brown, M. Couto and A. Jahn. They refined the results in [8,12] and determined the structures of the finite dual of affine commutative-by-finite Hopf algebras. As the Hopf algebras of GK-dimension one considered in this paper are affine commutative-by-finite, the conclusions gotten in [4] can apply to our case directly. We remark that a number of formulas obtained in Sections 3–5 in our paper were essentially the same as those in [4,Section 7.5]. As pointed out in [4], the results in [4] and the present paper are largely complementary. Indeed, we can use our results to answer a question appeared in [4,Remarks 7.1(3)] (see Remark 5.1 in detail).
The organization of this paper is as follows: We gather in Section 2 necessary concepts and techniques, including the classification result for H, some equivalent definitions of H∘ for computation, and invertible matrices used for later proofs. Sections 3–5 are devoted to giving the structures for T∞(n,v,ξ)∘, B(n,ω,γ)∘ and D(m,d,ξ)∘ respectively. Finally in Section 6, constructions and related properties of desired non-degenerate Hopf pairings on 𝕜D∞, T∞(n,v,ξ), B(n,ω,γ) and D(m,d,ξ) are studied. We pose a question at the last section too.
Notation
We end this section by introducing some notations used in this paper. Let ⌊∙∙⌋ stand for the floor function, that is, for any natural numbers m,n, ⌊mn⌋ denotes the biggest integer which is not bigger than mn.
Moreover, let n be a positive integer. We usually denote condition 0≤i≤n−1 simply by "i∈n". This can be referred to some convention in set theory that n={0,1,⋯,n−1} as a set. Moreover, the cartesian product of {0,1,⋯,m−1} and {0,1,⋯,n−1} are denoted by m×n, which is always ordered lexicographically.
Another notation which will be used is that: for an algebra H, we write I◃lH if I is a left ideal of H, and denote the principal left ideal generated by h∈H by (h).
In this section, we recall the classification result on affine prime regular Hopf algebras of GK-dimension one and give necessary tools and results for the determination of their finite duals. About general background knowledge, the reader is referred to [21,26] for Hopf algebras and [5,18] for exposition about noetherian Hopf algebras.
At first, let us recall the classification of affine prime regular Hopf algebras of GK-dimension one given in [32,Theorem 8.3].
Lemma 2.1. Any prime regular Hopf algebra of GK-dimension one must be isomorphic to one of the following:
(1) Connected algebraic groups of dimension one: 𝕜[x] and 𝕜[x±1];
(2) Infinite dihedral group algebra 𝕜D∞;
(3) Infinite dimensional Taft algebras T∞(n,v,ξ), where n,v are integers satisfying 0≤v≤n−1, and ξ is a primitive nth root of 1;
(4) Generalized Liu's algebras B(n,ω,γ), where n, ω are positive integers and γ is a primitive nth root of 1;
(5) The Hopf algebras D(m,d,ξ), where m,d are positive integers satisfying (1+m)d is even and ξ is a primitive 2mth root of 1.
Detailed structures of T∞(n,v,ξ), B(n,ω,γ) and D(m,d,ξ) are recalled as follows:
Definition 2.2. Let n be a positive integer, 0≤v≤n−1, and ξ be a primitive nth root of 1. As an algebra, T∞(n,v,ξ) is generated by g and x with relations
gn=1,xg=ξgx. |
Then T∞(n,v,ξ) becomes a Hopf algebra with comultiplication, counit and antipode given by
Δ(g)=g⊗g,Δ(x)=1⊗x+x⊗gv,ε(g)=1,ε(x)=0,S(g)=gn−1,S(x)=−ξ−vgn−vx. |
Definition 2.3. Let n and ω be positive integers, and γ be a primitive nth root of 1. As an algebra, B(n,ω,γ) is generated by x±1, g and y with relations
{xx−1=x−1x=1,xg=gx,xy=yx,yg=γgy,yn=1−xω=1−gn. |
Then B(n,ω,γ) becomes a Hopf algebra with comultiplication, counit and antipode given by
Δ(x)=x⊗x,Δ(g)=g⊗g,Δ(y)=1⊗y+y⊗g,ε(x)=ε(g)=1,ε(y)=0,S(x)=x−1,S(g)=g−1,S(y)=−γ−1g−1y. |
Definition 2.4. Let m,d be positive integers such that (1+m)d is even and ξ a primitive 2mth root of unity. Define
ω:=md,γ:=ξ2. |
As an algebra, D(m,d,ξ) is generated by x±1, g, y and u0,u1,⋯,um−1 with relations
xx−1=x−1x=1,gx=xg,yx=xy,yg=γgy,ym=1−xω=1−gm,uix=x−1ui,yui=ϕiui+1=ξxduiy,uig=γix−2dgui,uiuj={(−1)−jξ−jγj(j+1)21mx−1+m2dϕiϕi+1⋯ϕm−2−jyi+jg(i+j≤m−2)(−1)−jξ−jγj(j+1)21mx−1+m2dyi+jg(i+j=m−1)(−1)−jξ−jγj(j+1)21mx−1+m2dϕi⋯ϕm−1ϕ0⋯ϕm−2−jyi+j−mg(i+j≥m) |
where ϕi:=1−γ−i−1xd and i,j∈m.
Then D(m,d,ξ) becomes a Hopf algebra with comultiplication, counit and the antipode given by
Δ(x)=x⊗x,Δ(g)=g⊗g,Δ(y)=y⊗g+1⊗y,Δ(ui)=m−1∑j=0γj(i−j)uj⊗x−jdgjui−j,ε(x)=ε(g)=ε(u0)=1,ε(y)=ε(ul)=0,S(x)=x−1,S(g)=g−1,S(y)=−yg−1=−γ−1g−1y,S(ui)=(−1)iξ−iγ−i(i+1)2xid+32(1−m)dgm−i−1ui, |
for i∈m and 1≤l≤m−1.
In this subsection, we recall the concept of finite dual of a Hopf algebra, and list some lemmas in order to compute finite duals for the cases we are concerned with. The definition of finite dual is well-known and can be found in [26,Chapter 6] or [21,Chapter 9], for example.
Definition 2.5. Let H be a Hopf algebra over 𝕜 and denote its dual space by H∗. The finite dual of H is defined as
H∘:={f∈H∗∣f vanishes on a cofinite ideal I⊲H}. |
It is well-known that H∘ has a Hopf algebra structure naturally [21,Theorem 9.1.3]. We want to simplify H∘ a little. It is not hard to see that H∘ has the following alternative description: Let I be the family of all the cofinite left ideals of H. Then by [21,Lemma 9.1.1], we have H∘=∑I∈II⊥, where I⊥={f∈H∗∣f(I)=0}. Therefore, we have
Lemma 2.6. Let I0 be a subset of I. Suppose for any I∈I, there exist some I1,I2,⋯,Im∈I0 such that ⋂mi=1Ii⊆I. Then H∘=∑I∈I0I⊥.
Proof. The condition implies that for any I∈I, there exist some I1,I2,⋯,Im∈I0 such that ∑mi=1I⊥i=(⋂mi=1Ii)⊥⊇I⊥. Thus
H∘=∑I∈II⊥⊆∑I∈I0I⊥⊆H∘, |
and the claim is then verified.
Now we turn to a common feature of Hopf algebras in our cases. Specifically:
● For the remaining of this subsection, we assume that H is a Hopf algebra which is a finitely generated right module over some subalgebra P, such that P is a polynomial algebra 𝕜[x] or a Laurent polynomial algebra 𝕜[x±1].
We would note in subsequent sections that this assumption holds for all affine prime regular Hopf algebras of GK-dimension one. This assumption helps us to figure all the cofinite left ideals of H:
Lemma 2.7. For any left ideal I⊲lH, I is cofinite in H if and only if I∩P≠0.
Proof. Note that whenever P is 𝕜[x] or 𝕜[x±1], I∩P is an ideal of P and thus a principal ideal generated by some polynomial q(x)∈P. It follows that
I∩P≠0⟺q(x)≠0⟺I∩Pis cofinite inP. |
If I is cofinite in H, then the following relations of 𝕜-vector spaces
H/I⊇(P+I)/I≅P/(I∩P) |
implies that P/(I∩P) must be finite-dimensional. In other words, I∩P≠0 holds.
Conversely if I∩P≠0, then clearly P/(I∩P) is finite-dimensional, and so is (P+I)/I. However, the assumption above this lemma implies that H=∑mi=1hiP for some h1,h2,⋯,hm∈H. As I is a left ideal of H, we can know that (P+I)/I is a finite-dimensional subspace of the quotient left H-module H/I, and
H/I=m∑i=1hi(P+I/I) |
holds. It follows that H/I is also finite-dimensional.
Combining Lemmas 2.6 and 2.7, in order to compute H∘, it is sufficient to find enough appropriate polynomials p(x)'s, such that each I∈I must contain some principal left ideal (p(x)). Concerning our cases of T∞(n,v,ξ), B(n,ω,γ) and D(m,d,ξ), we can choose such p(x)'s respectively as follows.
Lemma 2.8. Let H be an example from any of the above three classes of Hopf algebras. Let I∈I, and m,n be positive integers.
(1) There is a polynomial p(x)∈P=𝕜[x] with form:
p(x)=N∏α=1(xm−λα)nrα | (2.1) |
for some N≥1, rα∈N and distinct λα∈𝕜, such that I contains the cofinite left ideal generated by p(x) in H;
(2) There is a polynomial p(x)∈P=𝕜[x±1] with form:
p(x)=N∏α=1(xn−λα)rα | (2.2) |
for some N≥1, rα∈N and distinct λα∈𝕜∗, such that I contains the cofinite left ideal generated by p(x) in H;
(3) There is a polynomial p(x)∈P=𝕜[x±1] with form:
p(x)=(xm−1)r′(xm+1)r"N∏α=1(xm−λα)rα(xm−λ−1α)rα | (2.3) |
for some N≥1, r′,r",rα∈N, and λα∈𝕜∗∖{±1} which are distinct and not inverses of each other, such that I contains the cofinite left ideal generated by p(x) in H.
Proof. We prove (2) for example, since (1) and (3) hold according to the same reason. It can be known from the proof of Lemma 2.7 that I∩P is a principal ideal of P generated by some non-zero polynomial q(x). Without loss of generality, assume that
q(x)=(x−μ1)(x−μ2)⋯(x−μM) |
for some M≥1 and μ1,μ2,⋯,μM∈𝕜∗. Now define a polynomial
p(x)=(xn−μn1)(xn−μn2)⋯(xn−μnM), |
which gives the desired form (2.2). Since q(x)∣p(x), I contains the left ideal of H generated by p(x) in H.
This lemma suggests us to consider the following subfamilies I0⊆I respectively.
Corollary 2.9. Let m,n be positive integers.
(1) For P=𝕜[x], denote
I0={((xm−λ)nr)⊲lH∣λ∈𝕜,r∈N}. |
If I0 satisfies the condition in Lemma 2.6, then H∘=∑I∈I0(H/I)∗;
(2) For P=𝕜[x±1], denote
I0={((xn−λ)r)⊲lH∣λ∈𝕜∗,r∈N}. |
If I0 satisfies the condition in Lemma 2.6, then H∘=∑I∈I0(H/I)∗;
(3) For P=𝕜[x±1], denote
I0={((xm−1)r),((xm+1)r),((xm−λ)r(xm−λ−1)r)⊲lH∣λ∈𝕜∗∖{±1},r∈N}. |
If I0 satisfies the condition in Lemma 2.6, then H∘=∑I∈I0(H/I)∗.
We end this subsection by providing the following lemma on the intersection of principal left ideals in I:
Lemma 2.10. Let N be a positive integer, and let P be 𝕜[x] or 𝕜[x±1]. Suppose that q1(x),q2(x),⋯,qN(x)∈P are coprime with each other. Then
N⋂α=1(qα(x))=(N∏α=1qα(x)) |
as left ideals in H.
Proof. It is sufficient to prove the case when N=2. First there exist p1(x),p2(x)∈P such that
q1(x)p1(x)+q2(x)p2(x)=1. |
Now for any h∈(q1(x))∩(q2(x)), we have h=h1q1(x)=h2q2(x) for some h1,h2∈H. Then
h2=h2(q1(x)p1(x)+q2(x)p2(x))=h2q1(x)p1(x)+h1q1(x)p2(x)=(h2p1(x)+h1p2(x))q1(x), |
and thus h=h2q2(x)=(h2p1(x)+h1p2(x))q1(x)q2(x)∈(q1(x)q2(x)).
The following primary fact on the Kronecker products is required, which is well-known in matrix theory.
Lemma 2.11. (1) Let Γ1 and Γ2 be finite totally ordered sets, and their Cartesian product Γ1×Γ2 be ordered lexicographically. Then for any two square matrices
A=(ai,i′∣i,i′∈Γ1)andB=(bj,j′∣j,j′∈Γ2) |
over 𝕜, A and B are both invertible if and only if their Kronecker product
(ai,i′bj,j′∣(i,j),(i′,j′)∈Γ1×Γ2) |
is invertible;
(2) Let n1,n2,⋯,nm be positive integers. Then the square matrices
Ak=(a(k)ik,i′k∣ik,i′k∈nk)(1≤k≤m) |
over 𝕜 are all invertible, if and only if their Kronecker product
(a(1)i1,i′1a(2)i2,i′2⋯a(m)im,i′m∣(i1,i2,⋯,im),(i′1,i′2,⋯,i′m)∈n1×n2×⋯×nm) |
is invertible.
A generalized version of above lemma is also required, and we provide a proof for convenience.
Lemma 2.12. Let Γ1 and Γ2 be finite totally ordered sets, and their Cartesian product Γ1×Γ2 be ordered lexicographically. Suppose A:=(ai,i′∣i,i′∈Γ1) is invertible. Then the following matrix
C:=(ai,i′bi′,j,j′∣(i,j),(i′,j′)∈Γ1×Γ2) |
is invertible, if and only if Bi′:=(bi′,j,j′∣j,j′∈Γ2) is invertible for each i′∈Γ1.
Proof. For clarity, denote the cardinalities of sets by m:=|Γ1| and n:=|Γ2|. Clearly, C can be divided into blocks of order m:
C=(ai,i′Yi′∣i,i′∈Γ1). |
Denote A−1:=(˜ai,i′∣i,i′∈Γ1), then Kronecker product of A−1 and In (the identity matrix of order n) can be written as a block matrix
(˜ai,i′In∣i,i′∈Γ1), | (2.4) |
which is invertible according to Lemma 2.11. One can directly verify that the product of (2.4) and C is equal to
(∑k∈Γ1˜ai,kak,i′Bi′∣i,i′∈Γ1)=(δi,i′Bi′∣i,i′∈Γ1), |
and the claim is followed.
Within the work of [9], they proved the invertibility of a certain form of matrices. This plays a crucial role on the process of constructing (𝕜D∞)∘. Our construction for D(m,d,ξ)∘ in this paper would also rely on this fact. However, we provide here another form of that result for simplicity.
Lemma 2.13. ([9,Corollary 2]) Let r be a positive integer, and λ∈𝕜∗∖{±1}. The 2r×2r matrix
(λ(−1)e(2s′+e′)(2s′+e′)s∣(e,s),(e′,s′)∈2×r) | (2.5) |
is invertible.
Proof. Denote M:=2r−1. It follows from [9,Corollary 2] that the 2r×2r matrix
(10⋯010⋯0λλ⋯λλ−1λ−1⋯λ−1λ22λ2⋯2r−1λ2λ−22λ−2⋯2r−1λ−2λ33λ3⋯3r−1λ3λ−33λ−3⋯3r−1λ−3⋮⋮⋱⋮⋮⋮⋱⋮λMMλM⋯Mr−1λMλ−MMλ−M⋯Mr−1λ−M) | (2.6) |
is invertible, which can be obtained from the transpose of (2.5) through elementary operations of type 2. Thus (2.5) is also invertible.
This result has a slightly generalized version, which is required in this paper as well:
Corollary 2.14. Let r be a positive integer, and λ∈𝕜∗∖{±1}. Then:
(1) The 2r×2r matrix
(λ(−1)e(2s′+e′)(a+2s′+e′)s∣(e,s),(e′,s′)∈2×r) | (2.7) |
is invertible for any a∈𝕜;
(2) The 2r×2r matrix
(λ(−1)e(b+2s′+e′)(a+2s′+e′)s∣(e,s),(e′,s′)∈2×r) | (2.8) |
is invertible for any a∈𝕜 and b∈Q.
Proof. (1) Denote M:=2r−1. Firstly, (2.6) can be obtained from the following matrix through elementary column operations of type 3:
(1a⋯ar−11a⋯ar−1λ(a+1)λ⋯(a+1)r−1λλ−1(a+1)λ−1⋯(a+1)r−1λ−1λ2(a+2)λ2⋯(a+2)r−1λ2λ−2(a+2)λ−2⋯(a+2)r−1λ−2λ3(a+3)λ3⋯(a+3)r−1λ3λ−3(a+3)λ−3⋯(a+3)r−1λ−3⋮⋮⋱⋮⋮⋮⋱⋮λM(a+M)λM⋯(a+M)r−1λMλ−M(a+M)λ−M⋯(a+M)r−1λ−M). | (2.9) |
The process is by deleting powers of λ or λ−1 with coefficients containing a, from the rth column to the 2nd column, and the 2rth column to the (r+1)th column. Thus (2.9) is also invertible.
However, (2.7) can be obtained from the transpose of (2.9) through elementary operations of type 2. This implies the invertibility of (2.7).
(2) The similar argument follows that (2.8) can be obtained from the transpose of the following matrix through elementary column operations:
(λbaλb⋯ar−1λbλ−baλ−b⋯ar−1λ−bλb+1(a+1)λb+1⋯(a+1)r−1λb+1λ−(b+1)(a+1)λ−(b+1⋯(a+1)r−1λ−(b+1)λb+2(a+2)λb+2⋯(a+2)r−1λb+2λ−(b+2)(a+2)λ−(b+2)⋯(a+2)r−1λ−(b+2)λb+3(a+3)λb+3⋯(a+3)r−1λb+3λ−(b+3)(a+3)λ−(b+3)⋯(a+3)r−1λ−(b+3)⋮⋮⋱⋮⋮⋮⋱⋮λb+M(a+M)λb+M⋯(a+M)r−1λb+Mλ−(b+M)(a+M)λ−(b+M)⋯(a+M)r−1λ−(b+M)) |
and this matrix has the same determinant as (2.9).
We end this section by listing two other concepts for later use. The one is the well-known quantum binomial coefficients for a parameter q∈𝕜∗, which is defined as
(lk)q:=l!qk!q(l−k)!q |
for integers l≥k≥0, where l!q:=1q2q⋯lq and lq:=1+q+⋯+ql−1.
The other one is a part of the second Stirling number (see (13.13) in [15]), written as
r∑t=0(rt)(−1)r−tts, | (2.10) |
which is in fact zero if 0≤s<r.
This section is devoted to providing the structure of T∞(n,v,ξ)∘, which is isomorphic to the Hopf algebra T∞∘(n,v,ξ) constructed by generators and relations in Subsection 3.1. For the purpose, we choose certain elements in T∞(n,v,ξ)∘ and study their properties in Subsection 3.2. The desired isomorphism is shown in Subsection 3.3 finally.
Let n be a positive integer, 0≤v≤n−1 and ξ be a primitive nth root of 1. For simplicity we denote m:=ngcd(n,v). The Hopf algebra T∞∘(n,v,ξ) is constructed as follows. As an algebra, it is generated by Ψλ(λ∈𝕜),Ω,F1,F2 with relations
Ψλ1Ψλ2=Ψλ1+λ2,Ψ0=1,Ωn=1,Fm1=0,ΩΨλ=ΨλΩ,F2Ω=ΩF2,F1Ω=ξvΩF1,F2Ψλ=ΨλF2,F1Ψλ=ΨλF1,F1F2=F2F1 |
for all λ,λ1,λ2∈𝕜. The comultiplication, counit and antipode are given by
Δ(Ω)=Ω⊗Ω,Δ(F1)=1⊗F1+F1⊗Ω,Δ(F2)=1⊗F2+F2⊗Ωm+m−1∑k=1F[k]1⊗ΩkF[m−k]1,Δ(Ψλ)=(n/m)−1∑c=0(Ψλξmc⊗Ψλςc)(1⊗1+λm−1∑k=1F[k]1⊗ΩkF[m−k]1),ε(Ω)=ε(Ψλ)=1,ε(F1)=ε(F2)=0,S(Ω)=Ωn−1,S(F1)=−ξ−vΩn−1F1,S(F2)=−F2,S(Ψλ)=(n/m)−1∑c=0Ψ−λξ−mcςc, |
for λ∈𝕜, where F[k]1:=1k!ξvFk1 for 1≤k≤m−1, and
ςc:=mn(1+ξ−mcΩm+ξ−2mcΩ2m+⋯+ξ−(n−m)cΩn−m)(c∈n/m). |
The fact that T∞∘(n,v,ξ) is a Hopf algebra would be stated and proved in Subsection 3.3 as Theorem 3.4(1). Here we note that
{ΨλΩjFs2Fl1∣λ∈𝕜,j∈n,s∈N,l∈m} |
is a linear basis, due to an application of the Diamond lemma [2].
Let n be a positive integer, 0≤v≤n−1, and ξ be a primitive nth root of 1. According to the definition of T∞(n,v,ξ) recalled in Definition 2.2, we know that it has a linear basis {gjxl∣j∈n,l∈N}. Denote m:=ngcd(n,v), and then ξv is a primitive mth root of 1. We define following elements in T∞(n,v,ξ)∗:
ψλ:gjxl↦∞∑u=0δl,umλu,ω:gjxl↦δl,0ξj,E1:gjxl↦δl,1,E2:gjxl↦δl,m | (3.1) |
for any j∈n, l∈N and λ∈𝕜. We remark that these definitions make sense for all j∈Z as well due to a direct computation.
One can verify that ψλ vanishes on the principal left ideal (xm−λ) for any λ∈𝕜; ω vanishes on (x); E2 vanishes on (xm+1); E1 vanishes on (x2). Clearly, these left ideals are cofinite in T∞(n,v,ξ), and hence all the elements in (3.1) lie in T∞(n,v,ξ)∘.
Lemma 3.1. Following equations hold in T∞(n,v,ξ)∘:
ψλ1ψλ2=ψλ1+λ2,ψ0=1,ωn=1,Em1=0,ωψλ=ψλω,E2ω=ωE2,E1ω=ξvωE1,E2ψλ=ψλE2,E1ψλ=ψλE1,E1E2=E2E1 |
for all λ,λ1,λ2∈𝕜.
Proof. Note that
Δ(gjxl)=(g⊗g)j(1⊗x+x⊗gv)l=l∑k=0(lk)ξvgjxk⊗gj+kvxl−k. |
We prove the lemma by checking values at the basis elements gjxl(j∈n,l∈N):
⟨ψλ1ψλ2,gjxl⟩=∞∑u=0δl,umu∑i=0(umim)ξv⟨ψλ1,gjxim⟩⟨ψλ2,gj+imvx(u−i)m⟩=∞∑u=0δl,umu∑i=0(ui)λi1λu−i2=⟨ψλ1+λ2,gjxl⟩,⟨ψ0,gjxl⟩=∞∑u=0δl,um0u=δl,0=⟨ε,gjxl⟩,⟨ωψλ,gjxl⟩=∞∑u=0δl,um⟨ω,gj⟩⟨ψλ,gjxum⟩=∞∑u=0δl,um⟨ψλ,gjxum⟩⟨ω,gj+umv⟩=⟨ψλω,gjxl⟩,⟨E2ω,gjxl⟩=δl,m⟨E2,gjxm⟩⟨ω,gj+mv⟩=δl,m⟨ω,gj⟩⟨E2,gjxm⟩=⟨ωE2,gjxl⟩,⟨E1ω,gjxl⟩=δl,1⟨E1,gjx⟩⟨ω,gj+v⟩=δl,1ξv⟨ω,gj⟩⟨E1,gjx⟩=⟨ξvωE1,gjxl⟩,⟨E2ψλ,gjxl⟩=∞∑u=0δl,(u+1)m((u+1)mm)ξv⟨E2,gjxm⟩⟨ψλ,gj+mvxum⟩=∞∑u=0δl,(u+1)m((u+1)mum)ξv⟨ψλ,gjxum⟩⟨E2,gj+umvxm⟩=⟨ψλE2,gjxl⟩,⟨E1ψλ,gjxl⟩=∞∑u=0δl,um+1(um+11)ξv⟨E1,gjx⟩⟨ψλ,gj+vxum⟩=∞∑u=0δl,um+1(um+1um)ξv⟨ψλ,gjxum⟩⟨E1,gj+umvx⟩=⟨ψλE1,gjxl⟩. |
Also, we find by induction on k∈n and k′∈m that
⟨ωk,gjxl⟩=δl,0⟨ωk,gj⟩=δl,0⟨ω,gj⟩k=δl,0ξjk,⟨Ek′1,gjxl⟩=δl,k′(k′1)ξv⟨E1,gjx⟩⟨Ek′−11,gj+vxk′−1⟩=δl,k′k′!ξv(k′−1)!ξv⟨Ek′−11,gj+vxk′−1⟩=δl,k′k′!ξv(k′−1)!ξv(k′−1)!ξv(k′−2)!ξv⟨Ek′−21,gj+2vxk′−2⟩=⋯=δl,k′k′!ξv. |
Thus ωn=1 and Em1=0 hold.
Furthermore if denote E[k′]1:=1k′!ξvEk′1, we verify that for any k′∈m, k,j∈n and l∈N,
⟨ωkE[k′]1,gjxl⟩=δl,k′⟨ωk,gj⟩⟨E[k′]1,gjxk′⟩=δl,k′ξjk,⟨ψλωkE[k′]1,gjxl⟩=∞∑u=0δl,um+k′(um+k′um)ξv⟨ψλ,gjxum⟩⟨ωkE[k′]1,gj+umvxk′⟩=∞∑u=0δl,um+k′λuξjk. |
If we denote E[s]2:=1s!Es2 for s≥1, followings can be obtained by induction:
⟨Es2,gjxl⟩=δl,sm(smm)ξv⟨E2,gjxm⟩⟨Es−12,gj+mvx(s−1)m⟩=δl,sms⟨Es−12,gj+mvx(s−1)m⟩=δl,sms((s−1)mm)ξv⟨E2,gj+mvxm⟩⟨Es−22,gj+2mvx(s−2)m⟩=δl,sms(s−1)⟨Es−22,gj+2mvx(s−2)m⟩=⋯=δl,sms!,⟨E[s]2,gjxl⟩=δl,sn,⟨E[s]2E[k′]1,gjxl⟩=δl,sm+k′(sm+k′sm)ξv⟨E[s]2,gjxsm⟩⟨E[k′]1,gj+smvxk′⟩=δl,sm+k′,⟨ωkE[s]2E[k′]1,gjxl⟩=δl,sm+k′⟨ωk,gj⟩⟨E[s]2E[k′]1,gjxsm+k′⟩=δl,sm+k′ξjk. |
Lemma 3.2. Following equations hold in T∞(n,v,ξ)∘:
Δ(ω)=ω⊗ω,Δ(E1)=1⊗E1+E1⊗ω,Δ(E2)=1⊗E2+E2⊗ωm+m−1∑k=1E[k]1⊗ωkE[m−k]1,Δ(ψλ)=(n/m)−1∑c=0(ψλξmc⊗ψλσc)(1⊗1+λm−1∑k=1E[k]1⊗ωkE[m−k]1),ε(ω)=ε(ψλ)=1,ε(E1)=ε(E2)=0,S(ω)=ωn−1,S(E1)=−ξ−vωn−1E1,S(E2)=−E2,S(ψλ)=(n/m)−1∑c=0ψ−λξ−mcσc, |
for λ∈𝕜, where E[k]1:=1k!ξvEk1 for 1≤k≤m−1, and
σc:=mn(1+ξ−mcωm+ξ−2mcω2m+⋯+ξ−(n−m)cωn−m)(c∈n/m). |
Proof. Note that (gjxl)(gj′xl′)=ξj′lgj+j′xl+l′(j,j′∈n,l,l′∈N). We also prove the lemma by checking values at each gjxl⊗gj′xl′.
⟨ω,ξj′lgj+j′xl+l′⟩=δl+l′,0ξj′l+j+j′=δl,0ξjδl′,0ξj′=⟨ω⊗ω,gjxl⊗gj′xl′⟩,⟨E1,ξj′lgj+j′xl+l′⟩=δl+l′,1ξj′l=δl,0δl′,1+δl,1δl′,0ξj′=⟨1⊗E1+E1⊗ω,gjxl⊗gj′xl′⟩,⟨E2,ξj′lgj+j′xl+l′⟩=δl+l′,mξj′l=δl,0δl′,m+δl,mδl′,0ξj′m+m−1∑k=1δl,kδl′,m−kξj′k=⟨1⊗E2+E2⊗ωm+m−1∑k=1E[k]1⊗ωkE[m−k]1,gjxl⊗gj′xl′⟩. |
It is not hard to find that for each c∈n/m, j∈n and l∈N:
⟨ψλσc,gjxl⟩={∞∑u=0δl,umλu,ifj≡c(modn/m);0,otherwise. |
Now compute that
⟨ψλ,ξj′lgj+j′xl+l′⟩=∞∑u=0δl+l′,umλuξj′l=∞∑u,u′=0(δl,umδl′,u′mλu+u′ξj′um+m−1∑k=1δl,um+kδl′,u′m+(m−k)λu+u′+1ξj′(um+k))=∞∑u=0δl,um(λξj′m)u∞∑u′=0δl′,u′mλu′+λm−1∑k=1∞∑u=0δl,um+k(λξj′m)u∞∑u′=0δl′,u′m+(m−k)λu′ξj′k=⟨ψλξj′m⊗ψλ+λm−1∑k=1ψλξj′mE[k]1⊗ψλωkE[m−k]1,gjxl⊗gj′xl′⟩=⟨(n/m)−1∑c=0(ψλξmc⊗ψλσc+λm−1∑k=1ψλξmcE[k]1⊗ψλσcωkE[m−k]1),gjxl⊗gj′xl′⟩=⟨(n/m)−1∑c=0(ψλξmc⊗ψλσc)(1⊗1+λm−1∑k=1E[k]1⊗ωkE[m−k]1),gjxl⊗gj′xl′⟩. |
The expressions of the counit and the antipode are clear.
Proposition 3.3. As an algebra, T∞(n,v,ξ)∘ is generated by ψλ, ω, E2 and E1 for λ∈𝕜.
Proof. Clearly T∞(n,v,ξ) has a polynomial subalgebra P=𝕜[x]. Choose
I0={((xm−λ)nr)⊲lT∞(n,v,ξ)∣λ∈𝕜,r∈N}. |
First for any cofinite left ideal I of T∞(n,v,ξ), according to Lemma 2.8(1), there must be some non-zero polynomial
p(x)=N∏α=1(xm−λα)nrα |
for some N≥1, rα∈N and distinct λα∈𝕜, which generates a cofinite left ideal contained in I. It can be known by Lemma 2.10 that
N⋂α=1((xm−λα)nrα)=(p(x))⊆I |
as left ideals of T∞(n,v,ξ), and thus Lemma 2.6 or Corollary 2.9(1) can be applied to obtain that T∞(n,v,ξ)∘=∑I∈I0(T∞(n,v,ξ)/I)∗.
Now we try to prove that for each I∈I0, the subspace (T∞(n,v,ξ)/I)∗ can be spanned by some products of ψλ,ω,E2,E1(λ∈𝕜). Specifically, suppose I=((xm−λ)nr) for some r∈N and λ∈𝕜, and we aim to show that
{ψλωjEs2El1∣j∈n,s∈nr,l∈m} | (3.2) |
is a linear basis of n2mr-dimensional space (T∞(n,v,ξ)/I)∗.
Evidently I=𝕜{gjxl(xm−λ)nr∣j∈n,l∈N}. Also, T∞(n,v,ξ)/I has a linear basis
{gjxl(xm−λ)s∣j∈n,l∈m,s∈nr}, |
and hence dim((T∞(n,v,ξ)/I)∗)=dim(T∞(n,v,ξ)/I)=n2mr. Next we show that all elements in (3.2) vanish on I. Recall that
E[s]2E[l]1=1s!⋅l!ξvEs2El1:gj′xl′↦δl′,sm+l(l∈m,s,l′∈N,j′∈n). |
Therefore, for any j,j′∈n, s∈nr, l∈m and l′∈N, we have
⟨ψλωjE[s]2E[l]1,gj′xl′(xm−λ)nr⟩=⟨ψλωjE[s]2E[l]1,nr∑t=0(nrt)gj′xl′+tm(−λ)nr−t⟩=nr∑t=0(nrt)(−λ)nr−t⟨ψλ⊗ωjE[s]2E[l]1,Δ(gj′xl′+tm)⟩=nr∑t=s(nrt)(−λ)nr−tδl′,l(l+tm(t−s)m)ξv⟨ψλ,gj′x(t−s)m⟩⟨ωjE[s]2E[l]1,gj′+(t−s)mvxl+sm⟩=nr∑t=s(nrt)(−λ)nr−tδl′,l(tt−s)λt−sξ(j′+(t−s)mv)j=λnr−sδl′,lξj′jnr∑t=s(nrt)(−1)nr−t(ts)=λnr−sδl′,lξj′jnr∑t=s(nr−st−s)(−1)nr−t=0. |
In other words, ⟨ψλωjE[s]2E[l]1,I⟩=0 for any j∈n, s∈nr and l∈m, which follows that the elements in (3.2) belong to I⊥=(T∞(n,v,ξ)/I)∗.
Finally we prove that the elements in (3.2) are linearly independent. Choose a lexicographically ordered set of elements {gj′xl′+s′m∈T∞(n,v,ξ)∣(j′,s′,l′)∈n×nr×m}. Our goal is to show that the n2mr×n2mr square matrix
A:=(⟨ψλωjE[s]2E[l]1,gj′xl′+s′m⟩∣(j,s,l),(j′,s′,l′)∈n×nr×m) |
is invertible, which would imply the linear independence of (3.2). We try to compute
⟨ψλωjE[s]2E[l]1,gj′xl′+s′m⟩=⟨ψλ⊗ωjE[s]2E[l]1,Δ(gj′xl′+s′m)⟩. |
One finds that ⟨ψλωjE[s]2E[l]1,gj′xl′+s′m⟩=0 if s>s′. Otherwise when s≤s′,
⟨ψλωjE[s]2E[l]1,gj′xl′+s′m⟩=δl′,l(l+s′m(s′−s)m)ξv⟨ψλ,gj′x(s′−s)m⟩⟨ωjE[s]2E[l]1gj′+(s′−s)mvxl+sm⟩=δl′,l(s′s)λs′−sξj′j. |
The computation follows that the matrix A is the Kronecker product of following three invertible matrices:
● a Vandermonde matrix (ξj′j∣j,j′∈n),
● an nr×nr upper triangular matrix with entries (s′s)λs′−s for s≤s′, and
● an identity matrix matrix (δl′,l∣l,l′∈m).
Thus A is invertible by Lemma 2.11, and hence (3.2) is a basis of (T∞(n,v,ξ)/I)∗. We conclude that
T∞(n,v,ξ)∘=∑I∈I0(T∞(n,v,ξ)/I)∗=𝕜{ψλωjEs2El1∣λ∈𝕜,j∈n,s∈N,l∈m}. |
Theorem 3.4. (1) T∞∘(n,v,ξ) constructed in Subsection 3.1 is a Hopf algebra;
(2) As a Hopf algebra, T∞(n,v,ξ)∘ is isomorphic to T∞∘(n,v,ξ).
Proof. Consider the following map
Θ:T∞∘(n,v,ξ)→T∞(n,v,ξ)∘,Ψλ↦ψλ,Ω↦ω,F1↦E1,F2↦E2, |
where λ∈𝕜. This is an epimorphism of algebras by Lemma 3.1 and Proposition 3.3. Furthermore, Θ would become an isomorphism of Hopf algebras with desired coalgebra structure and antipode, as long as it is injective (since T∞(n,v,ξ)∘ is in fact a Hopf algebra).
In order to show that Θ is injective, it is enough to show the linear independence of
{ψλωjEs2El1∣λ∈𝕜,j∈n,s∈N,l∈m} |
in T∞(n,v,ξ)∘. By the linear independence of elements in (3.2), we only need to show that any finite sum of form
N∑α=1(T∞(n,v,ξ)/((xm−λα)nrα))∗=N∑α=1((xm−λα)nrα)⊥ | (3.3) |
is direct as long as λα's are distinct from each other. But this is clear:
((xm−λα′)nr)⊥∩[∑α≠α′((xm−λα)nr)⊥]=((xm−λα′)nr)⊥∩[⋂α≠α′((xm−λα)nr)]⊥=((xm−λα′)nr)⊥∩(∏α≠α′(xm−λα)nr)⊥=[((xm−λα′)nr)+(∏α≠α′(xm−λα)nr)]⊥=T∞(n,v,ξ)⊥=0. |
Remark 3.5. As a special case, the connected algebraic groups of dimension one H1=𝕜[x] is equal to T∞(1,0,1). Therefore, Theorem 3.4 gives H1∘, which is well-known (see [21,Example 9.1.7] for example).
The concept and definition of a generalized Liu's algebra B(n,ω,γ) is given in [5,Section 3.4]. In this section, we use the same procedure as Section 3 to determine the structure of B(n,ω,γ)∘.
Let n and ω be positive integers, and γ be a primitive nth root of 1. For convenience, λ1m always denotes an mth root of λ∈𝕜∗ for a positive integer m throughout the paper.
The Hopf algebra B∘(n,ω,γ) is constructed as follows. As an algebra, it is generated by
Ψλ1ω,λ1n,F1,F2(λ1ω,λ1n∈𝕜∗) |
(where the parameters of Ψ are meant to range through {(α,β)∈𝕜∗×𝕜∗∣αω=βn}) with relations
Ψλ1ω1,λ1n1Ψλ1ω2,λ1n2=Ψ(λ1ω1λ1ω2),(λ1n1λ1n2),Ψ1,1=1,Fn1=0,F2Ψλ1ω,λ1n=Ψλ1ω,λ1nF2,F1Ψλ1ω,λ1n=λ1nΨλ1ω,λ1nF1,F1F2=F2F1+1nF1 |
for λ1ω,λ1n,λ1ω1,λ1n1,λ1ω2,λ1n2∈𝕜∗. The comultiplication, counit and antipode are given by
Δ(F1)=1⊗F1+F1⊗Ψ1,γ,Δ(F2)=1⊗F2+F2⊗1−n−1∑k=1F[k]1⊗Ψk1,γF[n−k]1,Δ(Ψλ1ω,λ1n)=(Ψλ1ω,λ1n⊗Ψλ1ω,λ1n)(1⊗1+(1−λ)n−1∑k=1F[k]1⊗Ψk1,γF[n−k]1),ε(Ψλ1ω,λ1n)=1,ε(F1)=ε(F2)=0,S(F1)=−γn−1Ψn−11,γF1,S(F2)=−F2,S(Ψλ1ω,λ1n)=Ψλ−1ω,λ−1n |
for λ1ω,λ1n∈𝕜∗, where F[k]1:=1k!γFk1 for 1≤k≤n−1.
The fact that B∘(n,ω,γ) is a Hopf algebra would be stated and proved in Subsection 4.3 as Theorem 4.4(1). Let η be a primitive ωth root of 1. Here we note that
{Ψλ1ω,λ1nΨiη,1Ψj1,γFs2Fl1∣λ1ω,λ1nare fixed roots ofλ∈𝕜∗,i∈ω,j∈n,s∈N,l∈n} |
is a linear basis, due to an application of the Diamond lemma [2].
Let n and ω be positive integers, and γ be a primitive nth root of 1. According to the definition of B(n,ω,γ) recalled in Definition 2.3, we know that it has a linear basis {xigjyl∣i∈ω,j∈Z,l∈n}. Define following elements in B(n,ω,γ)∗:
ψλ1ω,λ1n:xigjyl↦δl,0λiωλjn,E1:xigjyl↦δl,1,E2:xigjyl↦δl,0(iω+jn) | (4.1) |
for any i∈ω, j∈Z, l∈n and λ1ω,λ1n∈𝕜∗. We remark that these definitions make sense for all i∈Z as well.
One can verify that ψλ1ω,λ1n vanishes on the principal left ideal (gn−λ) for any λ1ω,λ1n∈𝕜∗; E2 vanishes on ((gn−1)2); E1 vanishes on (gn−1). Therefore, these elements lie in B(n,ω,γ)∘.
Lemma 4.1. Following equations hold in B(n,ω,γ)∘:
ψλ1ω1,λ1n1ψλ1ω2,λ1n2=ψ(λ1ω1λ1ω2),(λ1n1λ1n2),ψ1,1=1,En1=0,E2ψλ1ω,λ1n=ψλ1ω,λ1nE2,E1ψλ1ω,λ1n=λ1nψλ1ω,λ1nE1,E1E2=E2E1+1nE1 |
for all λ1ω,λ1n,λ1ω1,λ1n1,λ1ω2,λ1n2∈𝕜∗.
Proof. Note that
Δ(xigjyl)=(x⊗x)i(g⊗g)j(1⊗y+y⊗g)l=l∑k=0(lk)γxigjyk⊗xigj+kyl−k. |
We prove the lemma through checking their values on the basis xigjyl(i∈ω,j∈Z,l∈n):
⟨ψλ1ω1,λ1n1ψλ1ω2,λ1n2,xigjyl⟩=δl,0⟨ψλ1ω1,λ1n1,xigj⟩⟨ψλ1ω2,λ1n2,xigj⟩=δl,0λiω1λiω2λjn1λjn2=⟨ψλ1ω1λ1ω2,λ1n1λ1n2,xigjyl⟩,⟨ψ1,1,xigjyl⟩=δl,01i1j=⟨ε,xigjyl⟩,⟨E2ψλ1ω,λ1n,xigjyl⟩=δl,0⟨E2,xigj⟩⟨ψλ1ω,λ1n,xigj⟩=δl,0(iω+jn)λiωλjn=δl,0⟨ψλ1ω,λ1n,xigj⟩⟨E2,xigj⟩=⟨ψλ1ω,λ1nE2,xigjyl⟩,⟨E1ψλ1ω,λ1n,xigjyl⟩=δl,1⟨E1,xigjy⟩⟨ψλ1ω,λ1n,xigj+1⟩=δl,1λiωλj+1n=λ1nδl,1λiωλjn=λ1nδl,1⟨ψλ1ω,λ1n,xigj⟩⟨E1,xigjy⟩=⟨λ1nψλ1ω,λ1nE1,xigjyl⟩,⟨E1E2,xigjyl⟩=δl,1⟨E1,xigjy⟩⟨E2,xigj+1⟩=δl,1(iω+j+1n)=δl,1(iω+jn)+δl,11n=δl,1⟨E2,xigj⟩⟨E1,xigjy⟩+δl,11n⟨E1,xigjy⟩=⟨E2E1+1nE1,xigjyl⟩. |
Similar to the case in Lemma 3.1, it can be found by induction on k∈n that
⟨Ek1,xigjyl⟩=δl,kk!γ, |
and thus En1=0.
Furthermore if denote E[k]1:=1k!γEk1, we can also verify that for any λ1ω,λ1n∈𝕜∗ and k∈n,
⟨ψλ1ω,λ1nE[k]1,xigjyl⟩=δl,k⟨ψλ1ω,λ1n,xigj⟩⟨E[k]1,xigjyk⟩=δl,kλiωλjn. | (4.2) |
Lemma 4.2. Following equations hold in B(n,ω,γ)∘:
Δ(E1)=1⊗E1+E1⊗ψ1,γ,Δ(E2)=1⊗E2+E2⊗1−n−1∑k=1E[k]1⊗ψk1,γE[n−k]1,Δ(ψλ1ω,λ1n)=(ψλ1ω,λ1n⊗ψλ1ω,λ1n)(1⊗1+(1−λ)n−1∑k=1E[k]1⊗ψk1,γE[n−k]1),ε(ψλ1ω,λ1n)=1,ε(E1)=ε(E2)=0,S(E1)=−γn−1ψn−11,γE1,S(E2)=−E2,S(ψλ1ω,λ1n)=ψλ−1ω,λ−1n |
for λ1ω,λ1n∈𝕜∗, where E[k]1:=1k!γEk1 for 1≤k≤n−1.
Proof. Note that (xigjyl)(xi′gj′yl′)=γj′lxi+i′gj+j′yl+l′(i,i′∈ω,j,j′∈Z,l,l′∈n). We also prove the lemma through checking their values on the basis xigjyl⊗xi′gj′yl′.
⟨E1,γj′lxi+i′gj+j′yl+l′⟩=δl+l′,1γj′l⟨E1,xi+i′gj+j′y⟩+δl+l′,n+1γj′l⟨E1,xi+i′gj+j′yn+1⟩=δl+l′,1γj′l=δl,0δl′,1+δl,1δl′,0γj′=⟨1⊗E1+E1⊗ψ1,γ,xigjyl⊗xi′gj′yl′⟩,⟨E2,γj′lxi+i′gj+j′yl+l′⟩=γj′l(δl+l′,0⟨E2,xi+i′gj+j′⟩+δl+l′,n⟨E2,xi+i′gj+j′−xi+i′gj+j′+n⟩)=δl,0δl′,0(i+i′ω+j+j′n)+δl+l′,nγj′l(i+i′ω+j+j′n−i+i′ω−j+j′+nn)=δl,0δl′,0(i+i′ω+j+j′n)+n−1∑k=1δl,kδl′,n−kγj′k(−1)=δl,0δl′,0(i′ω+j′n)+δl,0(iω+jn)δl′,0−n−1∑k=1δl,kδl′,n−kγj′k=⟨1⊗E2+E2⊗1−n−1∑k=1E[k]1⊗ψk1,γE[n−k]1,xigjyl⊗xi′gj′yl′⟩, |
and
⟨ψλ1ω,λ1n,γj′lxi+i′gj+j′yl+l′⟩=δl+l′,0⟨ψλ1ω,λ1n,xi+i′gj+j′⟩+δl+l′,nγj′l⟨ψλ1ω,λ1n,xi+i′gj+j′−xi+i′gj+j′+n⟩)=δl+l′,0λi+i′ωλj+j′n+δl+l′,nγj′l(λi+i′ωλj+j′n−λi+i′ωλj+j′+nn)=δl,0δl′,0λi+i′ωλj+j′n+δl+l′,nγj′l(1−λ)λi+i′ωλj+j′n=(δl,0λiωλjn)(δl′,0λi′ωλj′n)+(1−λ)n−1∑k=1(δl,kλiωλjn)(δl′,n−kλi′ωλj′nγj′k)=⟨ψλ1ω,λ1n⊗ψλ1ω,λ1n+(1−λ)n−1∑k=1ψλ1ω,λ1nE[k]1⊗ψλ1ω,λ1nψk1,γE[n−k]1,xigjyl⊗xi′gj′yl′⟩. |
The expressions of the counit and the antipode are clear.
Proposition 4.3. As an algebra, B(n,ω,γ)∘ is generated by ψλ1ω,λ1n, E1 and E2 for λ1ω,λ1n∈𝕜∗.
Proof. Clearly B(n,ω,γ) has a Laurent polynomial subalgebra P=𝕜[g±1]. Choose
I0={((gn−λ)r)⊲lB(n,ω,γ)∣λ∈𝕜∗,r∈N}. |
At first for any cofinite left ideal I of B(n,ω,γ), according to Lemma 2.8(2), there must be some non-zero polynomial
p(g)=N∏α=1(gn−λα)rα |
for some N≥1, rα∈N and distinct λα∈𝕜∗, which generates a cofinite left ideal contained in I. It can be known by Lemma 2.10 that
N⋂α=1((gn−λα)rα)=(p(g))⊆I |
as left ideals of B(n,ω,γ), and thus Lemma 2.6 or Corollary 2.9(2) can be applied to obtain that B(n,ω,γ)∘=∑I∈I0(B(n,ω,γ)/I)∗.
Now we try to prove that for each I∈I0, the subspace (B(n,ω,γ)/I)∗ can be spanned by some products of ψλ1ω,λ1n,E2,E1(λ1ω,λ1n∈𝕜∗).
For the purpose, suppose I=((gn−λ)r) for some r∈N and λ∈𝕜∗. Let η be a primitive ωth root of 1, and λ1ω,λ1n be fixed roots of λ respectively. We aim to show that
{ψλ1ω,λ1nψiη,1ψj1,γEs2El1∣i∈ω,j∈n,s∈r,l∈n} | (4.3) |
is a linear basis of ωn2r-dimensional space (B(n,ω,γ)/I)∗.
Evidently I=𝕜{xigj(gn−λ)ryl∣i∈ω,j∈Z,l∈n}, since gn is central. Also, B(n,ω,γ)/I has a linear basis
{xigj(gn−λ)syl∣i∈ω,j∈n,s∈r,l∈n}, |
and hence dim((B(n,ω,γ)/I)∗)=dim(B(n,ω,γ)/I)=ωn2r. Next we show that all elements in (4.3) vanish on I. Recall that
E[l]1=1l!γEl1:xi′gj′yl′↦δl′,l(i′,j′∈Z,l,l′∈n). |
Therefore, for any s∈r, l,l′∈n, i′∈ω and j′∈Z, and for arbitrary roots λ1ω and λ1n of λ, we have
⟨ψλ1ω,λ1nEs2E[l]1,xi′gj′(gn−λ)ryl′⟩=⟨ψλ1ω,λ1nEs2E[l]1,r∑t=0(rt)xi′gj′+nt(−λ)r−tyl′⟩=r∑t=0(rt)(−λ)r−t⟨ψλ1ω,λ1nEs2⊗E[l]1,Δ(xi′gj′+ntyl′)⟩=δl′,lr∑t=0(rt)(−λ)r−t⟨ψλ1ω,λ1nEs2,xi′gj′+nt⟩⟨E[l]1,xi′gj′+ntyl⟩=δl′,lr∑t=0(rt)(−λ)r−tλi′ωλj′+ntn(i′ω+j′+ntn)s=δl′,lλrλi′ωλj′nr∑t=0(rt)(−1)r−ts∑u=0(su)(i′ω+j′n)uts−u=δl′,lλrλi′ωλj′ns∑u=0(su)(i′ω+j′n)ur∑t=0(rt)(−1)r−tts−u=0, | (4.4) |
since ∑rt=0(rt)(−1)r−tts−u=0 when s−u<r is a part of the second Stirling number (2.10).
In other words, ⟨ψλ1ω,λ1nEs2E[l]1,I⟩=0 for any s∈r, l∈n, and arbitrary roots λ1ω,λ1n of λ. It follows that the elements in (4.3) belong to I⊥=(B(n,ω,γ)/I)∗, as λ1ωηi and λ1nγj are still roots of λ.
Finally we prove that (4.3) are linearly independent. Choose a lexicographically ordered set of elements {xi′gj′+s′nyl′∈B(n,ω,γ)∣(i′,j′,s′,l′)∈ω×n×r×n}. Our goal is to show that the n2r×n2r square matrix
A:=(⟨ψλ1ω,λ1nψiη,1ψj1,γEs2E[l]1,xi′gj′+s′nyl′⟩∣(i,j,s,l),(i,′j′,s′,l′)∈ω×n×r×n) |
is invertible for fixed λ1ω,λ1n, which would imply the linear independence of (4.3). We try to compute
⟨ψλ1ω,λ1nψiη,1ψj1,γEs2E[l]1,xi′gj′+s′nyl′⟩=⟨ψλ1ωηi,λ1nγjEs2⊗E[l]1,Δ(xi′gj′+s′nyl′)⟩=δl′,l⟨ψλ1ωηi,λ1nγjEs2,xi′gj′+s′n⟩⟨E[l]1,xi′gj′+s′nyl⟩=δl′,l(λ1ωηi)i′(λ1nγj)j′+s′n(i′ω+j′+s′nn)s=δl′,lλi′ω+j′n+s′ηii′γjj′(i′ω+j′n+s′)s, |
and observe that det(A) equals to
(λ1ω)nr(ω−1)ω2(λ1n)ωr(n−1)n2λωn(r−1)r2det(δl′,lηii′γjj′(i′ω+j′n+s′)s). |
However, there are following two facts:
● The matrix (δl′,lηii′γjj′∣(i,j,l),(i′,j′,l′)∈ω×n×n) is invertible by Lemma 2.11, since it is the Kronecker product of three invertible matrices;
● For each (i′,j′)∈ω×n, the matrix ((i′ω+j′n+s′)s∣s,s′∈r) is an invertible Vandermonde matrix.
Thus (δl′,lηii′γjj′(i′ω+j′n+s′)s∣(i,j,s,l),(i,′j′,s′,l′)∈ω×n×r×n) is invertible according to Lemma 76762.12, which follows that A is invertible. As a consequence, (4.3) is a basis of (B(n,ω,γ)/I)∗.
Therefore,
B(n,ω,γ)∘=∑I∈I0(B(n,ω,γ)/I)∗=𝕜{ψλ1ω,λ1nEs2El1∣λ1ω,λ1n∈𝕜∗,s∈N,l∈n}. |
Theorem 4.4. (1) B∘(n,ω,γ) constructed in Subsection 4.1 is a Hopf algebra;
(2) As a Hopf algebra, B(n,ω,γ)∘ is isomorphic to B∘(n,ω,γ).
Proof. Consider the following map
Θ:B∘(n,ω,γ)→B(n,ω,γ)∘,Ψλ1ω,λ1n↦ψλ1ω,λ1n,F1↦E1,F2↦E2, |
where λ1ω,λ1n∈𝕜∗. This is an epimorphism of algebras by Lemma 4.1 and Proposition 4.3. Furthermore, Θ would become an isomorphism of Hopf algebras with desired coalgebra structure and antipode, as long as it is injective (since B(n,ω,γ)∘ is in fact a Hopf algebra).
In order to show that Θ is injective, we aim to show the linear independence of
{ψλ1ω,λ1nψiη,1ψj1,γEs2El1∣λ∈𝕜∗,i∈ω,j∈n,s∈N,l∈n} |
in B(n,ω,γ)∘, where λ1ω and λ1n are fixed roots of each λ∈𝕜∗. By linear independence of elements in (4.3), we only need to show that any finite sum of form
N∑α=1(B(n,ω,γ)/((gn−λα)r))∗=N∑α=1((gn−λα)r))⊥ | (4.5) |
is direct, as long as λα's are distinct from each other. But this is clear:
((gn−λα′)r)⊥∩[∑α≠α′((gn−λα)r)⊥]=((gn−λα′)r)⊥∩[⋂α≠α′((gn−λα)r)]⊥=((gn−λα′)r)⊥∩(∏α≠α′(gn−λα)r)⊥=[((gn−λα′)r)+(∏α≠α′(gn−λα)r))]⊥=B(n,ω,γ)⊥=0. | (4.6) |
Remark 4.5. As a special case, the connected algebraic groups of dimension one H2=𝕜[g±1] is equal to B(1,1,1). Therefore, Theorem 4.4 gives H2∘ which is well-known too (see [21,Example 9.2.7] for example).
Remark 4.6. Let us contrast the structure of B(n,ω,γ)∘ with the Hopf algebra introduced in [1]: Choose a set I:={(α,β)∈𝕜∗×𝕜∗∣αω=βn}. Then a particular case of the Hopf algebra
D:=D(n,γ,(λ1n)(λ1ω,λ1n)∈I,1) |
defined in [1,Theorem 3.4] is generated by u,x,a±1λ1ω,λ1n(λ1ω,λ1n∈𝕜∗) with relations
un=1,xn=0,a±1λ1ω,λ1na∓1λ1ω,λ1n=1,ux=γxu,uaλ1ω,λ1n=aλ1ω,λ1nu,aλ1ω,λ1nx=λ1nxaλ1ω,λ1n,aλ1ω1,λ1n1aλ1ω2,λ1n2=aλ1ω2,λ1n2aλ1ω1,λ1n1 |
as an algebra for λ1ω,λ1n,λ1ω1,λ1n1,λ1ω2,λ1n2∈𝕜∗. The comultiplication, counit and antipode on D is defined by
Δ(u)=u⊗u,Δ(x)=u⊗x+x⊗1,Δ(a±1λ1ω,λ1n)=(1⊗1+(1−λ±1)n−1∑k=11k!γ(n−k)!γxn−kuk⊗xk)(a±1λ1ω,λ1n⊗a±1λ1ω,λ1n)ε(u)=1,ε(x)=0,ε(a±1λ1ω,λ1n)=1,S(u)=un−1,S(x)=−un−1x,S(a±1λ1ω,λ1n)=a∓1λ1ω,λ1n |
for λ1ω,λ1n∈𝕜∗. Denote by ¯D the quotient Hopf algebra of D modulo the ideal generated by
a1,1−1,u−a1,γ,aλ1ω1,λ1n1aλ1ω2,λ1n2−aλ1ω1λ1ω2,λ1n1λ1n2(λ1ω1,λ1n1,λ1ω2,λ1n2∈𝕜∗). |
It can be found that ¯Dopcop is isomorphic to the Hopf subalgebra of B(n,ω,γ)∘ generated by ψλ1ω,λ1n,E1(λ1ω,λ1n∈𝕜∗) via the isomorphism
ψλ1ω,λ1n↦aλ1ω,λ1n,x↦E1(λ1ω,λ1n∈𝕜∗). |
We apply the same way used Sections 3 and 4 to give structure of D(m,d,ξ)∘ in this section.
Let m and d be positive integers such that (1+m)d is even, and ξ be a primitive 2mth root of 1. Define
ω:=md,γ:=ξ2. |
The Hopf algebra D∘(m,d,ξ) is constructed as follows. As an algebra, it is generated by
Zλ1ω,λ1m,Xλ1ω,λ1m,F1,F2,(λ1ω,λ1m∈𝕜∗) |
(where the parameters of Z and X are meant to range through {(α,β)∈𝕜∗×𝕜∗∣αω=βm}) with relations
Zλ1ω1,λ1m1Zλ1ω2,λ1m2=Z(λ1ω1λ1ω2),(λ1m1λ1m2),Xλ1ω1,λ1m1Xλ1ω2,λ1m2=X(λ1ω1λ1ω2),(λ1m1λ1m2),Zλ1ω1,λ1m1Xλ1ω2,λ1m2=Xλ1ω1,λ1m1Zλ1ω2,λ1m2=0,Z1,1+X1,1=1,Fm1=1(1−γ)mX1,1,F2Zλ1ω,λ1m=Zλ1ω,λ1mF2,F1Zλ1ω,λ1m=λ1mZλ1ω,λ1mF1,F2Xλ1ω,λ1m=Xλ1ω,λ1mF2,F1Xλ1ω,λ1m=λ−dωλ1mXλ1ω,λ1mF1,F1F2=F2F1+1mZ1,1F1 |
for λ1ω,λ1m,λ1ω1,λ1m1,λ1ω2,λ1m2∈𝕜∗. Denote F[k]1:=1k!γFk1 for 1≤k≤m−1. The comultiplication is given by
Δ(F1)=1⊗F1+F1⊗(Z1,γ+ξX1,γ),Δ(F2)=(Z1,1−X1,1)⊗F2+F2⊗1−m−1∑k=1(Z1,1−X1,1)F[k]1⊗(Z1,γ+ξX1,γ)−m+kF[m−k]1,Δ(Zλ1ω,λdω)=Zλ1ω,λdω⊗Zλ1ω,λdω+(1−λ)m−1∑k=1Zλ1ω,λdωF[k]1⊗Zλ1ω,λdωZk1,γF[m−k]1+λ(1−m)d/2ω(1−λ)(θ−10Xλ1ω,λdω⊗Xλ−1ω,λ−dω+m−1∑k=1θ−1m−kXλ1ω,λdωF[k]1⊗Xλ−1ω,λ−dωξkXk1,γF[m−k]1),Δ(Xλ1ω,λdω)=Zλ1ω,λdω⊗Xλ1ω,λdω−θ0m−1∑k=1θ1⋯θk−1Zλ1ω,λdωE[k]1⊗Xλ1ω,λdωξkXk1,γF[m−k]1+Xλ1ω,λdω⊗Zλ−1ω,λ−dω−θ0m−1∑k=1λ−(m−k)dωθ1⋯θm−k−1Xλ1ω,λdωE[k]1⊗Zλ−1ω,λ−dωZk1,γF[m−k]1, |
where λ1ω∈𝕜∗ and θ0=1−λdω1/m,θk=1−γkλdω1−γk(1≤k≤m−1). Note that θ0θ1⋯θm−1=1−λ holds (see [13,Proposition IV.2.7]) and thus (1−λ)θ−1k for 0≤k≤m−1 is well-defined. The coproducts on Zλ1ω,λ1m and Xλ1ω,λ1m for arbitrary λ1ω and λ1m are given by
Δ(Zλ1ω,λ1m)=Δ(Zλ1ω,λdω)Δ(Z1,γ+ξX1,γ)k, |
Δ(Xλ1ω,λ1m)=ξ−kΔ(Xλ1ω,λdω)Δ(Z1,γ+ξX1,γ)k, |
where k is a non-negative integer such that λ1m=λdωγk and Z1,γ+ξX1,γ is defined to be a group-like element.
The counit is given by
ε(Zλ1ω,λ1m)=1,ε(Xλ1ω,λ1m)=0,ε(F1)=ε(F2)=0. |
The antipode is given by
S(F1)=−γ−1(Z1,γ−1+ξ−1X1,γ−1)F1,S(F2)=−Z1,1F2+X1,1F2+1−m2mX1,1,S(Zλ1ω,λ1m)=Zλ−1ω,λ−1m,S(Xλ1ω,λ1m)=λ(1−m)d/2ωγ−kXλ1ω,λdωγ−kwhenλ1m=λdωγk. |
The fact that D∘(m,d,ξ) is a Hopf algebra would be stated and proved in Subsection 5.3 as Theorem 5.7(1). Let η be an primitive ωth root of 1. Here we note that
{Zλ1ω,λ1mZiη,1Zj1,γFs2Fl1,Xλ1ω,λ1mXiη,1Xj1,γFs2Fl1∣λ1ω,λ1nare fixed roots ofλ∈𝕜∗,i∈ω,j∈m,s∈N,l∈m} |
is a linear basis, due to an application of the Diamond lemma [2].
Remark 5.1. It is asked in [4,Remarks 7.1(3)] whether the cocycle σ non-trivial in the formula of [4,Proposition 7.2(V)]:
D(m,d,ξ)∘=(𝕜C2#σTf(m,1,ξ2))#(𝕜[f]⊗𝕜(𝕜×)). |
Our results can be used to find that σ is trivial in fact.
Let m,d be positive integers such that (1+m)d is even and ξ a primitive 2mth root of unity. Define
ω:=md,γ:=ξ2. |
According to the definition of D(m,d,ξ) recalled in Definition 2.4, we know that it has a linear basis {xigjyl,xigjul∣i∈ω,j∈Z,l∈m} by [31,Lemma 3.3] and [32,Eq 4.7]. Define following elements in D(m,d,ξ)∗:
ζλ1ω,λ1m:{xigjyl↦δl,0λiωλjmxigjul↦0,χλ1ω,λ1m:{xigjyl↦0xigjul↦δl,0λiωλjm,E1:{xigjyl↦δl,1xigjul↦ξ1−γ−1δl,1,E2:{xigjyl↦δl,0(iω+jm)xigjul↦δl,0(iω+jm), | (5.1) |
for any i∈ω, j∈Z, l∈m and λ1ω,λ1m∈𝕜∗. We remark that these definitions make sense for all i∈Z as well, due to direct computations.
One can verify that ζλ1ω,λ1m and χλ1ω,λ1m both vanish on the principal left ideal ((gm−λ)(gm−λ−1)) for any λ1ω,λ1m∈𝕜∗; E2 vanishes on ((gm−1)2); E1 vanishes on (gm−1). Therefore, these elements belong to D(m,d,ξ)∘.
Lemma 5.2. Following equations hold in D(m,d,ξ)∘:
ζλ1ω1,λ1m1ζλ1ω2,λ1m2=ζ(λ1ω1λ1ω2),(λ1m1λ1m2),χλ1ω1,λ1m1χλ1ω2,λ1m2=χ(λ1ω1λ1ω2),(λ1m1λ1m2),ζλ1ω1,λ1m1χλ1ω2,λ1m2=χλ1ω1,λ1m1ζλ1ω2,λ1m2=0,ζ1,1+χ1,1=1,Em1=1(1−γ)mχ1,1,E2ζλ1ω,λ1m=ζλ1ω,λ1mE2,E1ζλ1ω,λ1m=λ1mζλ1ω,λ1mE1,E2χλ1ω,λ1m=χλ1ω,λ1mE2,E1χλ1ω,λ1m=λ−dωλ1mχλ1ω,λ1mE1,E1E2=E2E1+1mζ1,1E1 |
for all λ1ω,λ1m,λ1ω1,λ1m1,λ1ω2,λ1m2∈𝕜∗.
Proof. Note that
Δ(xigjyl)=l∑k=0(lk)γxigjyk⊗xigj+kyl−k,Δ(xigjul)=m−1∑k=0γk(l−k)xigjuk⊗xi−kdgj+kul−k. |
We prove the lemma through checking their values on the basis xigjyl,xigjul(i∈ω,j∈Z,l∈m):
⟨ζλ1ω1,λ1m1ζλ1ω2,λ1m2,xigjyl⟩=δl,0⟨ζλ1ω1,λ1m1,xigj⟩⟨ζλ1ω2,λ1m2,xigj⟩=δl,0λiω1λiω2λjm1λjm2=⟨ζλ1ω1λ1ω2,λ1m1λ1m2,xigjyl⟩,⟨ζλ1ω1,λ1m1ζλ1ω2,λ1m2,xigjul⟩=0=⟨ζλ1ω1λ1ω2,λ1m1λ1m2,xigjul⟩,⟨χλ1ω1,λ1m1χλ1ω2,λ1m2,xigjyl⟩=0=⟨χλ1ω1λ1ω2,λ1m1λ1m2,xigjyl⟩,⟨χλ1ω1,λ1m1χλ1ω2,λ1m2,xigjul⟩=δl,0⟨χλ1ω1,λ1m1,xigju0⟩⟨χλ1ω2,λ1m2,xigju0⟩=δl,0λiω1λiω2λjm1λjm2=⟨ζλ1ω1λ1ω2,λ1m1λ1m2,xigjul⟩,⟨ζλ1ω1,λ1m1χλ1ω2,λ1m2,xigjyl⟩=⟨ζλ1ω1,λ1m1χλ1ω2,λ1m2,xigju0⟩=0,⟨χλ1ω1,λ1m1ζλ1ω2,λ1m2,xigjyl⟩=⟨χλ1ω1,λ1m1ζλ1ω2,λ1m2,xigju0⟩=0,⟨ζ1,1+χ1,1,xigjyl⟩=δl,01i1j=⟨ε,xigjyl⟩,⟨ζ1,1+χ1,1,xigjul⟩=δl,01i1j=⟨ε,xigjul⟩,⟨E2ζλ1ω,λ1m,xigjyl⟩=δl,0⟨E2,xigj⟩⟨ζλ1ω,λ1m,xigj⟩=δl,0(iω+jm)λiωλjm=δl,0⟨ζλ1ω,λ1m,xigj⟩⟨E2,xigj⟩=⟨ζλ1ω,λ1mE2,xigjyl⟩,⟨E2ζλ1ω,λ1m,xigjul⟩=0=⟨ζλ1ω,λ1mE2,xigjul⟩,⟨E2χλ1ω,λ1m,xigjyl⟩=0=⟨χλ1ω,λ1mE2,xigjyl⟩,⟨E2χλ1ω,λ1m,xigjul⟩=δl,0⟨E2,xigju0⟩⟨ζλ1ω,λ1m,xigju0⟩=δl,0(iω+jm)λiωλjm=δl,0⟨ζλ1ω,λ1m,xigju0⟩⟨E2,xigju0⟩=⟨ζλ1ω,λ1mE2,xigjul⟩,⟨E1ζλ1ω,λ1m,xigjyl⟩=δl,1⟨E1,xigjy⟩⟨ζλ1ω,λ1m,xigj+1⟩=δl,1λiωλj+1m=λ1mδl,1λiωλjm=λ1mδl,1⟨ζλ1ω,λ1m,xigj⟩⟨E1,xigjy⟩=⟨λ1mζλ1ω,λ1mE1,xigjyl⟩,⟨E1ζλ1ω,λ1m,xigjul⟩=0=⟨λ1mζλ1ω,λ1mE1,xigjul⟩,⟨E1χλ1ω,λ1m,xigjyl⟩=0=⟨λ−dωλ1mχλ1ω,λ1mE1,xigjyl⟩,⟨E1χλ1ω,λ1m,xigjul⟩=δl,1⟨E1,xigju1⟩⟨χλ1ω,λ1m,xi−dgj+1u0⟩=δl,1ξ1−γ−1λi−dωλj+1m=λ−dωλ1mδl,1λiωλjmξ1−γ−1=λ−dωλ1mδl,1⟨χλ1ω,λ1m,xigju0⟩⟨E1,xigju1⟩=⟨λ−dωλ1mχλ1ω,λ1mE1,xigjul⟩,⟨E1E2,xigjyl⟩=δl,1⟨E1,xigjy⟩⟨E2,xigj+1⟩=δl,1(iω+j+1m)=δl,1(iω+jm)+δl,11m=δl,1⟨E2,xigj⟩⟨E1,xigjy⟩+δl,11m⟨E1,xigjy⟩=⟨E2E1+1mE1,xigjyl⟩=⟨E2E1+1mζ1,1E1,xigjyl⟩,⟨E1E2,xigjul⟩=δl,1⟨E1,xigju1⟩⟨E2,xi−dgj+1u0⟩=δl,1ξ1−γ−1(i−dω+j+1m)=δl,1(iω+jm)ξ1−γ−1=δl,1⟨E2,xigju0⟩⟨E1,xigju1⟩=⟨E2E1,xigjul⟩=⟨E2E1+1mζ1,1E1,xigjul⟩. |
Similarly to the case in Lemmas 3.1 and 4.1, it can be found by induction on k∈n that ⟨Ek1,xigjyl⟩=δl,kk!γ and
⟨Ek1,xigjul⟩=δl,kγk−1⟨E1,xigju1⟩⟨Ek−11,xi−dgj+1uk−1⟩=δl,kγk−1ξ1−γ−1⟨Ek−11,xi−dgj+1uk−1⟩=δl,kγ(k−1)+(k−2)(ξ1−γ−1)2⟨Ek−21,xi−2dgj+2uk−2⟩=⋯=δl,kγ(k−1)+(k−2)+⋯+1(ξ1−γ−1)k−1⟨E1,xi−(k−1)dgj+(k−1)u1⟩=δl,kγ(k−1)+(k−2)+⋯+1(ξ1−γ−1)k=δl,kξk2(1−γ−1)k. |
Thus Em1=ξm2(1−γ−1)mχ1,1=(−1)m(1−γ−1)mχ1,1=1(1−γ)mχ1,1.
It can be found in the proof above that
E[k]1:=1k!γEk1:{xigjyl↦δl,kxigjul↦1k!γξk2(1−γ−1)kδl,k=ξk(1−γ−1)(1−γ−2)⋯(1−γ−k)δl,k. |
Similarly to Eq (4.2), we can also verify that for any λ1ω,λ1m∈𝕜∗ and k∈m,
ζλ1ω,λ1mE[k]1:{xigjyl↦δl,kλiωλjmxigjul↦0,χλ1ω,λ1mE[k]1:{xigjyl↦0xigjul↦ξk(1−γ−1)(1−γ−2)⋯(1−γ−k)δl,kλiωλjm. |
Lemma 5.3. Following equations hold in D(m,d,ξ)∘:
Δ(E1)=1⊗E1+E1⊗(ζ1,γ+ξχ1,γ),Δ(E2)=(ζ1,1−χ1,1)⊗E2+E2⊗1−m−1∑k=1(ζ1,1−χ1,1)E[k]1⊗(ζ1,γ+ξχ1,γ)−m+kE[m−k]1,ε(E1)=ε(E2)=0, |
where E[k]1:=1k!γEk1 for 1≤k≤m−1. We remark that ζ1,1−χ1,1=(ζ1,γ+ξχ1,γ)m.
Proof. Note that for each i,i′∈ω, j,j′∈Z and l,l′∈m,
(xigjyl)(xi′gj′yl′)=γj′lxi+i′gj+j′yl+l′,(xigjul)(xi′gj′ul′)=γj′lxi−i′−2dj′gj+j′ulul′,(xigjyl)(xi′gj′ul′)=γj′lxi+i′ϕl′ϕl′+1⋯ϕl′+l−1gj+j′ul+l′,(xigjul)(xi′gj′yl′)=ξ−l′γj′lxi−i′−2dj′−dl′ϕlϕl+1⋯ϕl+l′−1gj+j′ul+l′. |
We also prove the lemma by through checking their values on the basis.
⟨E1,(xigjyl)(xi′gj′yl′)⟩=⟨E1,γj′lxi+i′gj+j′yl+l′⟩=δl+l′,1γj′l⟨E1,xi+i′gj+j′y⟩+δl+l′,m+1γj′l⟨E1,xi+i′gj+j′ym+1⟩=δl+l′,1γj′l=δl,0δl′,1+δl,1δl′,0γj′,=⟨ζ1,1⊗ζ1,1E1+ζ1,1E1⊗ζ1,γ,xigjyl⊗xi′gj′yl′⟩,⟨E1,(xigjul)(xi′gj′ul′)⟩=⟨E1,γj′lxi−i′−2dj′gj+j′ulul′⟩=δl+l′,1γj′l⟨E1,xi−i′−2dj′gj+j′ulul′⟩=δl,0δl′,1⟨E1,xi−i′−2dj′gj+j′u0u1⟩+δl,1δl′,0γj′⟨E1,xi−i′−2dj′gj+j′u1u0⟩=δl,0δl′,1⟨E1,u0u1⟩+δl,1δl′,0γj′⟨E1,u1u0⟩=δl,0δl′,1⟨E1,−ξmϕ0ϕ1⋯ϕm−3yg⟩+δl,1δl′,0γj′⟨E1,1mϕ1ϕ2⋯ϕm−2yg⟩=δl,0δl′,1−ξγm(1−γ−1)(1−γ−2)⋯(1−γ−(m−2))+δl,1δl′,0γj′γm(1−γ−2)(1−γ−3)⋯(1−γ−(m−1))=δl,0δl′,1−ξγ1−γ+δl,1δl′,0γj′γ1−γ−1=δl,0(ξ1−γ−1δl′,1)+ξ(ξ1−γ−1δl,1)(δl′,0γj′)=⟨χ1,1⊗χ1,1E1+χ1,1E1⊗ξχ1,γ,xigjul⊗xi′gj′ul′⟩,⟨E1,(xigjyl)(xi′gj′ul′)⟩=⟨E1,γj′lxi+i′ϕl′ϕl′+1⋯ϕl′+l−1gj+j′ul+l′⟩=δl,0δl′,1⟨E1,xi+i′gj+j′u1⟩+δl,1δl′,0γj′⟨E1,xi+i′ϕ0gj+j′u1⟩=δl,0δl′,1⟨E1,u1⟩+δl,1δl′,0γj′⟨E1,ϕ0u1⟩=δl,0δl′,1ξ1−γ−1+δl,1δl′,0γj′(1−γ−1)ξ1−γ−1=δl,0(ξ1−γ−1δl′,1)+ξδl,1(δl′,0γj′)=⟨ζ1,1⊗χ1,1E1+ζ1,1E1⊗ξχ1,γ,xigjyl⊗xi′gj′ul′⟩,⟨E1,(xigjul)(xi′gj′yl′)⟩=⟨E1,ξ−l′γj′lxi−i′−2dj′−dl′ϕlϕl+1⋯ϕl+l′−1gj+j′ul+l′⟩=δl,0δl′,1ξ−1⟨E1,xi−i′−2dj′−dϕ0gj+j′u1⟩+δl,1δl′,0γj′⟨E1,xi−i′−2dj′gj+j′u1⟩=δl,0δl′,1ξ−1⟨E1,ϕ0u1⟩+δl,1δl′,0γj′⟨E1,u1⟩=δl,0δl′,1ξ−1(1−γ−1)ξ1−γ−1+δl,1δl′,0γj′ξ1−γ−1=δl,0δl′,1+(ξ1−γ−1δl,1)(δl′,0γj′)=⟨χ1,1⊗ζ1,1E1+χ1,1E1⊗ζ1,γ,xigjul⊗xi′gj′yl′⟩. |
It can be concluded that \Delta(E_1) = 1\otimes E_1+E_1\otimes(\zeta_{1, \gamma}+\xi\chi_{1, \gamma}) .
\begin{array}{l} && \langle E_2, (x^ig^jy^l)(x^{i'}g^{j'}y^{l'})\rangle \\ & = & \langle E_2, \gamma^{j'l}x^{i+i'}g^{j+j'}y^{l+l'}\rangle \\ & = & \gamma^{j'l}(\delta_{l+l', 0}\langle E_2, x^{i+i'}g^{j+j'}\rangle +\delta_{l+l', m}\langle E_2, x^{i+i'}g^{j+j'}-x^{i+i'}g^{j+j'+m}\rangle) \\ & = & \delta_{l, 0}\delta_{l', 0}(\frac{i+i'}{{\omega}}+\frac{j+j'}{m}) \\ & & +\delta_{l+l', m}\gamma^{j'l}(\frac{i+i'}{{\omega}}+\frac{j+j'}{m}-\frac{i+i'}{{\omega}}-\frac{j+j'+m}{m}) \\ & = & \delta_{l, 0}\delta_{l', 0}(\frac{i+i'}{{\omega}}+\frac{j+j'}{m}) +\sum\limits_{k = 1}^{m-1}\delta_{l, k}\delta_{l', m-k}\gamma^{j'k}(-1) \\ & = & \delta_{l, 0}\delta_{l', 0}(\frac{i'}{{\omega}}+\frac{j'}{m}) +\delta_{l, 0}(\frac{i}{{\omega}}+\frac{j}{m})\delta_{l', 0} -\sum\limits_{k = 1}^{m-1}\delta_{l, k}\delta_{l', m-k}\gamma^{j'k} \\ & = & \langle\zeta_{1, 1}\otimes\zeta_{1, 1}E_2+\zeta_{1, 1}E_2\otimes\zeta_{1, 1} -\sum\limits_{k = 1}^{m-1}\zeta_{1, 1}E_1^{[k]}\otimes\zeta_{1, \gamma}^kE_1^{[m-k]}, x^ig^jy^l\otimes x^{i'}g^{j'}y^{l'}\rangle, \\ && \langle E_2, (x^ig^ju_l)(x^{i'}g^{j'}u_{l'})\rangle \\ & = & \langle E_2, \gamma^{j'l}x^{i-i'-2dj'}g^{j+j'}u_lu_{l'}\rangle \\ & = & \delta_{l, 0}\delta_{l', 0} \langle E_2, \frac{1}{m}x^{i-i'-2dj'-\frac{(1+m)}{2}d}\phi_0\phi_1\cdots\phi_{m-2}g^{j+j'+1}\rangle \\ & & +\sum\limits_{k = 1}^{m-1}\delta_{l, k}\delta_{l', m-k}\gamma^{j'k} \langle E_2, (-1)^{-m+k}\xi^{(m-k)^2}\frac{1}{m}x^{i-i'-2dj'-\frac{(1+m)}{2}d} \phi_k\cdots\phi_{m-1}\phi_0\cdots\phi_{k-2}g^{j+j'+1}\rangle \\ & = & \delta_{l, 0}\delta_{l', 0}\frac{1}{m} [(\frac{i-i'-2dj'}{{\omega}}-\frac{(1+m)}{2{\omega}}d+\frac{j+j'+1}{m})m+\frac{m-1}{2}] \\ & & +\sum\limits_{k = 1}^{m-1}\delta_{l, k}\delta_{l', m-k}\gamma^{j'k} (-1)^{-m+k}\xi^{(m-k)^2}\frac{1}{m} [-\frac{1}{m}(1-\gamma^{-k-1})\cdots(1-\gamma^{-m+1})(1-\gamma^{-1})\cdots(1-\gamma^{-k+1})] \\ & = & \delta_{l, 0}\delta_{l', 0}(\frac{i-i'}{{\omega}}+\frac{j-j'}{m}) \\ & & +\sum\limits_{k = 1}^{m-1}\delta_{l, k}\delta_{l', m-k}\gamma^{j'k} (-1)^{-m+k}\xi^{(m-k)^2}\frac{1}{m} (-\frac{1}{m}\frac{m}{1-\gamma^{-k}}) \\ & = & -\delta_{l, 0}\delta_{l', 0}(\frac{i'}{{\omega}}+\frac{j'}{m}) +\delta_{l, 0}(\frac{i}{{\omega}}+\frac{j}{m})\delta_{l', 0} \\ & & -\sum\limits_{k = 1}^{m-1} \frac{\xi^k}{(1-\gamma^{-1})\cdots(1-\gamma^{-k})}\delta_{l, k} \xi^k\frac{\xi^{m-k}}{(1-\gamma^{-1})\cdots(1-\gamma^{-(m-k)})}\delta_{l', m-k}\gamma^{j'k} \\ & = & \langle-\chi_{1, 1}\otimes\chi_{1, 1}E_2+\chi_{1, 1}E_2\otimes\chi_{1, 1} -\sum\limits_{k = 1}^{m-1}\chi_{1, 1}E_1^{[k]}\otimes\xi^k\chi_{1, \gamma}^kE_1^{[m-k]}, x^ig^ju_l\otimes x^{i'}g^{j'}u_{l'}\rangle, \\ && \langle E_2, (x^ig^jy^l)(x^{i'}g^{j'}u_{l'})\rangle \\ & = & \langle E_2, \gamma^{j'l}x^{i+i'}\phi_{l'}\phi_{l'+1}\cdots\phi_{l'+l-1}g^{j+j'}u_{l+l'}\rangle \\ & = & \delta_{l, 0}\delta_{l', 0}\langle E_2, x^{i+i'}g^{j+j'}u_0\rangle \\ & & +\sum\limits_{k = 1}^{m-1}\delta_{l, k}\delta_{l', m-k}\gamma^{j'k} \langle E_2, x^{i+i'}\phi_{m-k}\phi_{m-k+1}\cdots\phi_{m-1}g^{j+j'}u_0\rangle \\ & = & \delta_{l, 0}\delta_{l', 0}(\frac{i+i'}{{\omega}}+\frac{j+j'}{m}) \\ & & +\sum\limits_{k = 1}^{m-1}\delta_{l, k}\delta_{l', m-k}\gamma^{j'k} (-\frac{1}{m})(1-\gamma^{k-1})(1-\gamma^{k-2})\cdots(1-\gamma) \\ & = & \delta_{l, 0}\delta_{l', 0}(\frac{i+i'}{{\omega}}+\frac{j+j'}{m}) \\ & & -\sum\limits_{k = 1}^{m-1}\delta_{l, k}\delta_{l', m-k}\gamma^{j'k}\frac{1}{m} \frac{m}{(1-\gamma^{-1})(1-\gamma^{-2})\cdots(1-\gamma^{-(m-k)})} \\ & = & \delta_{l, 0}\delta_{l', 0}(\frac{i'}{{\omega}}+\frac{j'}{m}) +\delta_{l, 0}(\frac{i}{{\omega}}+\frac{j}{m})\delta_{l', 0} \\ & & +\sum\limits_{k = 1}^{m-1}\xi^k\delta_{l, k} [\frac{\xi^{m-k}}{(1-\gamma^{-1})(1-\gamma^{-2})\cdots(1-\gamma^{-(m-k)})}\delta_{l', m-k}\gamma^{j'k}]\\ & = & \langle\zeta_{1, 1}\otimes\chi_{1, 1}E_2+\zeta_{1, 1}E_2\otimes\chi_{1, 1} +\sum\limits_{k = 1}^{m-1}\zeta_{1, 1}E_1^{[k]}\otimes\xi^k\chi_{1, \gamma}^kE_1^{[m-k]}, x^ig^jy^l\otimes x^{i'}g^{j'}u_{l'}\rangle, \\ && \langle E_2, (x^ig^ju_l)(x^{i'}g^{j'}y^{l'})\rangle \\ & = & \langle E_2, \xi^{-l'}\gamma^{j'l}x^{i-i'-2dj'-dl'}\phi_l\phi_{l+1}\cdots\phi_{l+l'-1}g^{j+j'}u_{l+l'}\rangle \\ & = & \delta_{l, 0}\delta_{l', 0}\langle E_2, x^{i-i'-2dj'}g^{j+j'}u_0\rangle \\ & & +\sum\limits_{k = 1}^{m-1}\delta_{l, k}\delta_{l', m-k}\xi^{-(m-k)}\gamma^{j'k} \langle E_2, x^{i-i'-2dj'-d(m-k)}\phi_k\phi_{k+1}\cdots\phi_{m-1}g^{j+j'}u_0\rangle \\ & = & \delta_{l, 0}\delta_{l', 0}(\frac{i-i'-2dj'}{{\omega}}+\frac{j+j'}{m}) \\ & & +\sum\limits_{k = 1}^{m-1}\delta_{l, k}\delta_{l', m-k}\xi^{-(m-k)}\gamma^{j'k} (-\frac{1}{m})(1-\gamma^{m-k-1})(1-\gamma^{m-k-2})\cdots(1-\gamma) \\ & = & \delta_{l, 0}\delta_{l', 0}(\frac{i-i'}{{\omega}}+\frac{j-j'}{m}) \\ & & -\sum\limits_{k = 1}^{m-1}\delta_{l, k}\delta_{l', m-k}\xi^{-(m-k)}\gamma^{j'k}\frac{1}{m} \frac{m}{(1-\gamma^{-1})(1-\gamma^{-2})\cdots(1-\gamma^{-k})} \\ & = & -\delta_{l, 0}\delta_{l', 0}(\frac{i'}{{\omega}}+\frac{j'}{m}) +\delta_{l, 0}(\frac{i}{{\omega}}+\frac{j}{m})\delta_{l', 0} \\ & & +\sum\limits_{k = 1}^{m-1} [\frac{\xi^k}{(1-\gamma^{-1})(1-\gamma^{-2})\cdots(1-\gamma^{-k})}\delta_{l, k}] (\delta_{l', m-k}\gamma^{j'k}) \\ & = & \langle-\chi_{1, 1}\otimes\zeta_{1, 1}E_2+\chi_{1, 1}E_2\otimes\zeta_{1, 1} +\sum\limits_{k = 1}^{m-1}\chi_{1, 1}E_1^{[k]}\otimes\zeta_{1, \gamma}^kE_1^{[m-k]}, x^ig^ju_l\otimes x^{i'}g^{j'}y^{l'}\rangle. \end{array} |
It can be concluded that
\Delta(E_2) = (\zeta_{1, 1}-\chi_{1, 1})\otimes E_2+E_2\otimes 1 -\sum\limits_{k = 1}^{m-1} (\zeta_{1, 1}-\chi_{1, 1})E_1^{[k]}\otimes(\zeta_{1, \gamma}+\xi\chi_{1, \gamma})^{-m+k}E_1^{[m-k]}. |
The counit is clear.
Lemma 5.4. Following equations hold in D(m, d, \xi)^\circ :
\Delta(\zeta_{1, \gamma}+\xi\chi_{1, \gamma}) = (\zeta_{1, \gamma}+\xi\chi_{1, \gamma})\otimes(\zeta_{1, \gamma}+\xi\chi_{1, \gamma}), \;\; \varepsilon(\zeta_{1, \gamma}+\xi\chi_{1, \gamma}) = 1, |
and
\begin{array}{l} & \Delta(\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}) = \zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}\otimes\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}} +(1-{\lambda})\sum\limits_{k = 1}^{m-1}\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}E_1^{[k]} \otimes\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}\zeta_{1, \gamma}^kE_1^{[m-k]} \\ & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; +{\lambda}^\frac{(1-m)d/2}{{\omega}}(1-{\lambda}) ({\theta}_0^{-1}\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}\otimes\chi_{{\lambda}^\frac{-1}{{\omega}}, {\lambda}^\frac{-d}{{\omega}}}\\ & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; +\sum\limits_{k = 1}^{m-1}{\theta}_{m-k}^{-1} \chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}E_1^{[k]} \otimes\chi_{{\lambda}^\frac{-1}{{\omega}}, {\lambda}^\frac{-d}{{\omega}}}\xi^k\chi_{1, \gamma}^kE_1^{[m-k]}) \\ & \Delta(\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}) = \zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}\otimes\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}} -{\theta}_0\sum\limits_{k = 1}^{m-1}{\theta}_1\cdots{\theta}_{k-1} \zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}E_1^{[k]} \otimes\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}\xi^k\chi_{1, \gamma}^kE_1^{[m-k]} \\ & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; +\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}\otimes\zeta_{{\lambda}^\frac{-1}{{\omega}}, {\lambda}^\frac{-d}{{\omega}}} \\ & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; -{\theta}_0\sum\limits_{k = 1}^{m-1}{\lambda}^\frac{-(m-k)d}{{\omega}}{\theta}_1\cdots{\theta}_{m-k-1} \chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}E_1^{[k]} \otimes\zeta_{{\lambda}^\frac{-1}{{\omega}}, {\lambda}^\frac{-d}{{\omega}}}\zeta_{1, \gamma}^kE_1^{[m-k]}, \\ & \varepsilon(\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}) = 1, \;\; \varepsilon(\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}) = 0, \end{array} |
where {\lambda}^\frac{1}{{\omega}}\in{𝕜}^\ast , E_1^{[k]}: = \frac{1}{k!_\gamma}E_1^k , and \theta_0 = \frac{1-{\lambda}^\frac{d}{{\omega}}}{1/m}, \; \theta_k = \frac{1-\gamma^k{\lambda}^\frac{d}{{\omega}}}{1-\gamma^k}\; (1\leq k\leq m-1).
Proof. We also prove the lemma through checking their values on the basis.
\begin{array}{l} &&\langle\zeta_{1, \gamma}+\xi\chi_{1, \gamma}, (x^ig^jy^l)(x^{i'}g^{j'}y^{l'})\rangle \\ & = & \langle\zeta_{1, \gamma}, \gamma^{j'l}x^{i+i'}g^{j+j'}y^{l+l'}\rangle \\ & = & \delta_{l+l', 0}\langle\zeta_{1, \gamma}, x^{i+i'}g^{j+j'}\rangle +\delta_{l+l', m}\langle\zeta_{1, \gamma}, \gamma^{j'l}x^{i+i'}g^{j+j'}-x^{i+i'}g^{j+j'+m}\rangle \\ & = & \delta_{l, 0}\delta_{l', 0}\gamma^{j+j'} \; = \; (\delta_{l, 0}\gamma^j)(\delta_{l', 0}\gamma^{j'}) \\ & = & \langle\zeta_{1, \gamma}\otimes\zeta_{1, \gamma}, x^ig^jy^l\otimes x^{i'}g^{j'}y^{l'}\rangle, \\ &&\langle\zeta_{1, \gamma}+\xi\chi_{1, \gamma}, (x^ig^ju_l)(x^{i'}g^{j'}u_{l'})\rangle \\ & = & \langle\zeta_{1, \gamma}, \gamma^{j'l}x^{i-i'-2dj'}g^{j+j'}u_lu_{l'}\rangle \\ & = & \delta_{l, 0}\delta_{l', 0}\langle\zeta_{1, \gamma}, \frac{1}{m}x^{i-i'-2dj'-\frac{(1+m)}{2}d}\phi_0\phi_1\cdots\phi_{m-2}g^{j+j'+1}\rangle \\ & & +\delta_{l+l', m}\gamma^{j'l} \langle\zeta_{1, \gamma}, (-1)^{-l'}\xi^{{l'}^2}\frac{1}{m}x^{i-i'-2dj'-\frac{(1+m)}{2}d} \phi_l\cdots\phi_{m-1}\phi_0\cdots\phi_{l-2}g^{j+j'+1}\rangle \\ & = & \delta_{l, 0}\delta_{l', 0}\frac{1}{m}\langle\zeta_{1, \gamma}, x^{i-i'-2dj'-\frac{(1+m)}{2}d}\phi_0\phi_1\cdots\phi_{m-2}g^{j+j'+1}\rangle \\ & = & \delta_{l, 0}\delta_{l', 0}\gamma^{j+j'+1} \; = \; (\xi\delta_{l, 0}\gamma^j)(\xi\delta_{l', 0}\gamma^{j'}) \\ & = & \langle\xi\chi_{1, \gamma}\otimes\xi\chi_{1, \gamma}, x^ig^ju_l\otimes x^{i'}g^{j'}u_{l'}\rangle, \\ &&\langle\zeta_{1, \gamma}+\xi\chi_{1, \gamma}, (x^ig^jy^l)(x^{i'}g^{j'}u_{l'})\rangle \\ & = & \langle\xi\chi_{1, \gamma}, \gamma^{j'l}x^{i+i'}\phi_{l'}\phi_{l'+1}\cdots\phi_{l'+l-1}g^{j+j'}u_{l+l'}\rangle \\ & = & \delta_{l, 0}\delta_{l', 0}\xi\langle\chi_{1, \gamma}, x^{i+i'}g^{j+j'}u_0\rangle +\delta_{l+l', m}\xi\langle\chi_{1, \gamma}, \gamma^{j'l}x^{i+i'}\phi_{l'}\phi_{l'+1}\cdots\phi_{m-1}g^{j+j'}u_m\rangle \\ & = & \delta_{l, 0}\delta_{l', 0}\xi\gamma^{j+j'} \; = \; (\delta_{l, 0}\gamma^j)(\xi\delta_{l', 0}\gamma^{j'}) \\ & = & \langle\zeta_{1, \gamma}\otimes\xi\chi_{1, \gamma}, x^ig^jy^l\otimes x^{i'}g^{j'}u_{l'}\rangle, \\ &&\langle\zeta_{1, \gamma}+\xi\chi_{1, \gamma}, (x^ig^ju_l)(x^{i'}g^{j'}y^{l'})\rangle \\ & = & \langle\xi\chi_{1, \gamma}, \xi^{-l'}\gamma^{j'l}x^{i-i'-2dj'-dl'}\phi_l\phi_{l+1}\cdots\phi_{l+l'-1}g^{j+j'}u_{l+l'}\rangle \\ & = & \delta_{l, 0}\delta_{l', 0}\xi\langle\chi_{1, \gamma}, x^{i-i'-2dj'}g^{j+j'}u_0\rangle \\ & & +\delta_{l+l', m}\xi\langle\chi_{1, \gamma}, \xi^{-l'}\gamma^{j'l}x^{i-i'-2dj'-dl'}\phi_l\phi_{l+1}\cdots\phi_{m-1}g^{j+j'}u_m\rangle \\ & = & \delta_{l, 0}\delta_{l', 0}\xi\gamma^{j+j'} \; = \; (\xi\delta_{l, 0}\gamma^j)(\delta_{l', 0}\gamma^{j'}) \\ & = & \langle\xi\chi_{1, \gamma}\otimes\zeta_{1, \gamma}, x^ig^ju_l\otimes x^{i'}g^{j'}y^{l'}\rangle. \end{array} |
It can be concluded that
\Delta(\zeta_{1, \gamma}+\xi\chi_{1, \gamma}) = (\zeta_{1, \gamma}+\xi\chi_{1, \gamma})\otimes(\zeta_{1, \gamma}+\xi\chi_{1, \gamma}). |
Now we compute that for any {\lambda}^\frac{1}{{\omega}}\in{𝕜}^\ast ,
\begin{array}{l} && \langle\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}, (x^ig^jy^l)(x^{i'}g^{j'}y^{l'})\rangle \\ & = & \langle\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}, \gamma^{j'l}x^{i+i'}g^{j+j'}y^{l+l'}\rangle \\ & = & \delta_{l, 0}\delta_{l', 0}\langle\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}, x^{i+i'}g^{j+j'}\rangle +\sum\limits_{k = 1}^{m-1}\delta_{l, k}\delta_{l', m-k}\langle\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}, \gamma^{j'k}(x^{i+i'}g^{j+j'}-x^{i+i'}g^{j+j'+m})\rangle \\ & = & \delta_{l, 0}\delta_{l', 0}{\lambda}^\frac{i+i}{{\omega}}{\lambda}^\frac{(j+j')d}{{\omega}} +\sum\limits_{k = 1}^{m-1}\delta_{l, k}\delta_{l', m-k}\gamma^{j'k} ({\lambda}^\frac{i+i'}{{\omega}}{\lambda}^\frac{(j+j')d}{{\omega}}-{\lambda}^\frac{i+i'}{{\omega}}{\lambda}^\frac{(j+j'+m)d}{{\omega}}) \\ & = & \delta_{l, 0}\delta_{l', 0}{\lambda}^\frac{i+i'}{{\omega}}{\lambda}^\frac{(j+j')d}{{\omega}} +(1-{\lambda})\sum\limits_{k = 1}^{m-1}\delta_{l, k}\delta_{l', m-k}\gamma^{j'k} {\lambda}^\frac{i+i'}{{\omega}}{\lambda}^\frac{(j+j')d}{{\omega}} \\ & = & (\delta_{l, 0}{\lambda}^\frac{i}{{\omega}}{\lambda}^\frac{jd}{{\omega}})(\delta_{l', 0}{\lambda}^\frac{i'}{{\omega}}{\lambda}^\frac{j'd}{{\omega}}) +(1-{\lambda})\sum\limits_{k = 1}^{m-1}(\delta_{l, k}{\lambda}^\frac{i}{{\omega}}{\lambda}^\frac{jd}{{\omega}}) (\delta_{l', m-k}\gamma^{j'k}{\lambda}^\frac{i'}{{\omega}}{\lambda}^\frac{j'd}{{\omega}}) \\ & = & \langle\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}\otimes\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}} +(1-{\lambda})\sum\limits_{k = 1}^{m-1}\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}E_1^{[k]} \otimes\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}\zeta_{1, \gamma}^kE_1^{[m-k]}, x^ig^jy^l\otimes x^{i'}g^{j'}y^{l'}\rangle, \\ && \langle\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}, (x^ig^ju_l)(x^{i'}g^{j'}u_{l'})\rangle \\ & = & \langle\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}, \gamma^{j'l}x^{i-i'-2dj'}g^{j+j'}u_lu_{l'}\rangle \\ & = & \delta_{l, 0}\delta_{l', 0}\langle\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}, \frac{1}{m}x^{i-i'-2dj'-\frac{(1+m)}{2}d}\phi_0\phi_1\cdots\phi_{m-2}g^{j+j'+1}\rangle \\ & & +\sum\limits_{k = 1}^{m-1}\delta_{l, k}\delta_{l', m-k}\gamma^{j'k} \langle\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}}, (-1)^{-(m-k)}\xi^{(m-k)^2}\frac{1}{m}x^{i-i'-2dj'-\frac{(1+m)}{2}d} \phi_k\cdots\phi_{m-1}\phi_0\cdots\phi_{k-2}g^{j+j'+1}\rangle \\ & = & \delta_{l, 0}\delta_{l', 0}\frac{1}{m}{\lambda}^{\frac{i-i'-2dj'}{{\omega}}-\frac{(1+m)d}{2{\omega}}} (1-\gamma^{-1}{\lambda}^\frac{d}{{\omega}})(1-\gamma^{-2}{\lambda}^\frac{d}{{\omega}}) \cdots(1-\gamma^{-(m-1)}{\lambda}^\frac{d}{{\omega}}){\lambda}^\frac{(j+j'+1)d}{{\omega}} \\ & & +\sum\limits_{k = 1}^{m-1}\delta_{l, k}\delta_{l', m-k}\gamma^{j'k} (-1)^{-(m-k)}\xi^{(m-k)^2}\frac{1}{m}{\lambda}^{\frac{i-i'-2dj'}{{\omega}}-\frac{(1+m)d}{2{\omega}}} \\ & & \;\;\;\;\;\;\;\;\;\;\times(1-\gamma^{-(k+1)}{\lambda}^\frac{d}{{\omega}})\cdots(1-\gamma^{-m}{\lambda}^\frac{d}{{\omega}}) (1-\gamma^{-1}{\lambda}^\frac{d}{{\omega}})\cdots(1-\gamma^{-(k-1)}{\lambda}^\frac{d}{{\omega}}){\lambda}^\frac{(j+j'+1)d}{{\omega}}\\ & = & \delta_{l, 0}\delta_{l', 0}\frac{1}{m}{\lambda}^\frac{i-i'}{{\omega}}{\lambda}^\frac{(j-j')d}{{\omega}}{\lambda}^\frac{(1-m)d/2}{{\omega}} \frac{1-{\lambda}}{1-{\lambda}^\frac{d}{{\omega}}} \\ & & +\sum\limits_{k = 1}^{m-1}\delta_{l, k}\delta_{l', m-k}\gamma^{j'k} (-1)^{-(m-k)}\xi^{(m-k)^2}\frac{1}{m}{\lambda}^\frac{i-i'}{{\omega}}{\lambda}^\frac{(j-j')d}{{\omega}} {\lambda}^\frac{(1-m)d/2}{{\omega}}\frac{1-{\lambda}}{1-\gamma^{-k}{\lambda}^\frac{d}{{\omega}}} \\ & = & (1-{\lambda}){\lambda}^\frac{(1-m)d/2}{{\omega}} [\frac{1/m}{1-{\lambda}^\frac{d}{{\omega}}}\delta_{l, 0}\delta_{l', 0}{\lambda}^\frac{i-i'}{{\omega}}{\lambda}^\frac{(j-j')d}{{\omega}} \\ & & \;\;\;\;\;\;\;\;\; +\sum\limits_{k = 1}^{m-1}\frac{1-\gamma^{-k}}{1-\gamma^{-k}{\lambda}^\frac{d}{{\omega}}} \frac{\xi^k}{(1-\gamma^{-1})\cdots(1-\gamma^{-k})}\delta_{l, k} \frac{\xi^{m-k}}{(1-\gamma^{-1})\cdots(1-\gamma^{-(m-k)})}\delta_{l', m-k} {\lambda}^\frac{i-i'}{{\omega}}{\lambda}^\frac{(j-j')d}{{\omega}}\xi^k\gamma^{j'k}] \\ & = & \langle{\lambda}^\frac{(1-m)d/2}{{\omega}}(1-{\lambda})(\theta_0^{-1} \chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}\otimes\chi_{{\lambda}^\frac{-1}{{\omega}}, {\lambda}^\frac{-d}{{\omega}}} \\ && \;\;\;\;\;\;\;\;\;+\sum\limits_{k = 1}^{m-1}\theta_{m-k}^{-1} \chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}E_1^{[k]} \otimes\chi_{{\lambda}^\frac{-1}{{\omega}}, {\lambda}^\frac{-d}{{\omega}}}\xi^k\chi_{1, \gamma}^kE_1^{[m-k]}), \\ & &\;\; x^ig^ju_l\otimes x^{i'}g^{j'}u_{l'}\rangle, \\ && \langle\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}, (x^ig^jy^l)(x^{i'}g^{j'}u_{l'})\rangle \\ & = & \langle\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}, \gamma^{j'l}x^{i+i'}\phi_{l'}\phi_{l'+1}\cdots\phi_{l'+l-1}g^{j+j'}u_{l+l'}\rangle \; = \; 0, \\ && \langle\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}, (x^ig^ju_l)(x^{i'}g^{j'}y^{l'})\rangle \\ & = & \langle\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}, \xi^{-l'}\gamma^{j'l}x^{i-i'-2dj'-dl'}\phi_l\phi_{l+1}\cdots\phi_{l+l'-1}g^{j+j'}u_{l+l'}\rangle \; = \; 0. \end{array} |
It can be concluded that
\begin{array}{l} & \Delta(\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}) = \zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}\otimes\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}} +(1-{\lambda})\sum\limits_{k = 1}^{m-1}\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}E_1^{[k]} \otimes\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}\zeta_{1, \gamma}^kE_1^{[m-k]} \\ & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; +{\lambda}^\frac{(1-m)d/2}{{\omega}}(1-{\lambda}) ({\theta}_0^{-1}\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}\otimes\chi_{{\lambda}^\frac{-1}{{\omega}}, {\lambda}^\frac{-d}{{\omega}}}\\ & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; +\sum\limits_{k = 1}^{m-1}{\theta}_{m-k}^{-1} \chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}E_1^{[k]} \otimes\chi_{{\lambda}^\frac{-1}{{\omega}}, {\lambda}^\frac{-d}{{\omega}}}\xi^k\chi_{1, \gamma}^kE_1^{[m-k]}). \end{array} |
On the other hand,
\begin{array}{l} && \langle\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}, (x^ig^jy^l)(x^{i'}g^{j'}y^{l'})\rangle \; = \; 0, \\ && \langle\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}, (x^ig^ju_l)(x^{i'}g^{j'}u_{l'})\rangle \; = \; 0, \\ && \langle\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}, (x^ig^jy^l)(x^{i'}g^{j'}u_{l'})\rangle \\ & = & \langle\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}, \gamma^{j'l}x^{i+i'}\phi_{l'}\phi_{l'+1}\cdots\phi_{l'+l-1}g^{j+j'}u_{l+l'}\rangle \\ & = & \delta_{l, 0}\delta_{l', 0} \langle\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}, x^{i+i'}g^{j+j'}u_0\rangle \\ & & +\sum\limits_{k = 1}^{m-1}\delta_{l, k}\delta_{l', m-k}\langle\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}, \gamma^{j'k}x^{i+i'}\phi_{m-k}\phi_{m-k+1}\cdots\phi_{m-1}g^{j+j'}u_0\rangle \\ & = & \delta_{l, 0}\delta_{l', 0}{\lambda}^\frac{i+i'}{{\omega}}{\lambda}^\frac{(j+j')d}{{\omega}} +\sum\limits_{k = 1}^{m-1}\delta_{l, k}\delta_{l', m-k}\gamma^{j'k}{\lambda}^\frac{i+i'}{{\omega}} (1-\gamma^{k-1}{\lambda}^\frac{d}{{\omega}})(1-\gamma^{k-2}{\lambda}^\frac{d}{{\omega}})\cdots(1-{\lambda}^\frac{d}{{\omega}}) {\lambda}^\frac{(j+j')d}{m} \\ & = & \delta_{l, 0}\delta_{l', 0}{\lambda}^\frac{i+i'}{{\omega}}{\lambda}^\frac{(j+j')d}{{\omega}} \\ & & -\sum\limits_{k = 1}^{m-1} (1-{\lambda}^\frac{d}{{\omega}})(1-\gamma{\lambda}^\frac{d}{{\omega}})\cdots(1-\gamma^{k-1}{\lambda}^\frac{d}{{\omega}}) \frac{m}{(1-\gamma)\cdots(1-\gamma^{k-1})} \\ & & \;\;\;\;\;\;\;\;\;\; \times \delta_{l, k}\frac{\xi^{m-k}}{(1-\gamma^{-1})(1-\gamma^{-2})\cdots(1-\gamma^{-(m-k)})} \delta_{l', m-k}\xi^k\gamma^{j'k}{\lambda}^\frac{i+i'}{{\omega}}{\lambda}^\frac{(j+j')d}{{\omega}} \\ & = & \langle\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}\otimes\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}} -{\theta}_0\sum\limits_{k = 1}^{m-1}{\theta}_1\cdots{\theta}_{k-1} \zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}E_1^{[k]} \otimes\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}\xi^k\chi_{1, \gamma}^kE_1^{[m-k]}, x^ig^jy^l\otimes x^{i'}g^{j'}u_{l'}\rangle, \\ && \langle\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}, (x^ig^ju_l)(x^{i'}g^{j'}y^{l'})\rangle \\ & = & \langle\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}, \xi^{-l'}\gamma^{j'l}x^{i-i'-2dj'-dl'}\phi_l\phi_{l+1}\cdots\phi_{l+l'-1}g^{j+j'}u_{l+l'}\rangle \\ & = & \delta_{l, 0}\delta_{l', 0} \langle\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}, x^{i-i'-2dj'}g^{j+j'}u_0\rangle \\ & & +\sum\limits_{k = 1}^{m-k}\delta_{l, k}\delta_{l', m-k}\langle\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}, \xi^{-(m-k)}\gamma^{j'k}x^{i-i'-2dj'-d(m-k)}\phi_k\phi_{k+1}\cdots\phi_{m-1}g^{j+j'}u_0\rangle \\ & = & \delta_{l, 0}\delta_{l', 0}{\lambda}^\frac{i-i'-2dj'}{{\omega}}{\lambda}^\frac{(j+j')d}{{\omega}} +\sum\limits_{k = 1}^{m-k}\delta_{l, k}\delta_{l', m-k} \xi^{-(m-k)}\gamma^{j'k}{\lambda}^\frac{i-i'-2dj'-d(m-k)}{{\omega}} \\ & & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \times(1-\gamma^{m-k-1}{\lambda}^\frac{d}{{\omega}})(1-\gamma^{m-k-2}{\lambda}^\frac{d}{{\omega}})\cdots(1-{\lambda}^\frac{d}{{\omega}}) {\lambda}^\frac{(j+j')d}{{\omega}} \\ & = & \delta_{l, 0}\delta_{l', 0}{\lambda}^\frac{i-i'}{{\omega}}{\lambda}^\frac{(j-j')d}{{\omega}} +\sum\limits_{k = 1}^{m-k}\delta_{l, k}\delta_{l', m-k} \xi^{-(m-k)}\gamma^{j'k}{\lambda}^\frac{i-i'}{{\omega}}{\lambda}^\frac{(j-j')d}{{\omega}}{\lambda}^\frac{-(m-k)d}{{\omega}} \\ & & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \times (1-\gamma^{m-k-1}{\lambda}^\frac{d}{{\omega}})(1-\gamma^{m-k-2}{\lambda}^\frac{d}{{\omega}})\cdots(1-{\lambda}^\frac{d}{{\omega}}) \\ & = & \delta_{l, 0}\delta_{l', 0}{\lambda}^\frac{i-i'}{{\omega}}{\lambda}^\frac{(j-j')d}{{\omega}} \\ & & -\sum\limits_{k = 1}^{m-k}{\lambda}^\frac{-(m-k)d}{{\omega}} (1-{\lambda}^\frac{d}{{\omega}})(1-\gamma{\lambda}^\frac{d}{{\omega}})\cdots(1-\gamma^{m-k-1}{\lambda}^\frac{d}{{\omega}}) \frac{m}{(1-\gamma)\cdots(1-\gamma^{m-k-1})} \\ & & \;\;\;\;\;\;\;\;\;\; \times \frac{\xi^k}{(1-\gamma^{-1})\cdots(1-\gamma^{-k})} \delta_{l, k}\delta_{l', m-k}\gamma^{j'k}{\lambda}^\frac{i-i'}{{\omega}}{\lambda}^\frac{(j-j')d}{{\omega}} \\ & = & \langle\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}\otimes\zeta_{{\lambda}^\frac{-1}{{\omega}}, {\lambda}^\frac{-d}{{\omega}}} -{\theta}_0\sum\limits_{k = 1}^{m-1}{\lambda}^\frac{-(m-k)d}{{\omega}}{\theta}_1\cdots{\theta}_{m-k-1} \chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}E_1^{[k]} \otimes\zeta_{{\lambda}^\frac{-1}{{\omega}}, {\lambda}^\frac{-d}{{\omega}}}\zeta_{1, \gamma}^kE_1^{[m-k]}, x^ig^ju_l\otimes x^{i'}g^{j'}y^{l'}\rangle. \end{array} |
It can be concluded that
\begin{array}{l} & \Delta(\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}) = \zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}\otimes\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}} -{\theta}_0\sum\limits_{k = 1}^{m-1}{\theta}_1\cdots{\theta}_{k-1} \zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}E_1^{[k]} \otimes\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}\xi^k\chi_{1, \gamma}^kE_1^{[m-k]} \\ & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; +\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}\otimes\zeta_{{\lambda}^\frac{-1}{{\omega}}, {\lambda}^\frac{-d}{{\omega}}} \\ & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; -{\theta}_0\sum\limits_{k = 1}^{m-1}{\lambda}^\frac{-(m-k)d}{{\omega}}{\theta}_1\cdots{\theta}_{m-k-1} \chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}}E_1^{[k]} \otimes\zeta_{{\lambda}^\frac{-1}{{\omega}}, {\lambda}^\frac{-d}{{\omega}}}\zeta_{1, \gamma}^kE_1^{[m-k]}. \end{array} |
The counit is clear.
Lemma 5.5. Following equations hold in D(m, d, \xi)^\circ :
\begin{array}{l} S(\zeta_{1, \gamma}+\xi\chi_{1, \gamma}) & = & \zeta_{1, \gamma^{-1}}+\xi^{-1}\chi_{1, \gamma^{-1}}, \\ S(E_1) & = & -\gamma^{-1}(\zeta_{1, \gamma^{-1}}+\xi^{-1}\chi_{1, \gamma^{-1}})E_1, \\ S(E_2) & = & -\zeta_{1, 1}E_2+\chi_{1, 1}E_2+\frac{1-m}{2m}\chi_{1, 1}, \\ S(\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}}) & = & \zeta_{{\lambda}^\frac{-1}{{\omega}}, {\lambda}^\frac{-1}{m}}, \\ S(\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}}) & = & {\lambda}^\frac{(1-m)d/2}{{\omega}}\gamma^{-k}\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}\gamma^{-k}}\;\; \mathit{\text{when}}\;\;{\lambda}^\frac{1}{m} = {\lambda}^\frac{d}{{\omega}}\gamma^k. \end{array} |
Proof. We also prove the lemma through checking their values on the basis.
\begin{array}{l} \langle S(E_1), x^ig^jy^l\rangle & = & \delta_{l, 1}\langle E_1, -\gamma^{-j-1}x^{-i}g^{-j-1}y\rangle \\ & = & -\delta_{l, 1}\gamma^{-j-1} \; = \; \langle-\gamma^{-1}\zeta_{1, \gamma^{-1}}E_1, x^ig^jy^l\rangle, \\ \langle S(E_1), x^ig^ju_l\rangle & = & \delta_{l, 1}\langle E_1, -\xi^{-1}\gamma^{-j-1}x^{i+d+\frac{3}{2}(1-m)d+2dj}g^{m-2-j}u_1\rangle \\ & = & -\xi^{-1}\gamma^{-j-1}\frac{\xi}{1-\gamma^{-1}}\delta_{l, 1} \; = \; \langle-\gamma^{-1}\xi^{-1}\chi_{1, \gamma^{-1}}E_1, x^ig^ju_l\rangle. \end{array} |
It can be concluded that
S(E_1) = -\gamma^{-1}(\zeta_{1, \gamma^{-1}}+\xi^{-1}\chi_{1, \gamma^{-1}})E_1. |
Also,
\begin{array}{l} \langle S(E_2), x^ig^jy^l\rangle & = & \delta_{l, 0}\langle E_2, x^{-i}g^{-j}\rangle \\ & = & -\delta_{l, 0}(\frac{i}{{\omega}}+\frac{j}{m}) \; = \; \langle -\zeta_{1, 1}E_2, x^ig^jy^l\rangle, \\ \langle S(E_2), x^ig^ju_l\rangle & = & \delta_{l, 0}\langle E_2, x^{i+2dj+\frac{3}{2}(1-m)d}g^{m-1-j}u_0\rangle \\ & = & \delta_{l, 0}(\frac{i+2dj}{{\omega}}+\frac{3(1-m)d/2}{{\omega}}+\frac{m-1-j}{m}) \\ & = & \delta_{l, 0}(\frac{i}{{\omega}}+\frac{j}{m}+\frac{1-m}{2m}) \; = \; \langle\chi_{1, 1}E_2+\frac{1-m}{2m}\chi_{1, 1}, x^ig^ju_l\rangle. \end{array} |
It can be concluded that
S(E_2) = -\zeta_{1, 1}E_2+\chi_{1, 1}E_2+\frac{1-m}{2m}\chi_{1, 1}. |
Moreover,
\begin{array}{l} \langle S(\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}}), x^ig^jy^l\rangle & = & \delta_{l, 0}\langle\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}}, x^{-i}g^{-j}\rangle \; = \; \delta_{l, 0}{\lambda}^\frac{-i}{{\omega}}{\lambda}^\frac{-j}{m} \\ & = & \langle\zeta_{{\lambda}^\frac{-1}{{\omega}}, {\lambda}^\frac{-1}{m}}, x^ig^jy^l\rangle, \\ \langle S(\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}}), x^ig^ju_l\rangle & = & 0. \end{array} |
When {\lambda}^\frac{1}{m} = {\lambda}^\frac{d}{{\omega}}\gamma^k holds where k is a non-negative integer,
\begin{array}{l} \langle S(\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}}), x^ig^jy^l\rangle & = & 0, \\ \langle S(\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}}), x^ig^ju_l\rangle & = & \delta_{l, 0} \langle\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}}, x^{i+2dj+\frac{3}{2}(1-m)d}g^{m-1-j}u_0\rangle \\ & = & \delta_{l, 0}{\lambda}^\frac{i+2dj}{{\omega}}{\lambda}^\frac{3(1-m)d/2}{{\omega}}{\lambda}^\frac{m-1-j}{m} \\ & = & \delta_{l, 0}{\lambda}^\frac{i+2dj}{{\omega}}{\lambda}^\frac{3(1-m)d/2}{{\omega}}({\lambda}^\frac{d}{{\omega}}\gamma^k)^{m-1-j} \\ & = & {\lambda}^\frac{(1-m)d/2}{{\omega}}\gamma^{-k}\delta_{l, 0}{\lambda}^\frac{i}{{\omega}}({\lambda}^\frac{d}{{\omega}}\gamma^{-k})^j \\ & = & \langle{\lambda}^\frac{(1-m)d/2}{{\omega}}\gamma^{-k}\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}\gamma^{-k}}, x^ig^ju_l\rangle. \end{array} |
It can be concluded that
\begin{array}{l} S(\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}}) & = & \zeta_{{\lambda}^\frac{-1}{{\omega}}, {\lambda}^\frac{-1}{m}}, \\ S(\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}}) & = & {\lambda}^\frac{(1-m)d/2}{{\omega}}\gamma^{-k}\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{d}{{\omega}}\gamma^{-k}}\;\; \text{when}\;\;{\lambda}^\frac{1}{m} = {\lambda}^\frac{d}{{\omega}}\gamma^k. \end{array} |
Proposition 5.6. As an algebra, D(m, d, \xi)^\circ is generated by \zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}} , \chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}} , E_1 and E_2 for {\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}\in{𝕜}^\ast .
Proof. Clearly D(m, d, \xi) has a Laurent polynomial subalgebra P = {𝕜}[g^{\pm1}] . Choose
\mathcal{I}_0 = \{((g^m-1)^r), \;((g^m+1)^r), \;((g^m-{\lambda})^r(g^m-{\lambda}^{-1})^r)\lhd_l D(m, d, \xi) \mid {\lambda}\in{𝕜}^\ast\setminus\{\pm1\}, \;r\in{\mathbb{N}}\}. |
At first for any cofinite left ideal I of D(m, d, \xi) , according to Lemma 2.8(3), there must be some non-zero polynomial
p(g) = (g^m-1)^{r'}(g^m+1)^{r"} \prod\limits_{\alpha = 1}^N(g^m-{\lambda}_\alpha)^{r_\alpha}(g^m-{\lambda}_\alpha^{-1})^{r_\alpha} |
for some N\geq1 , r', r", r_\alpha\in{\mathbb{N}} and distinct {\lambda}_\alpha\in{𝕜}^\ast\setminus\{\pm1\} which are not inverses of each other, such that p(g) generates a cofinite left ideal contained in I . It can be known by Lemma 2.10 that
((g^m-1)^{r'})\cap((g^m+1)^{r"})\cap\left[\bigcap\limits_{\alpha = 1}^N ((g^m-{\lambda}_\alpha)^{r_\alpha}(g^m-{\lambda}_\alpha^{-1})^{r_\alpha})\right] = (p(g)) \subseteq I |
as left ideals of D(m, d, \xi) , and thus Lemma 2.6 or Corollary 2.9(3) can be applied to obtain that D(m, d, \xi)^\circ = \sum_{I\in\mathcal{I}_0}(D(m, d, \xi)/I)^\ast .
Now we try to prove that for each I\in\mathcal{I}_0 , the subspace (D(m, d, \xi)/I)^\ast can be spanned by some products of \zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}}, \chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}}, E_2, E_1\; ({\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}\in{𝕜}^\ast) . For simplicity, we always denote
\psi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}} : = \zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}}+\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}}: \left\{\begin{array}{ll} x^ig^jy^l\mapsto \delta_{l, 0}\lambda^\frac{i}{{\omega}}\lambda^\frac{j}{m} \\ x^ig^ju_l\mapsto \delta_{l, 0}\lambda^\frac{i}{{\omega}}\lambda^\frac{j}{m} \end{array}\right., \;\; |
where {\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}\in{𝕜}^\ast . Note that \psi_{{\lambda}_1^\frac{1}{{\omega}}, {\lambda}_1^\frac{1}{m}}\psi_{{\lambda}_2^\frac{1}{{\omega}}, {\lambda}_2^\frac{1}{m}} = \psi_{{\lambda}_1^\frac{1}{{\omega}}{\lambda}_2^\frac{1}{{\omega}}, {\lambda}_1^\frac{1}{m}{\lambda}_2^\frac{1}{m}} always holds. Moreover, let \eta be an primitive {\omega} th root of 1 .
We continue to show the spanning by a discussion on the form of I\in\mathcal{I}_0 :
Case 1. Suppose I = ((g^m-{\lambda})^r(g^m-{\lambda}^{-1})^r) for some r\in{\mathbb{N}} and {\lambda}\in{𝕜}^\ast\setminus\{\pm1\} . Let {\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m} be some fixed roots of {\lambda} respectively. In this case we aim to show that
\begin{equation} \{\psi_{{\lambda}^\frac{(-1)^e}{{\omega}}, {\lambda}^\frac{(-1)^e}{m}}(\zeta_{\eta, 1}+\chi_{\eta, 1})^i (\zeta_{1, \gamma}+\xi\chi_{1, \gamma})^kE_2^sE_1^l \mid e\in 2, \;i\in{\omega}, \;k\in 2m, \;s\in r, \;l\in m\} \end{equation} | (5.2) |
is a linear basis of 4{\omega} m^2r -dimensional space (D(m, d, \xi)/I)^\ast . We remark that (5.2) is linearly equivalent to
\begin{equation} \{\zeta_{{\lambda}^\frac{(-1)^e}{{\omega}}, {\lambda}^\frac{(-1)^e}{m}}\zeta_{\eta, 1}^i\zeta_{1, \gamma}^jE_2^sE_1^l, \; \chi_{{\lambda}^\frac{(-1)^e}{{\omega}}, {\lambda}^\frac{(-1)^e}{m}}\chi_{\eta, 1}^i\chi_{1, \gamma}^jE_2^sE_1^l \mid e\in 2, \;i\in{\omega}, \;j\in m, \;s\in r, \;l\in m\}. \end{equation} | (5.3) |
Evidently
I = {𝕜}\{x^ig^j(g^m-{\lambda})^r(g^m-{\lambda}^{-1})^ry^l, \;x^ig^j(g^m-{\lambda})^r(g^m-{\lambda}^{-1})^ru_l \mid i\in{\omega}, \;j\in{\mathbb{Z}}, \;l\in m\}, |
since yg^m = g^my and u_ig^m = g^{-m}u_i for each i\in m . Also, D(m, d, \xi)/I has a linear basis
\{x^ig^j(g^m-{\lambda})^s(g^m-{\lambda}^{-1})^sy^l, \;x^ig^j(g^m-{\lambda})^s(g^m-{\lambda}^{-1})^su_l \mid i\in{\omega}, \;j\in 2m, \;s\in r, \;l\in m\}, |
and hence {\operatorname{dim}}((D(m, d, \xi)/I)^\ast) = {\operatorname{dim}}(D(m, d, \xi)/I) = 4{\omega} m^2r . Next we show that all elements in (5.3) vanish on I . We make computations for any s\in r , l, l'\in m and j'\in{\mathbb{Z}} , especially for arbitrary roots {\lambda}^\frac{1}{{\omega}} and {\lambda}^\frac{1}{m} of {\lambda} : Clearly
\begin{eqnarray} \langle\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}}E_2^sE_1^l, x^{i'}g^{j'}(g^m-{\lambda})^ru_{l'}\rangle& = &0, \end{eqnarray} | (5.4) |
\begin{eqnarray} \langle\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}}E_2^sE_1^l, x^{i'}g^{j'}(g^m-{\lambda})^ry^{l'}\rangle& = &0, \end{eqnarray} | (5.5) |
and
\begin{eqnarray} \langle\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}}E_2^sE_1^l, x^{i'}g^{j'}(g^m-{\lambda})^ry^{l'}\rangle& = &0 \end{eqnarray} | (5.6) |
due to the same reason as Eq (4.4). Similarly,
\begin{eqnarray} & & \langle\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}}E_2^sE_1^l, x^{i'}g^{j'}(g^m-{\lambda})^ru_{l'}\rangle \\ & = & \langle\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}}E_2^sE_1^l, \sum\limits_{t = 0}^r\binom{r}{t}x^{i'}g^{j'+mt}(-{\lambda})^{r-t}u_{l'}\rangle \\ & = & \sum\limits_{t = 0}^r\binom{r}{t}(-{\lambda})^{r-t} \langle\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}}E_2^s\otimes E_1^l, \Delta(x^{i'}g^{j'+nt}u_{l'})\rangle \\ & = & \delta_{l', l}\sum\limits_{t = 0}^r\binom{r}{t}(-{\lambda})^{r-t} \langle\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}}E_2^s, x^{i'}g^{j'+mt}u_0\rangle \langle E_1^l, x^{i'}g^{j'+nt}u_l\rangle \\ & = & \delta_{l', l}\sum\limits_{t = 0}^r\binom{r}{t}(-{\lambda})^{r-t} {\lambda}^\frac{i'}{{\omega}}{\lambda}^\frac{j'+mt}{m}(\frac{i'}{{\omega}}+\frac{j'+mt}{m})^s \frac{\xi^{l^2}}{(1-\gamma^{-1})^l} \\ & = & \delta_{l', l}\frac{\xi^{l^2}}{(1-\gamma^{-1})^l}{\lambda}^r{\lambda}^\frac{i'}{{\omega}}{\lambda}^\frac{j'}{m} \sum\limits_{t = 0}^r\binom{r}{t}(-1)^{r-t} \sum\limits_{u = 0}^s\binom{s}{u}(\frac{i'}{{\omega}}+\frac{j'}{m})^ut^{s-u} \\ & = & \delta_{l', l}\frac{\xi^{l^2}}{(1-\gamma^{-1})^l}{\lambda}^r{\lambda}^\frac{i'}{{\omega}}{\lambda}^\frac{j'}{m} \sum\limits_{u = 0}^s\binom{s}{u}(\frac{i'}{{\omega}}+\frac{j'}{m})^u \sum\limits_{t = 0}^r\binom{r}{t}(-1)^{r-t}t^{s-u} \\ & = & 0, \end{eqnarray} | (5.7) |
since \sum_{t = 0}^r\binom{r}{t}(-1)^{r-t}t^{s-u} = 0 when s-u < r is a part of the second Stirling number (2.10).
Of course, these equations still hold if we substitute {\lambda}^\frac{1}{{\omega}} and {\lambda}^\frac{1}{m} by {\lambda}^\frac{-1}{{\omega}} and {\lambda}^\frac{-1}{m} respectively. As a conclusion, all elements in (5.3) as well as (5.2) belong to I^\perp = (D(m, d, \xi)/I)^\ast .
Finally we prove that (5.2) are linearly independent. Choose a lexicographically ordered set of elements \{h_{e', i', k', s', l'}\in D(m, d, \xi)\mid (e', i', k', s', l')\in 2\times{\omega}\times 2m\times r\times m\} , where
h_{e', i', k', s', l'}: = \left\{\begin{array}{ll} \frac{1}{l'!_\gamma}x^{i'}g^{\frac{k'}{2}+2s'm+e'm}y^{l'}, & \text{when}\;\; 2\mid k'; \\ \frac{(1-\gamma^{-1})^{l'}}{\xi^{{l'}^2}}x^{i'}g^{\frac{k'-1}{2}+2s'm+e'm}u_{l'}, & \text{when}\;\; 2\nmid k'. \end{array}\right. |
Our goal is to show that the 4{\omega} m^2r\times 4{\omega} m^2r square matrix
\begin{array}{l} A&: = & (\langle\psi_{{\lambda}^\frac{(-1)^e}{{\omega}}, {\lambda}^\frac{(-1)^e}{m}}(\zeta_{\eta, 1}+\chi_{\eta, 1})^i (\zeta_{1, \gamma}+\xi\chi_{1, \gamma})^kE_2^sE_1^l, h_{e', i', k', s', l'}\rangle \\ & & \;\;\mid (e, i, k, s, l), (e', i', k', s', l')\in 2\times{\omega}\times 2m\times r\times m ) \end{array} |
is invertible for fixed {\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{n} , which would imply the linear independence of (5.2) as well as (5.3). We make following computations: When 2\mid k' ,
\begin{array}{l} & & \langle\psi_{{\lambda}^\frac{(-1)^e}{{\omega}}, {\lambda}^\frac{(-1)^e}{m}}(\zeta_{\eta, 1}+\chi_{\eta, 1})^i (\zeta_{1, \gamma}+\xi\chi_{1, \gamma})^kE_2^sE_1^l, h_{e', i', k', s', l'}\rangle \\ & = & \langle\zeta_{{\lambda}^\frac{(-1)^e}{{\omega}}, {\lambda}^\frac{(-1)^e}{m}}\zeta_{\eta, 1}^i\zeta_{1, \gamma}^kE_2^sE_1^l, \frac{1}{l'!_\gamma}x^{i'}g^{\frac{k'}{2}+2s'm+e'm}y^{l'}\rangle \\ & = & \delta_{l', l}\langle\zeta_{{\lambda}^\frac{(-1)^e}{{\omega}}\eta^i, {\lambda}^\frac{(-1)^e}{m}\gamma^k}E_2^s, x^{i'}g^{\frac{k'}{2}+2s'm+e'm}\rangle \frac{1}{l!_\gamma}\langle E_1^l, x^{i'}g^{\frac{k'}{2}+2s'm+e'm}y^l\rangle \\ & = & \delta_{l', l}{\lambda}^{(-1)^e(\frac{i'}{{\omega}}+\frac{k'}{2m}+2s'+e')}\eta^{ii'}\xi^{kk'} (\frac{i'}{{\omega}}+\frac{k'}{2m}+2s'+e')^s. \end{array} |
When 2\nmid k' ,
\begin{array}{l} & & \langle\psi_{{\lambda}^\frac{(-1)^e}{{\omega}}, {\lambda}^\frac{(-1)^e}{m}}(\zeta_{\eta, 1}+\chi_{\eta, 1})^i (\zeta_{1, \gamma}+\xi\chi_{1, \gamma})^kE_2^sE_1^l, h_{e', i', k', s', l'}\rangle \\ & = & \langle\chi_{{\lambda}^\frac{(-1)^e}{{\omega}}, {\lambda}^\frac{(-1)^e}{m}}\chi_{\eta, 1}^i(\xi\chi_{1, \gamma})^k E_2^sE_1^l, \frac{(1-\gamma^{-1})^{l'}}{\xi^{{l'}^2}}x^{i'}g^{\frac{k'-1}{2}+2s'm+e'm}u_{l'}\rangle \\ & = & \delta_{l', l}\langle\chi_{{\lambda}^\frac{(-1)^e}{{\omega}}\eta^i, {\lambda}^\frac{(-1)^e}{m}\gamma^k}\xi^kE_2^s, x^{i'}g^{\frac{k'-1}{2}+2s'm+e'm}u_0\rangle \frac{(1-\gamma^{-1})^l}{\xi^{l^2}}\langle E_1^l, x^{i'}g^{\frac{k'-1}{2}+2s'm+e'm}u_l\rangle \\ & = & \delta_{l', l}{\lambda}^{(-1)^e(\frac{i'}{{\omega}}+\frac{k'-1}{2m}+2s'+e')}\eta^{ii'}\xi^{kk'} (\frac{i'}{{\omega}}+\frac{k'-1}{2m}+2s'+e')^s. \end{array} |
One concludes that
\begin{array}{l} A&: = & (\delta_{l', l}{\lambda}^{(-1)^e(\frac{i'}{{\omega}}+\lfloor\frac{k'}{2}\rfloor\frac{1}{m}+2s'+e')} \eta^{ii'}\xi^{kk'}(\frac{i'}{{\omega}}+\lfloor\frac{k'}{2}\rfloor\frac{1}{m}+2s'+e')^s \\ & & \;\;\mid (e, i, k, s, l), (e', i', k', s', l')\in 2\times{\omega}\times 2m\times r\times m). \end{array} |
However, there are following two facts:
● The matrix (\delta_{l', l}\eta^{ii'}\xi^{kk'}\mid (i, k, l), (i', k', l')\in {\omega}\times 2m\times m) is invertible by Lemma 2.11, since it is the Kronecker product of three invertible matrices;
● For each (i', k')\in {\omega}\times 2m , the matrix
\left({\lambda}^{(-1)^e(\frac{i'}{{\omega}}+\lfloor\frac{k'}{2}\rfloor\frac{1}{m}+2s'+e')} (\frac{i'}{{\omega}}+\lfloor\frac{k'}{2}\rfloor\frac{1}{m}+2s'+e')^s \mid (e, s), (e', s')\in 2\times r\right) |
is invertible by Corollary 2.14(2).
Thus A is invertible according to Lemma 2.12, and (5.2) or (5.3) is a basis of (D(m, d, \xi)/I)^\ast .
Case 2. Suppose I = ((g^m-1)^r) for some r\in{\mathbb{N}} . In this case we aim to show that
\begin{equation} \{(\zeta_{\eta, 1}+\chi_{\eta, 1})^i(\zeta_{1, \gamma}+\xi\chi_{1, \gamma})^kE_2^sE_1^l \mid i\in{\omega}, \;k\in 2m, \;s\in r, \;l\in m\} \end{equation} | (5.8) |
is a linear basis of 2{\omega} m^2r -dimensional space (D(m, d, \xi)/I)^\ast . We remark that (5.8) is linearly equivalent to
\begin{equation} \{\zeta_{\eta, 1}^i\zeta_{1, \gamma}^jE_2^sE_1^l, \;\chi_{\eta, 1}^i\chi_{1, \gamma}^jE_2^sE_1^l \mid i\in{\omega}, \;j\in m, \;s\in r, \;l\in m\}. \end{equation} | (5.9) |
Evidently
I = {𝕜}\{x^ig^j(g^m-1)^ry^l, \;x^ig^j(g^m-1)^ru_l \mid i\in{\omega}, \;j\in{\mathbb{Z}}, \;l\in m\}, |
since yg^m = g^my and u_ig^m = g^{-m}u_i for each i\in m . Also, D(m, d, \xi)/I has a linear basis
\{x^ig^j(g^m-1)^sy^l, \;x^ig^j(g^m-1)^su_l \mid i\in{\omega}, \;j\in m, \;s\in r, \;l\in m\}, |
and hence {\operatorname{dim}}((D(m, d, \xi)/I)^\ast) = {\operatorname{dim}}(D(m, d, \xi)/I) = 2{\omega} m^2r . Similar computations to Eqs (5.4) to (5.7) follows that all elements in (5.9) vanish on I , and hence all elements in (5.9) as well as (5.8) belong to I^\perp = (D(m, d, \xi)/I)^\ast .
Finally we prove that (5.8) are linearly independent. Our goal is to show that the 2{\omega} m^2r\times 2{\omega} m^2r square matrix
\begin{array}{l} A&: = & (\langle(\zeta_{\eta, 1}+\chi_{\eta, 1})^i(\zeta_{1, \gamma}+\xi\chi_{1, \gamma})^kE_2^sE_1^l, h_{0, i', k', s', l'}\rangle \\ & & \;\;\mid (i, k, s, l), (i', k', s', l')\in {\omega}\times 2m\times r\times m) \\ & = & \left(\delta_{l', l}\eta^{ii'}\xi^{kk'}(\frac{i'}{{\omega}}+\lfloor\frac{k'}{2}\rfloor\frac{1}{m}+2s')^s \mid (i, k, s, l), (i', k', s', l')\in {\omega}\times 2m\times r\times m\right) \end{array} |
is invertible. This is true due to the same argument in the proof of Proposition 4.3 that A is invertible there.
Case 3. Suppose I = ((g^m+1)^r) for some r\in{\mathbb{N}} . Let (-1)^\frac{1}{{\omega}}, (-1)^\frac{1}{m} be some fixed roots of -1 respectively. In this case we aim to show that
\begin{equation} \{\psi_{(-1)^\frac{1}{{\omega}}, (-1)^\frac{1}{m}}(\zeta_{\eta, 1}+\chi_{\eta, 1})^i (\zeta_{1, \gamma}+\xi\chi_{1, \gamma})^kE_2^sE_1^l \mid i\in{\omega}, \;k\in 2m, \;s\in r, \;l\in m\} \end{equation} | (5.10) |
is a linear basis of 2{\omega} m^2r -dimensional space (D(m, d, \xi)/I)^\ast . We remark that (5.10) is linearly equivalent to
\begin{equation} \{\zeta_{(-1)^\frac{1}{{\omega}}, (-1)^\frac{1}{m}}\zeta_{\eta, 1}^i\zeta_{1, \gamma}^jE_2^sE_1^l, \; \chi_{(-1)^\frac{1}{{\omega}}, (-1)^\frac{1}{m}}\chi_{\eta, 1}^i\chi_{1, \gamma}^jE_2^sE_1^l \mid i\in{\omega}, \;j\in m, \;s\in r, \;l\in m\}. \end{equation} | (5.11) |
Evidently
I = {𝕜}\{x^ig^j(g^m+1)^ry^l, \;x^ig^j(g^m+1)^ru_l \mid i\in{\omega}, \;j\in{\mathbb{Z}}, \;l\in m\}, |
since yg^m = g^my and u_ig^m = g^{-m}u_i for each i\in m . Also, D(m, d, \xi)/I has a linear basis
\{x^ig^j(g^m+1)^sy^l, \;x^ig^j(g^m+1)^su_l \mid i\in{\omega}, \;j\in m, \;s\in r, \;l\in m\}, |
and hence {\operatorname{dim}}((D(m, d, \xi)/I)^\ast) = {\operatorname{dim}}(D(m, d, \xi)/I) = 2{\omega} m^2r . Similar computations to Eqs (5.4) to (5.7) follows that all elements in (5.11) vanish on I , and hence all elements in (5.11) as well as (5.10) belong to I^\perp = (D(m, d, \xi)/I)^\ast .
Finally we prove that (5.10) are linearly independent. Our goal is to show that the 2{\omega} m^2r\times 2{\omega} m^2r square matrix
\begin{array}{l} A&: = & (\langle\psi_{(-1)^\frac{1}{{\omega}}, (-1)^\frac{1}{m}} (\zeta_{\eta, 1}+\chi_{\eta, 1})^i(\zeta_{1, \gamma}+\xi\chi_{1, \gamma})^kE_2^sE_1^l, h_{0, i', k', s', l'}\rangle \\ & & \;\;\mid (i, k, s, l), (i', k', s', l')\in {\omega}\times 2m\times r\times m) \\ & = & ((-1)^{\frac{i'}{{\omega}}+\lfloor\frac{k'}{2}\rfloor\frac{1}{m}+2s'} \delta_{l', l}\eta^{ii'}\xi^{kk'}(\frac{i'}{{\omega}}+\lfloor\frac{k'}{2}\rfloor\frac{1}{m}+2s')^s \\ & & \;\; \mid (i, k, s, l), (i', k', s', l')\in {\omega}\times 2m\times r\times m) \end{array} |
is invertible. This is true due to the same argument in the proof of Proposition 4.3 that A is invertible there.
We summarize three cases above in a result that
\begin{array}{l} D(m, d, \xi)^\circ & = & \sum\limits_{I\in\mathcal{I}_0}(D(m, d, \xi)/I)^\ast \\ & = & {𝕜}\{\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}}E_2^sE_1^l, \; \chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}}E_2^sE_1^l \mid {\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}\in{𝕜}^\ast, \;s\in{\mathbb{N}}, \;l\in m\}. \end{array} |
Theorem 5.7. (1) D_\circ(m, d, \xi) constructed in Subsection 5.1 is a Hopf algebra;
(2) As a Hopf algebra, D(m, d, \xi)^\circ is isomorphic to D_\circ(m, d, \xi) .
Proof. Consider the following map
\begin{array}{l} \Theta &:& D_\circ(m, d, \xi)\rightarrow D(m, d, \xi)^\circ, \\ & & {\mathrm{Z}}_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}}\mapsto\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}}, \; {\mathrm{X}}_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}}\mapsto\chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}}, \; F_1\mapsto E_1, \;F_2\mapsto E_2, \end{array} |
where {\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}\in{𝕜}^\ast . This is a epimorphism of algebras by Lemma 5.2 and Proposition 5.6. Furthermore, \Theta would become an isomorphism of Hopf algebras with desired coalgebra structure and antipode, as long as it is injective (since D(m, d, \xi)^\circ is in fact a Hopf algebra).
In order to show that \Theta is injective, we aim to show the linear independence of
\{\zeta_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}}\zeta_{\eta, 1}^i\zeta_{1, \gamma}^jE_2^sE_1^l, \; \chi_{{\lambda}^\frac{1}{{\omega}}, {\lambda}^\frac{1}{m}}\chi_{\eta, 1}^i\chi_{1, \gamma}^jE_2^sE_1^l \mid {\lambda}\in{𝕜}^\ast, \;i\in{\omega}, \;j\in m, \;s\in{\mathbb{N}}, \;l\in m\} |
in D(m, d, \xi)^\circ , where {\lambda}^\frac{1}{{\omega}} and {\lambda}^\frac{1}{m} are fixed roots of each {\lambda}\in{𝕜}^\ast . By linear independence of elements in (5.3), (5.9) and (5.11), we only need to show that
This is due to the fact that any finite sum of form
\begin{array}{l} & & (D(m, d, \xi)/((g^m-1)^r))^\ast+(D(m, d, \xi)/((g^m+1)^r))^\ast \\ & & +\sum\limits_{\alpha = 1}^N(D(m, d, \xi)/((g^m-{\lambda}_\alpha)^r(g^m-{\lambda}_\alpha^{-1})^r)^\ast \\ & = & ((g^m-1)^r)^\perp+((g^m+1)^r)^\perp +\sum\limits_{\alpha = 1}^N((g^m-{\lambda}_\alpha)^r(g^m-{\lambda}_\alpha^{-1})^r)^\perp \end{array} |
is direct, as long as {\lambda}_\alpha 's are distinct and not inverses of each other. The reason is the same as Eq (4.6).
Remark 5.8. The infinite dihedral group \mathbb{D}_\infty is generated by g and x with relations
x^2 = 1, \;\;xgx = g^{-1}, |
and the finite dual of {𝕜}\mathbb{D}_\infty is established by [9]. Note that {𝕜}\mathbb{D}_\infty = D(1, 1, -1) and thus we cover the presentation of ({𝕜}\mathbb{D}_\infty)^\circ . For our convenience, we use our notion to write ({𝕜}\mathbb{D}_\infty)^\circ out: It is generated by
\zeta_{\lambda}:g^jx^k\mapsto \delta_{k, 0}{\lambda}^j, \;\;\chi_{\lambda}:g^jx^k\mapsto \delta_{k, 1}{\lambda}^j, \;\; E_2:g^jx^k\mapsto j\;\;\;(j\in{\mathbb{Z}}, \;k\in2) |
for {\lambda}\in{𝕜}^\ast , with relations
\begin{array}{l} & \zeta_{{\lambda}_1}\zeta_{{\lambda}_2} = \zeta_{{\lambda}_1{\lambda}_2}, \;\;\chi_{{\lambda}_1}\chi_{{\lambda}_2} = \chi_{{\lambda}_1{\lambda}_2}, \;\; \zeta_{{\lambda}_1}\chi_{{\lambda}_2} = \chi_{{\lambda}_1}\zeta_{{\lambda}_2} = 0, \;\;\chi_1+\zeta_1 = 1, \\ & E_2\zeta_{\lambda} = \zeta_{\lambda} E_2, \;\;E_2\chi_{\lambda} = \chi_{\lambda} E_2, \end{array} |
and
\begin{array}{l} & \Delta(\zeta_{\lambda}) = \zeta_{\lambda}\otimes\zeta_{\lambda}+\chi_{\lambda}\otimes\chi_{{\lambda}^{-1}}, \;\; \Delta(\chi_{\lambda}) = \zeta_{\lambda}\otimes\chi_{\lambda}+\chi_{\lambda}\otimes\zeta_{{\lambda}^{-1}}, \\ & \Delta(E_2) = (\zeta_1-\chi_1)\otimes E_2+E_2\otimes 1, \\ & \varepsilon(\zeta_{\lambda}) = 1, \;\;\varepsilon(\chi_{\lambda}) = \varepsilon(E_2) = 0, \;\; \\ & S(\zeta_{\lambda}) = \zeta_{{\lambda}^{-1}}, \;\;S(\chi_{\lambda}) = \chi_{{\lambda}^{-1}}, \;\;S(E_2) = -(\zeta_1-\chi_1)E_2. \end{array} |
This presentation is equivalent to that in [9,Section 3.1] and the equivalence is given through:
\zeta_{\lambda}\mapsto \frac{1}{2}(\phi_{\lambda}+\psi_{\lambda}), \;\; \chi_{\lambda}\mapsto \frac{1}{2}(\phi_{\lambda}-\psi_{\lambda})\;\;({\lambda}\in{𝕜}^\ast), \;\; E_2\mapsto F, |
where we used notions of [9] freely.
With the help of Sections 3–5, we show that there always is a non-degenerate Hopf pairing on each affine prime regular Hopf algebra of GK-dimension one. Using such Hopf pairing, we attempt to consider a infinite-dimensional version of some conclusions which are valid in the finite-dimensional situation.
As mentioned in the introduction, the notion of pairing of a bialgebra or a Hopf algebra was given by Majid [19]:
Definition 6.1. Let H and H^\bullet be Hopf algebras. A linear map \langle-, -\rangle:H^\bullet\otimes H\rightarrow {𝕜} is called a Hopf pairing (on H ), if
\begin{array}{ll} \mathrm{(i)}\;\;\;\;\langle ff', h\rangle = \sum\langle f, h_{(1)}\rangle\langle f', h_{(2)}\rangle, & \mathrm{(ii)}\;\;\;\;\langle f, hh'\rangle = \sum\langle f_{(1)}, h\rangle\langle f_{(2)}, h'\rangle, \\ \mathrm{(iii)}\;\;\langle 1, h\rangle = \varepsilon(h), & \mathrm{(iv)}\;\;\;\langle f, 1\rangle = \varepsilon(f), \\ \mathrm{(v)}\;\;\;\langle f, S(h)\rangle = \langle S(f), h\rangle \end{array} |
hold for all f, f'\in H^\bullet and h, h'\in H . Moreover, it is said to be non-degenerate, if for any f\in H^\bullet and any h\in H ,.
\langle f, H\rangle = 0~~ implies ~~ f = 0 , and ~~ \langle H^\bullet, h\rangle = 0 ~~ implies ~~ h = 0 |
Clearly, the definition follows that there are linear maps
\alpha:H^\bullet\rightarrow H^\ast, \;f\mapsto \langle f, -\rangle\;\;\;\text{and}\;\;\;\beta:H\rightarrow H^\bullet{}^\ast, \;h\mapsto \langle-, h\rangle. |
Furthermore, we know by (ii) in Definition 6.1 that for any f\in H^\bullet ,
M^\ast(\alpha(f)) = \sum\alpha(f_{(1)})\otimes\alpha(f_{(2)})\in H^\ast\otimes H^\ast, |
where M denotes the multiplication on H , and hence the image of \alpha is in fact contained in H^\circ by [21,Lemma 9.1.1].
As a conclusion, Definition 6.1 follows that there are two maps of Hopf algebras
\alpha:H^\bullet\rightarrow H^\circ, \;f\mapsto \langle f, -\rangle\;\;\;\text{and}\;\;\;\beta:H\rightarrow H^\bullet{}^\circ, \;h\mapsto \langle-, h\rangle, |
which are both injective if and only if the Hopf pairing \langle-, -\rangle is non-degenerate. Thus we might be able to construct non-degenerate Hopf pairing on H , as long as the structure of finite dual H^\circ is determined.
As examples, non-degenerate Hopf pairing on {𝕜}\mathbb{D}_\infty , T_\infty(n, v, \xi) , B(n, {\omega}, \gamma) and D(m, d, \xi) are constructed as follows, respectively.
Proposition 6.2. For each affine prime regular Hopf algebra H of GK-dimension one, we can construct a Hopf algebra H^\bullet and a non-degenerate Hopf pairing \langle-, -\rangle:H^\bullet\otimes H\rightarrow {𝕜} as follows: specifically, keeping the notions used in Sections 3–5, we have:
(1) The evaluation \langle-, -\rangle:({𝕜}\mathbb{D}_\infty)^\bullet\otimes{𝕜}\mathbb{D}_\infty\rightarrow {𝕜} is a non-degenerate Hopf pairing, where
\begin{array}{l} ({𝕜}\mathbb{D}_\infty)^\bullet & = & {𝕜}\{\zeta_1E_2^s, \;\chi_1E_2^s\mid s\in{\mathbb{N}}\} \\ & = & {𝕜}\{(\zeta_1-\chi_1)^kE_2^s\mid k\in2, \;s\in{\mathbb{N}}\} \;\subseteq\;({𝕜}\mathbb{D}_\infty)^\circ. \end{array} |
(2) The evaluation \langle-, -\rangle:T_\infty(n, v, \xi)^\bullet\otimes T_\infty(n, v, \xi)\rightarrow {𝕜} is a non-degenerate Hopf pairing, where
\begin{array}{l} T_\infty(n, v, \xi)^\bullet & = & {𝕜}\{\omega^jE_2^sE_1^l \mid j\in n, \;s\in{\mathbb{N}}, \;l\in m\} \\ & = & {𝕜}\{\omega^jE_2^{[s]}E_1^{[l]} \mid j\in n, \;s\in{\mathbb{N}}, \;l\in m\} \;\subseteq\; T_\infty(n, v, \xi)^\circ, \end{array} |
and m = \frac{n}{\gcd(n, v)} , E_2^{[s]}E_1^{[l]} = \frac{1}{s!\cdot l!_{\xi^v}}E_2^sE_1^l .
(3) The evaluation \langle-, -\rangle:B(n, \omega, \gamma)^\bullet\otimes B(n, \omega, \gamma)\rightarrow {𝕜} is a non-degenerate Hopf pairing, where
\begin{array}{l} B(n, \omega, \gamma)^\bullet & = & {𝕜}\{\psi_{1, \gamma}^jE_2^sE_1^l\mid j\in n, \;s\in{\mathbb{N}}, \;l\in n\} \\ & = & {𝕜}\{\psi_{1, \gamma}^jE_2^sE_1^{[l]}\mid j\in n, \;s\in{\mathbb{N}}, \;l\in n\} \;\subseteq\; B(n, \omega, \gamma)^\circ, \end{array} |
and E_1^{[l]} = \frac{1}{l!_\gamma}E_1^l .
(4) The evaluation \langle-, -\rangle:D(m, d, \xi)^\bullet\otimes D(m, d, \xi)\rightarrow {𝕜} is a non-degenerate Hopf pairing, where
\begin{array}{l} D(m, d, \xi)^\bullet & = & {𝕜}\{\zeta_{1, \gamma}^jE_2^sE_1^l, \;\chi_{1, \gamma}^jE_2^sE_1^l \mid i\in{\omega}, \;j\in m, \;s\in{\mathbb{N}}, \;l\in m\} \\ & = & {𝕜}\{(\zeta_{1, \gamma}+\xi\chi_{1, \gamma})^kE_2^sE_1^l \mid k\in 2m, \;s\in{\mathbb{N}}, \;l\in m\} \subseteq D(m, d, \xi)^\circ. \end{array} |
Proof. It is not hard to see that all H^{\bullet} given above are Hopf subalgebras of the corresponding H^{\circ} and the evaluation gives a Hopf pairing. Thus the remaining task is to show that the evaluation is non-degenerate. We only prove the non-degeneracy in (4) since the others can be proved similarly.
Clearly, the Hopf pairing \langle-, -\rangle:D(m, d, \xi)^\bullet\otimes D(m, d, \xi)\rightarrow {𝕜} gives two Hopf algebra maps
\alpha:D(m, d, \xi)^\bullet\rightarrow D(m, d, \xi)^\circ\;\;\text{and}\;\; \beta:D(m, d, \xi)\rightarrow D(m, d, \xi)^\bullet{}^\circ, |
and the map \alpha is just an inclusion. Therefore, to show that \langle-, -\rangle is non-degenerate, we only need to prove the injectivity of \beta . Recall that in Section 5 that D(m, d, \xi) = {𝕜}\{x^ig^jy^l, \; x^ig^ju_l\mid i\in{\omega}, \; j\in{\mathbb{Z}}, \; l\in m\} . Also,
\begin{array}{l} \langle(\zeta_{1, \gamma}+\xi\chi_{1, \gamma})^{k'}E_2^{s'}E_1^{l'}, x^ig^jy^l\rangle & = & l!_\gamma \cdot \delta_{l, l'} \xi^{2jk'}(\frac{i}{{\omega}}+\frac{j}{m})^{s'}, \\ \langle(\zeta_{1, \gamma}+\xi\chi_{1, \gamma})^{k'}E_2^{s'}E_1^{l'}, x^ig^ju_l\rangle & = & \frac{\xi^{l^2}}{(1-\gamma^{-1})^l} \cdot \delta_{l, l'} \xi^{(2j+1)k'}(\frac{i}{{\omega}}+\frac{j}{m})^{s'} \end{array} |
hold for any k'\in 2m, \; s'\in{\mathbb{N}}, \; l'\in m and i\in{\omega}, \; j\in{\mathbb{Z}}, \; l\in m . This means that we only need to show the linear independence of
\{\beta(x^ig^jy^l), \;\beta(x^ig^ju_l)\mid i\in{\omega}, \;j\in{\mathbb{Z}}, \;l\in m\} \subseteq D(m, d, \xi)^\bullet{}^\circ. |
Choose a lexicographically ordered set of elements for any positive integer N : \{h_{i, k, s, l}\in D(m, d, \xi)\mid (i, k, s, l)\in {\omega}\times 2m\times \{-N, \cdots, N\}\times m\} , where
h_{i, k, s, l}: = \left\{\begin{array}{ll} \frac{1}{l!_\gamma}x^ig^{\frac{k}{2}+sm}y^l, & \text{when}\;\; 2\mid k; \\ \frac{(1-\gamma^{-1})^l}{\xi^{l^2}}x^ig^{\frac{k-1}{2}+sm}u_l, & \text{when}\;\; 2\nmid k. \end{array}\right. |
The desired linear independence would be implied by the invertibility of the following 2{\omega} m^2(2N+1)\times2{\omega} m^2(2N+1) matrix
\begin{array}{l} A: & = & (\langle\beta(h_{i, k, s, l}), (\zeta_{1, \gamma}+\xi\chi_{1, \gamma})^{k'}E_2^{i'+s'{\omega}+N{\omega}}E_1^{l'}\rangle \\ & & \;\;\mid (i, k, s, l), (i', k', s', l')\in {\omega}\times 2m\times \{-N, \cdots, N\}\times m) \end{array} |
for any positive integer N . However,
\begin{array}{l} A & = & (\langle(\zeta_{1, \gamma}+\xi\chi_{1, \gamma})^{k'}E_2^{i'+s'{\omega}+N{\omega}}E_1^{l'}, h_{i, k, s, l}\rangle \mid (i, k, s, l), (i', k', s', l')\in {\omega}\times 2m\times \{-N, \cdots, N\}\times m) \\ & = & (\delta_{l, l'}\xi^{kk'}(\frac{i}{{\omega}}+\lfloor\frac{k}{2}\rfloor\frac{1}{m}+s)^{i'+s'{\omega}+N{\omega}} \mid (i, k, s, l), (i', k', s', l')\in {\omega}\times 2m\times \{-N, \cdots, N\}\times m) \end{array} |
is similar to the matrix
\begin{array}{l} && (\delta_{l, l'}\xi^{kk'}(\lfloor\frac{k}{2}\rfloor\frac{1}{m}+\frac{s{\omega}+i}{{\omega}})^{s'{\omega}+i'+N{\omega}} \mid (k, l, s, i), (k', l', s', i', )\in 2m\times m\times\{-N, \cdots, N\}\times{\omega}) \\ & = & (\delta_{l, l'}\xi^{kk'}(\lfloor\frac{k}{2}\rfloor\frac{1}{m}+\frac{t}{{\omega}})^{t'+N{\omega}} \mid (k, l, t), (k', l', t')\in 2m\times m\times\{-N{\omega}, \cdots, (N+1){\omega}-1\}), \end{array} |
which is invertible according to Lemma 2.12, because of the invertibility of
(\delta_{l, l'}\xi^{kk'}\mid (k, l), (k', l')\in 2m\times m) |
and
((\lfloor\frac{k}{2}\rfloor\frac{1}{m}+\frac{t}{{\omega}})^{t'+N{\omega}} \mid t, t'\in \{-N{\omega}, \cdots, (N+1){\omega}-1\}) |
for each k\in 2m .
For a general infinite-dimensional Hopf algebra H , there may be no H^{\bullet} such that there is a non-degenerate Hopf pairing between H and H^{\bullet} (see [21,Example 9.1.5(1)] for example). In fact, the existence of non-degenerate Hopf pairings over a Hopf algebra H is equivalent to require that H is residually finite-dimensional. Recall that an algebra H is said to be residually finite-dimensional, if there is a family \{\pi_i\} of finite-dimensional {𝕜} -representations of H such that \bigcap_i {\operatorname{Ker}}(\pi_i) = 0 . See [21,Definition 9.2.8] for example.
Lemma 6.3. Let H be a Hopf algebra over an arbitrary field {𝕜} . Then there is a non-degenerate Hopf pairing \langle-, -\rangle:H^\bullet\otimes H\rightarrow {𝕜} , if and only if H is residually finite-dimensional.
Proof. This is followed by [21,Proposition 9.2.10].
However, our constructions for H^\bullet in Proposition 6.2 satisfy further conditions, which might be crucial in a sense. Before that, we recall in [24,Definition 15.4.4] that a non-cosemisimple pointed Hopf algebra H is called minimal-pointed, if each of its proper Hopf subalgebras must be cosemisimple.
Proposition 6.4. Suppose that H is any of {𝕜}\mathbb{D}_\infty , T_\infty(n, v, \xi) , B({\omega}, n, \gamma) and D(m, d, \xi) , and H^\bullet denotes the Hopf algebra constructed in Proposition 6.2. Then H^\bullet is a minimal (under inclusion) Hopf subalgebra of H^\circ such that the evaluation \langle-, -\rangle:H^\bullet\otimes H\rightarrow {𝕜} is a non-degenerate Hopf pairing.
Proof. Since H is infinite-dimensional, it is sufficient to show that H^\bullet is the only infinite-dimensional Hopf subalgebra of itself.
(1) For the case when H = {𝕜}\mathbb{D}_\infty , we find that ({𝕜}\mathbb{D}_\infty)^\bullet is a minimal-pointed Hopf algebra (see [24,Proposition 15.4.6] and [23,Proposition 2]) with the finite-dimensional coradical. Thus ({𝕜}\mathbb{D}_\infty)^\bullet is the only infinite-dimensional Hopf subalgebra as desired.
(2) For the case when H = T_\infty(n, v, \xi) , note that K: = {𝕜}\{{\omega}^jE_1^l\mid j, l\in n\} is a finite-dimensional normal Hopf subalgebra of H^\bullet = T_\infty(n, v, \xi)^\bullet . The quotient Hopf algebra H^\bullet/K^+H^\bullet = {𝕜}[\overline{E_2}] is the polynomial Hopf algebra, which is minimal-pointed too. Therefore, any infinite-dimensional Hopf subalgebra of T_\infty(n, v, \xi)^\bullet must contain the element E_2 . By the coproduct of E_2 , it follows that T_\infty(n, v, \xi)^\bullet is a minimal infinite-dimensional Hopf algebra.
(3) The case when H = B(n, {\omega}, \gamma) is completely similar to the case T_\infty(n, v, \xi) , due to the facts that K: = {𝕜}\{\psi_{1, \gamma}^jE_1^l\mid j, l\in n\} is a finite-dimensional normal Hopf subalgebra, and the quotient H^\bullet/K^+H^\bullet is the polynomial Hopf algebra.
(4) The case when H = D(m, d, \xi) is similar, too. Note that the finite-dimensional normal Hopf subalgebra of D(m, d, \xi)^\bullet is chosen as a generalized Taft algebra {𝕜}\{(\zeta_{1, \gamma}+\xi\chi_{1, \gamma})^kE_1^l\mid k\in 2m, \; l\in m\} , and the responding quotient is also the polynomial Hopf algebra.
For possible future applications, we might consider Hopf pairings H^\bullet\otimes H\rightarrow {𝕜} with additional requirements (including the non-degeneracy), which are listed as follows:
(HP1) The pairing is non-degenerate;
(HP2) H^\bullet and H are both affine and noetherian;
(HP3) {\operatorname{GKdim}}(H^\bullet) = {\operatorname{GKdim}}(H) .
Furthermore, we might also wish H^\bullet would have certain properties dual to some of H , for examples in this paper:
(HP4) H^\bullet is indecomposable as a coalgebra, when H is prime as an algebra.
These considerations motivate us to discuss some properties of H^\bullet in the following. Recall that a coalgebra is indecomposable if and only if it is link-indecomposable (see [22,Corollary 2.2]). Besides, a ring is called regular if it has finite global dimension.
Proposition 6.5. For affine prime regular Hopf algebras H of GK-dimension one, consider Hopf algebras H^\bullet constructed in Proposition 6.2. We have:
(1) All the Hopf algebras H^\bullet have GK-dimension one.
(2) All the Hopf algebras H^\bullet are pointed and (link-)indecomposable as coalgebras.
(3) All the Hopf algebras H^\bullet are noetherian.
(4) The Hopf algebra ({𝕜}\mathbb{D}_\infty)^\bullet is regular while T_\infty(n, v, \xi)^\bullet , B(n, {\omega}, \gamma)^\bullet and D(m, d, \xi)^\bullet are not when n, m\geq 2 .
Proof. (1) and (2) are clear by our constructions for H^\bullet .
(3) This is because every H^\bullet constructed is a finitely generated module (on both sides) over its noetherian subalgebra {𝕜}[E_2] .
(4) The regularity of ({𝕜}\mathbb{D}_\infty)^\bullet is a direct consequence of the Hilbert Theorem on Syzygies (see [25,Theorem 8.36] e.g.). Recall that as an algebra, ({𝕜}\mathbb{D}_\infty)^\bullet is generated by \zeta_1-\chi_1 and E_2 with relations
\begin{array}{l} & (\zeta_1-\chi_1)^2 = 1, \;\;E_2(\zeta_1-\chi_1) = (\zeta_1-\chi_1)E_2. \end{array} |
Clearly, ({𝕜}\mathbb{D}_\infty)^\bullet is the polynomial algebra over the subalgebra
{𝕜}\{1, \zeta_1-\chi_1\}\cong{𝕜}{\mathbb{Z}}_2 |
with the indeterminate E_2 commuting with all the coefficients. However, {𝕜}{\mathbb{Z}}_2 is semisimple and hence regular. Thus {\operatorname{gldim}}(({𝕜}\mathbb{D}_\infty)^\bullet) = {\operatorname{gldim}}({𝕜}{\mathbb{Z}}_2)+1 < \infty , where {\operatorname{gldim}} denotes the global dimension.
Next we show that B(n, {\omega}, \gamma)^\bullet is not regular when n\geq2 and the non-regularity for T_\infty(n, v, \xi)^\bullet can be proved similarly. Recall that as an algebra, B(n, {\omega}, \gamma)^\bullet is generated by \psi_{1, \gamma} , E_2 and E_1 with relations
\psi_{1, \gamma}^n = 1, \;\;E_1^n = 0, \;\; E_2\psi_{1, \gamma} = \psi_{1, \gamma}E_2, \;\; E_1\psi_{1, \gamma} = \gamma\psi_{1, \gamma}E_1, \;\; E_1E_2 = E_2E_1+\frac{1}{n}E_1. |
Note in Proposition 6.2(3) that
\begin{array}{l} B(n, {\omega}, \gamma)^\bullet & = & {𝕜}\{\psi_{1, \gamma}^jE_2^sE_1^l \mid j\in n, \;s\in{\mathbb{N}}, \;l\in n\} \\ & = & {𝕜}\{E_2^sE_1^l\psi_{1, \gamma}^j \mid s\in{\mathbb{N}}, \;l\in n, \;j\in n\}. \end{array} |
Since B(n, {\omega}, \gamma)^\bullet is a free {𝕜}\langle E_1\rangle -module by Poincaré-Birkhoff-Witt theorem, if B(n, {\omega}, \gamma)^\bullet has finite global dimension, then the projective dimension \mathrm{prdim}_{{𝕜}\langle E_1\rangle}({𝕜}) < \infty . However, this is not possible since {𝕜}\langle E_1\rangle is a finite-dimensional commutative local, but not a field.
Finally, let's show the non-regularity of D(m, d, \xi)^\bullet . By Proposition 6.2(4), we find that
\begin{array}{l} D(m, d, \xi)^\bullet & = & {𝕜}\{\zeta_{1, \gamma}^jE_2^sE_1^l, \;\chi_{1, \gamma}^jE_2^sE_1^l \mid j\in m, \;s\in{\mathbb{N}}, \;l\in m\} \\ & = & {𝕜}\{\zeta_{1, \gamma}^jE_2^sE_1^l\mid j\in m, \;s\in{\mathbb{N}}, \;l\in m\} \\ & & \oplus\;{𝕜}\{\chi_{1, \gamma}^jE_2^sE_1^l\mid j\in m, \;s\in{\mathbb{N}}, \;l\in m\} \end{array} |
is a direct sum of ideals. The former ideal is isomorphic to B(m, {\omega}, \gamma)^\bullet . and thus it is not regular as a ring when m\geq 2 . Therefore,
{\operatorname{gldim}}(D(m, d, \xi)^\bullet)\geq {\operatorname{gldim}}(B(m, {\omega}, \gamma)^\bullet) = \infty. |
Thus concluding Propositions 6.2 and 6.5, we find that
Corollary 6.6. For affine prime regular Hopf algebras H of GK-dimension one, the Hopf pairings \langle-, -\rangle:H^\bullet\otimes H\rightarrow {𝕜} constructed in Proposition 6.2 satisfy the requirements (HP1) to (HP4).
Remark 6.7. Due to Proposition 6.4, if one verifies that H^\bullet is exactly the (link-)indecomposable component of H^\circ containing the unit element, then H^\bullet would be the unique Hopf subalgebra of H^\circ such that (HP1) to (HP4) hold.
Remark 6.8. Takeuchi [28] defined a quantum group G to be a triple (A, U, \langle-, -\rangle) where \langle-, -\rangle:A\otimes U\to {𝕜} is a Hopf pairing. By Proposition 6.2, for each affine prime regular Hopf algebra H of GK-dimension one, one can get a quantum group in Takeuchi's sense naturally.
A classical result by Larson and Radford [14,Theorem 3.3] states that a finite-dimensional Hopf algebra H in characteristic 0 is semisimple, if and only if H^\ast is semisimple. Note that the regularity can be regarded as an infinite-dimensional analogue for the semisimplicity. However according to Proposition 6.5(3), it is not true that H is regular if and only if H^\bullet is regular for a non-degenerate Hopf pairing H^\bullet\otimes H\rightarrow {𝕜} (or a quantum group). This might be a version negating the semisimplicity result by Larson and Radford in infinite-dimensional cases.
Question 6.9. For a general infinite-dimensional Hopf algebra H which is residually finite-dimensional, natural questions are: When does a minimal Hopf algebra H^{\bullet} forming a non-degenerate Hopf pairing over H exist? When are such minimal Hopf algebras H^\bullet unique and of the same GK-dimensions with H ?
In this paper, we determine the finite duals H^\circ of all the affine prime regular Hopf algebras H of GK-dimension one. All these finite duals are given by generators and relations. In addition, we choose respective Hopf subalgebras H^\bullet of these H^\circ which are also affine, noetherian and of GK-dimension one, such that the evaluation maps \langle-, -\rangle:H^\bullet\otimes H\rightarrow {𝕜} become non-degenerate Hopf pairings.
The first author was supported by Zhejiang Provincial Natural Science Foundation of China under No. LQ23A010003. The second author was supported by NSFC 12271243.
The authors would like to thank Doctor Ruipeng Zhu for discussions on additional techniques to estimate the regularity for Hopf algebras appeared, as well as the condition for the existence of non-degenerate Hopf pairings.
All authors declare no conflicts of interest in this paper.
[1] |
D. Bhatia, A. Gupta, P. Arora, Optimality via generalized approximate convexity and quasiefficiency, Optim. Lett., 7 (2013), 127–135. http://dx.doi.org/10.1007/s11590-011-0402-3 doi: 10.1007/s11590-011-0402-3
![]() |
[2] |
L. C. Ceng, S. C. Huang, Existence theorems for generalized vector variational inequalities with a variable ordering relation, J. Global Optim., 46 (2010), 521–535. http://dx.doi.org/10.1007/s10898-009-9436-9 doi: 10.1007/s10898-009-9436-9
![]() |
[3] |
L. C. Ceng, A. Latif, Characterizations of solution sets of set-valued generalized pseudoinvex optimization problems, J. Nonlinear Sci. Appl., 9 (2016), 6382–6395. http://dx.doi.org/10.22436/jnsa.009.12.38 doi: 10.22436/jnsa.009.12.38
![]() |
[4] |
L. C. Ceng, X. Li, X. Qin, Parallel proximal point methods for systems of vector optimization problems on Hadamard manifolds without convexity, Optimization, 69 (2020), 357–383. http://dx.doi.org/10.1080/02331934.2019.1625354 doi: 10.1080/02331934.2019.1625354
![]() |
[5] |
L. C. Ceng, B. S. Mordukhovich, J. C. Yao, Hybrid approximate proximal method with auxiliary variational inequalities for vector optimization, J. Optim. Theory Appl., 146 (2010), 267–303. http://dx.doi.org/10.1007/s10957-010-9667-4 doi: 10.1007/s10957-010-9667-4
![]() |
[6] |
L. C. Ceng, C. F. Wen, Y. C. Liou, On the existence of super efficient solutions and optimality conditions for set-valued vector optimization problems, Mathematics, 10 (2022), 3–16. http://dx.doi.org/10.3390/math10030316 doi: 10.3390/math10030316
![]() |
[7] |
L. C. Ceng, J. C. Yao, Existence theorems for generalized set-valued mixed quasi- variational inequalities in Banach spaces, J. Global Optim., 55 (2013), 27–51. http://dx.doi.org/10.1007/s10898-011-9811-1 doi: 10.1007/s10898-011-9811-1
![]() |
[8] | L. C. Ceng, J. C. Yao, Generalized vector equilibrium-like problems with application to vector optimization problems, Linear Nonlinear Anal., 2 (2016), 53–68. |
[9] | F. H. Clarke, Optimization and nonsmooth analysis, Wiley-Interscience, New York, 1983. |
[10] | V. F. Demyanov, Exhausters and convexificators–new tools in nonsmooth analysis, Quasidifferentiability and Related Topics, Kluwer Academic Publishers, Dordrecht, 2000, 85–137. |
[11] | V. F. Demyanov, Convexification and concavification of positively homogeneous functions by the same family of linear functions, Report, Universita di Pisa, 1994. |
[12] |
V. F. Demyanov, V. Jeyakumar, Hunting for a smaller convex subdifferential, J. Global Optim., 10 (1997), 305–326. http://dx.doi.org/10.1023/A:1008246130864 doi: 10.1023/A:1008246130864
![]() |
[13] |
J. Dutta, V. Vetrivel, On approximate minima in vector optimization, Numer. Funct. Anal. Optim., 22 (2001), 845–859. http://dx.doi.org/10.1081/NFA-100108312 doi: 10.1081/NFA-100108312
![]() |
[14] |
M. Golestani, S. Nobakhtian, Convexificator and strong Kuhn-Tucker conditions, Comput. Math. Appl., 64 (2012), 550–557. http://dx.doi.org/10.1016/j.camwa.2011.12.047 doi: 10.1016/j.camwa.2011.12.047
![]() |
[15] |
A. Gupta, A. Mehra, D. Bhatia, Approximate convexity in vector optimization, Bull. Aust. Math. Soc., 74 (2006), 207–218. http://dx.doi.org/10.1017/S0004972700035656 doi: 10.1017/S0004972700035656
![]() |
[16] |
D. Gupta, A. Mehra, Two types of approximate saddle points, Numer. Funct. Anal. Optim., 29 (2008), 532–550. http://dx.doi.org/10.1080/01630560802099274 doi: 10.1080/01630560802099274
![]() |
[17] |
P. Gupta, S. K. Mishra, On Minty variational principle for nonsmooth vector optimization problems with generalized approximate convexity, J. Math. Programm. Oper. Res., 67 (2018), 1157–1167. http://dx.doi.org/10.1080/02331934.2018.1466884 doi: 10.1080/02331934.2018.1466884
![]() |
[18] |
V. Jeyakumar, D. T. Luc, Nonsmooth calculus, minimality, and monotonicity of convexificators, J. Optim. Theory Appl., 101 (1999), 599–621. http://dx.doi.org/10.1023/A:1021790120780 doi: 10.1023/A:1021790120780
![]() |
[19] |
B. C. Joshi, On generalized approximate convex functions and variational inequalities, RAIRO-Oper. Res., 55 (2021), S2999–S3008. http://dx.doi.org/10.1051/ro/2020141 doi: 10.1051/ro/2020141
![]() |
[20] |
V. Laha, S. K. Mishra, On vector optimization problems and vector variational inequalities using convexifactor, J. Math. Program. Oper. Res., 66 (2017), 1837–1850. https://doi.org/10.1080/02331934.2016.1250268 doi: 10.1080/02331934.2016.1250268
![]() |
[21] |
X. F. Li, J. Z. Zhang, Necessary optimality conditions in terms of convexificators in Lipschitz optimization, J. Optim. Theory Appl., 131 (2006), 429–452. http://dx.doi.org/10.1007/s10957-006-9155-z doi: 10.1007/s10957-006-9155-z
![]() |
[22] |
X. F. Li, J. Z. Zhang, Stronger Kuhn-Tucker type conditions in nonsmooth multiobjective optimization: Locally Lipschitz case, J. Optim. Theory Appl., 127 (2005), 367–388. http://dx.doi.org/10.1007/s10957-005-6550-9 doi: 10.1007/s10957-005-6550-9
![]() |
[23] |
X. J. Long, N. J. Huang, Optimality conditions for efficiency on nonsmooth multiobjective programming problems, Taiwanese J. Math., 18 (2014), 687–699. http://dx.doi.org/10.11650/tjm.18.2014.3730 doi: 10.11650/tjm.18.2014.3730
![]() |
[24] |
D. V. Luu, Convexifcators and necessary conditions for efficiency, Optimization, 63 (2013), 321–335. http://dx.doi.org/10.1080/02331934.2011.648636 doi: 10.1080/02331934.2011.648636
![]() |
[25] |
D. V. Luu, Necessary and sufficient conditions for efficiency via convexificators, J. Optim. Theory Appl., 160 (2014), 510–526. http://dx.doi.org/10.1007/s10957-013-0377-6 doi: 10.1007/s10957-013-0377-6
![]() |
[26] |
S. K. Mishra, V. Laha, On approximately star-shaped functions and approximate vector variational inequalities, J. Optim. Theory Appl., 156 (2013), 278–293. http://dx.doi.org/10.1007/s10957-012-0124-4 doi: 10.1007/s10957-012-0124-4
![]() |
[27] |
S. K. Mishra, V. Laha, On Minty variational principle for nonsmooth vector optimization problems with approximate convexity, Optim. Lett., 10 (2016), 577–589. http://dx.doi.org/10.1007/s11590-015-0883-6 doi: 10.1007/s11590-015-0883-6
![]() |
[28] |
S. K. Mishra, B. B. Upadhyay, Some relations between vector variational inequality problems and nonsmooth vector optimization problems using quasi efficiency, Positivity, 17 (2013), 1071–1083. http://dx.doi.org/10.1007/s11117-013-0221-0 doi: 10.1007/s11117-013-0221-0
![]() |
[29] |
S. K. Mishra, S. Y. Wang, Vector variational-like inequalities and nonsmooth vector optimization problems, Nonlinear Anal., 64 (2006), 1939–1945. http://dx.doi.org/10.1016/j.na.2005.07.030 doi: 10.1016/j.na.2005.07.030
![]() |
[30] | B. B. Upadhyay, P. Mishra, R. N. Mohapatra, S. K. Mishra, On relationships between vector variational inequality and nonsmooth vector optimization problems via strict minimizers, Adv. Nonlinear Var. Inequal., 20 (2017), 1–12. |
[31] |
B. B. Upadhyay, P. Mishra, R. N. Mohapatra, S. K. Mishra, On the applications of nonsmooth vector optimization problems to solve generalized vector variational inequalities using convexificators, Adv. Intell. Syst. Comput., 991 (2020), 660–671. http://dx.doi.org/10.1007/978-3-030-21803-4_66 doi: 10.1007/978-3-030-21803-4_66
![]() |
[32] |
X. Q. Yang, X. Y. Zheng, Approximate solutions and optimality conditions of vector variational inequalities in Banach spaces, J. Glob. Optim., 40 (2008), 455–462. http://dx.doi.org/10.1007/s10898-007-9183-8 doi: 10.1007/s10898-007-9183-8
![]() |
[33] |
L. C. Zeng, Necessary optimality conditions for super minimizers in structural problems of multiobjective optimization, Asian-Eur. J. Math., 2 (2009), 321–358. https://doi.org/10.1142/S1793557109000261 doi: 10.1142/S1793557109000261
![]() |
1. | K. Brown, J. Stafford, 2024, 801, 978-1-4704-7632-8, 51, 10.1090/conm/801/16036 | |
2. | Yiwei Zheng, On the structure of irreducible Yetter-Drinfeld modules over D , 2024, 9, 2473-6988, 21321, 10.3934/math.20241035 | |
3. | Nicolás Andruskiewitsch, Sonia Luján Natale, Blas Torrecillas, A class of finite-by-cocommutative Hopf algebras, 2024, 1, 73, 10.5802/art.5 |