In this paper, all simple Yetter-Drinfeld modules and indecomposable projective Yetter-Drinfeld modules over a family of non-pointed 8m-dimension Hopf algebras of tame type with rank two, are construted and classified. The technique is Radford's method of constructing Yetter-Drinfeld modules over a Hopf algebra. Furthermore, the projective class rings of the category of Yetter-Drinfeld modules over this class of Hopf algebras are described explicitly by generators and relations.
Citation: Yaguo Guo, Shilin Yang. Projective class rings of a kind of category of Yetter-Drinfeld modules[J]. AIMS Mathematics, 2023, 8(5): 10997-11014. doi: 10.3934/math.2023557
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In this paper, all simple Yetter-Drinfeld modules and indecomposable projective Yetter-Drinfeld modules over a family of non-pointed 8m-dimension Hopf algebras of tame type with rank two, are construted and classified. The technique is Radford's method of constructing Yetter-Drinfeld modules over a Hopf algebra. Furthermore, the projective class rings of the category of Yetter-Drinfeld modules over this class of Hopf algebras are described explicitly by generators and relations.
The category of Yetter-Drinfeld modules over a Hopf algebra H was introduced firstly in [1], which provides a solution to the Yang-Baxter equation [2] when the antipode of H is bijective. In 1998, Andruskiewitsch and Schneider [3] introduced the liftingmethod which was extensively used in the classification of finite dimensional pointed and copointed Hopf algebras. It is remarked that the Yetter-Drinfeld modules play an important role in this process. More precisely, by determining the braiding in HHYD and indecomposable objects in the category of Yetter-Drinfeld modules over a Hopf algebra H, one can construct all finite dimensional Nichols algebras in HHYD, and then all finite dimensional Hopf algebras over H by the lifting method. There are a lot of works to classify finite dimensional Hopf algebras by lifting method, see for example [4,5,6,7,8,9,10,11,12,13]. Therefore, it is important to understand the structures of Yetter-Drinfeld modules for a finite dimensional Hopf algebra.
In 2003, Radford [14] gave an idea of constructing the simple Yetter-Drinfeld modules: any Yetter-Drinfeld H-module M is the form M=H⋅N for some simple subcomodule N of H⊗L, where L is a left H-module. In 2012, Zhu and Chen [15] gave the classification of all simple Yetter-Drinfeld modules over the Hopf-Ore extension A(n,0) of the dihedral group Dn for even or odd nummber n. In 2020, Yang and Zhang [16] classified all Hopf algebra structures on the quotient of Ore extensions H4[z;σ] of automorphism type for the Sweedler's 4-dimension Hopf algebra H4, thereby obtaining a family of non-pointed and non-seimsimple Hopf algebras H4n of rank one. The classifications of finite dimensional Hopf algebras over H8 and H12 were given in [8] and [9] respectively. The Green ring of H4n was determined by Chen and Yang et al. [17]. Xiong [18] (see also Zhang [19] in somewhat different idea) classified all simple Yetter-Drinfeld modules over H4n and gave the structures of projective class rings of the category of the Yetter-Drinfeld modules of H4n. In [20], we classified the Yetter-Drinfeld modules over H2n2 and gave the Grothendieck rings of the category of the Yetter-Drinfeld modules of H2n2, where H2n2 is a family of Kac-Paljutkin semisimple Hopf algebra of dimension 2n2.
Motivated by the above works, a family of non-pointed 8m-dimension Hopf algebras H8m of tame type with rank two are studied in [21], where m is even. It is pointed that H8m is a special biserial algebra, which is just one subclass of basic tame Hopf algebras with only one block in [22]. By the technique of special biserial algebras (see for example [23]), we construct and classify the isomorphism classes of all indecomposable modules of H8m, and determine the components of Auslander-Reiten quivers. Furthermore, we establish the tensor product of arbitrary simple (or projective) modules and indecomposable modules, and characterize the projective class rings and Grothendieck rings of H8m.
In this paper, we focus on the classification of simple (or indecomposable projective) Yetter-Drinfeld modules over H8m by Radford's method and the description of the projective class rings of the category of the Yetter-Drinfeld modules over H8m. In our further works, we hope that the classification of finite dimensional Hopf algebras over H8m is established.
Throughout this paper, K is assumed to be an algebraically closed field of characteristic zero, m≥2 is even and ω the 2m-th primitive root of unity. The Sweedler's notation
Δ(h)=∑(h)h(1)⊗h(2) |
for a Hopf algebra is used, and some other notations see [24].
In this section, let us recall some basic definition and results for the Hopf algebra H8m.
By definition, the Hopf algebra H8m as an algebra is generated by xi(i=1,2),z with the following relations
z2m=1,x2i=0,x1x2+x2x1=0,xiz=(−1)iωzxi. |
The co-multiplication, counit and antipode are given as follows:
Δ(xi)=1⊗xi+xi⊗zm,Δ(z)=(1⊗1+x2⊗x2zm)(z⊗z),ε(xi)=0,ε(z)=1,S(xi)=−xizm,S(z)=z−1. |
for i=1,2. It is easy to see that
{xs1xt2zi|i∈Z2m,s,t∈[0,1]} |
is a basis of H8m, and
Δ(zl)=zl⊗zl+alx2zl⊗x2zm+l | (2.1) |
for l∈Z2m, where al=1−ω−2l1−ω−2. In particular,
Δ(zm)=zm⊗zm. |
In [21], all finite dimensional simple modules of H8m were constructed and classified. There exist exactly 2m pairwise non-isomorphic 1-dimension simple H8m-modules Si with the basis {υi} for i∈Z2m. The actions are given by
x1⋅υi=0,x2⋅υi=0,z⋅υi=ωiυi. |
Let HM be the category of left H-modules. Recall that a left Yetter-Drinfeld H-module M for a finite dimensional Hopf algebra H is a left H-module (M,⋅) and a left H-comodule (M,ρ) satisfying
ρ(h⋅m)=∑h(1)m(−1)S(h(3))⊗h(2)⋅m(0),∀m∈M,h∈H, |
where S is the antipode of H and ρ(m)=∑(m)m(−1)⊗m(0). The category of left Yetter-Drinfeld H-modules is denoted by HHYD, whose morphisms are both H-linear and H-colinear maps (see [1]). Let V∈HHYD, the left dual V∗ is defined by
⟨h⋅f,v⟩=⟨f,S(h)v⟩,f(−1)⟨f(0),v⟩=S−1(v(−1))⟨f,v(0)⟩. | (2.2) |
According to Radford's results in [14], we have the following results.
Proposition 2.1. If V,W∈HHYD, then V⊗W∈HHYD. The actions and coactions are as follows:
h⋅(υ⊗ω)=∑(h)h(1)⋅υ⊗h(2)⋅ω,ρ(υ⊗ω)=∑(υ),(ω)υ(−1)ω(−1)⊗υ(0)⊗ω(0), |
where υ∈V,ω∈W,h∈H.
Lemma 2.2. Let L∈HM. Then, we have
(1) H⊗L∈HHYD; the module and comodule actions are given by
g⋅(h⊗l)=∑(g)g(1)hS(g(3))⊗g(2)⋅l, | (2.3) |
ρ(h⊗l)=∑(h)h(1)⊗h(2)⊗l,∀h,g∈H,l∈L. | (2.4) |
(2)If M is a simple Yetter-Drinfeld H-module, then M=H⋅N for some simple subcomodule N of H⊗L where L is a simple left H-module.
By Proposition 2.1 and Lemma 2.2, we can construct all simple Yetter-Drinfeld modules over any Hopf algebra H.
To classify the simple Yetter-Drinfeld modules over the Hopf algebra H8m, we firstly give all simple subcomodules of H8m⊗Si for all i∈Z2m.
Lemma 3.1. Let 0≠L be a subcomodule of H8m⊗Si, then there exists j∈Z2m such that zj⊗υi∈L or (zj+lx2zj)⊗υi∈L, where 0≠l∈K.
Proof. Let 0≠L be a subcomodule of H8m⊗Si and 0≠u=(∑2m−1j=0∑1s,t=0ks,t,ixs1xt2zi)⊗υi∈L, where ks,t,j∈K. By (2.4), we have
ρ(u)=2m−1∑j=0(α0,j⊗zj+α1,j⊗x1zj+α2,j⊗x2zj+α3,j⊗x1x2zj)⊗υi |
where
α0,j=k0,0,jzj+k1,1,jx1x2zj+k1,0,m+jx1zm+j+k0,1,m+jx2zm+j,α1,j=k1,0,jzj+k1,1,m+jx2zm+j,α2,j=k0,1,jzj−k1,0,jajx1x2zj+k0,0,m+jam+jx2zm+j−k1,1,m+jx1zm+j,α3,j=k1,1,jzj+k1,0,m+jam+jx2zm+j. |
Assume that j≠0,m (the discussions when j=0,m are similar). Note that there exist some j such that the vector θj=(k0,0,j,k1,0,j,k0,1,j,k1,1,j)≠0 and aj≠0. Now, we complete the proof by discussing the cases (i)-(vi).
(i) k1,1,j≠0,k1,0,j≠0.
(a) If α0,j and α2,j are linearly independent, we have zj⊗υi∈L. Moreover, x2zm+j⊗υi∈L since
ρ(zj⊗υi)=(zj⊗zj+ajx2zj⊗x2zm+j)⊗υi; |
(b) If α0,j and α2,j are linearly dependent, there exist a 0≠l∈K such that (zj+lx2zj)⊗υi∈L. Moreover, (ajx2zm+j+lzm+j)⊗υi∈L since
ρ((zj+lx2zj)⊗υi)=(zj⊗(zj+lx2zj)+x2zj⊗(ajx2zm+j+lzm+j))⊗υi; |
(ii) k1,1,j≠0,k1,0,j=0, we have zj⊗υi∈L;
(iii) k1,1,j=0,k1,0,j≠0, we have x2zj∈L. Moreover, zm+j⊗υi∈L since
ρ(x2zj⊗υi)=(zj⊗x2zj+x2zj⊗zm+j)⊗υi; |
(iv) k1,1,j=0,k1,0,j=0,k0,0,j≠0,k0,1,j≠0, similar to (i);
(v) k1,1,j=0,k1,0,j=0,k0,0,j≠0,k0,1,j=0, similar to (ii);
(vi) k1,1,j=0,k1,0,j=0,k0,0,j=0,k0,1,j≠0, similar to (iii).
Lemma 3.2. For i∈Z2m, all simple subcomodules of H8m⊗Si are as follows:
(1) U(i,j):=K{zj⊗υi}, where j=0,m;
(2) V(i,j):=K{zj⊗υi,x2zm+j⊗υi}, where j∈Z2mandj≠0,m;
(3) W(i,j,l):=K{(zj+lx2zj)⊗υi,(ajx2zm+j+lzm+j)⊗υi}, where 0≠l∈K,j∈Z2mandj≠0,m.
Proof. It is easy to see that U(i,j) is a simple subcomodule of H8m⊗Si for j=0,m. One also see that V(i,j) is a subcomodule of H8m⊗Si by (2.1). We show that V(i,j) is a simple. Indeed, let 0≠V be a subcomodule of V(i,j) and 0≠υ=(ki,jzj+li,jx2zm+j)⊗υi∈V, where ki,j,li,j∈K and (ki,j,li,j)≠0. By (2.4), we have
ρ(υ)=(ki,jzj+li,jx2zm+j)⊗zj⊗υi+(ki,jajx2zj+li,jzm+j)⊗x2zm+j⊗υi. |
One can check that ki,jzj+li,jx2zm+j,ki,jajx2zj+li,jzm+j are linearly independent. It follows that zj⊗υi,x2zm+j⊗υi∈V and V=V(i,j). Hence, V(i,j) is a simple subcomodule of H8m⊗Si. Similarly, we have W(i,j,l) is a simple subcomodule of H8m⊗Si for 0≠l∈K,j∈Z2m and j≠0,m. By Lemma3.1, the vector spaces defined in (1)–(3) are all simple subcomodules of H8m⊗Si.
Lemma 3.3. As Yetter-Drinfeld modules, H8m⋅V(i,j)=H8m⋅W(i,j,l), where 0≠l∈K,j∈Z2mandj≠0,m.
Proof. By (2.3), we have
x2⋅((zj+lx2zj)⊗υi)=((−1)i−ω−j)x2zm+j⊗υi∈H8m⋅W(i,j,l). |
By (2.4), we have
ρ(x2zm+j⊗υi)=x2zm+j⊗zj⊗υi+zm+j⊗x2zm+j⊗υi. |
Then zj⊗υi∈H8m⋅W(i,j,l). Hence
x1⋅(zj⊗υi)=((−1)i+(−1)j+1ω−j)x1zm+j⊗υi∈H8m⋅W(i,j,l),x1⋅(x2zm+j⊗υi)=((−1)i+(−1)j+1ω−j)x1x2zj⊗υi∈H8m⋅W(i,j,l). |
Therefore, as Yetter-Drinfeld modules, we have H8m⋅V(i,j)=H8m⋅W(i,j,l), where 0≠l∈K,j∈Z2m and j≠0,m.
By Lemma 3.2 and Lemma 3.3, we just need to consider the following Yetter-Drinfeld modules:
∙H8m⋅U(i,j), where i∈Z2m,j=0,m;
∙H8m⋅V(i,j), where i,j∈Z2m,j≠0,m.
Now, we consider M(i,j):=H8m⋅U(i,j), where i∈Z2m,j=0,m. There are two cases:
Case 1: Mi:=M(i,j) for i∈Z2m, where j=0 if i is even and j=m if i is odd. Mi are 1-dimension simple Yetter-Drinfeld modules. The actions on the basis {νi=zim⊗υi} and the coactions are given by
x1⋅νi=0,x2⋅νi=0,z⋅νi=ωiνi,ρ(νi)=zim⊗νi. |
Case 2: M(i,j) for i∈Z2m, where j=0 if i is odd and j=m if i is even. M(i,j) is a 4-dimension simple Yetter-Drinfeld module. The action on the basis {νi,j0=zj⊗υi,νi,j1=x1zm+j⊗υi,νi,j2=x2zm+j⊗υi,νi,j3=x1x2zj⊗υi} and the coaction are given by
x1⋅νi,j0=(−1)i2νi,j1,x2⋅νi,j0=(−1)i2νi,j2,z⋅νi,j0=ωiνi,j0,x1⋅νi,j1=0,x2⋅νi,j1=(−1)i+12νi,j3,z⋅νi,j1=−ωi−1νi,j1,x1⋅νi,j2=(−1)i2νi,j3,x2⋅νi,j2=0,z⋅νi,j2=ωi−1νi,j2,x1⋅νi,j3=0,x2⋅νi,j3=0,z⋅νi,j3=−ωi−2νi,j3. |
ρ(νi,j0)=zj⊗νi,j0,ρ(νi,j1)=x1zm+j⊗νi,j0+zm+j⊗νi,j1,ρ(νi,j2)=x2zm+j⊗νi,j0+zm+j⊗νi,j2,ρ(νi,j3)=x1x2zj⊗νi,j0+x2zj⊗νi,j1−x1zj⊗νi,j2+zj⊗νi,j3. |
For N(i,j):=H8m⋅V(i,j), i,j∈Z2m,j≠0,m. N(i,j) are 4-dimension simple Yetter-Drinfeld modules. The action on the basis {νi,j0=zj⊗υi,νi,j1=x1zm+j⊗υi,νi,j2=x2zm+j⊗υi,νi,j3=x1x2zj⊗υi} and the coaction are given by
x1⋅νi,j0=ai,jνi,j1,x2⋅νi,j0=bi,jνi,j2,z⋅νi,j0=ωiνi,j0,x1⋅νi,j1=0,x2⋅νi,j1=−bi,jνi,j3,z⋅νi,j1=−ωi−1νi,j1,x1⋅νi,j2=ai,jνi,j3,x2⋅νi,j2=0,z⋅νi,j2=ωi−1νi,j2,x1⋅νi,j3=0,x2⋅νi,j3=0,z⋅νi,j3=−ωi−2νi,j3. |
ρ(νi,j0)=zj⊗νi,j0+ajx2zj⊗νi,j2,ρ(νi,j1)=x1zm+j⊗νi,j0+zm+j⊗νi,j1−ajx1x2zm+j⊗νi,j2+ajx2zm+j⊗νi,j3,ρ(νi,j2)=x2zm+j⊗νi,j0+zm+j⊗νi,j2,ρ(νi,j3)=x1x2zj⊗νi,j0+x2zj⊗νi,j1−x1zj⊗νi,j2+zj⊗νi,j3. |
Here ai,j=(−1)i+(−1)j+1ω−j,bi,j=(−1)i−ω−j,aj=1−ω−2j1−ω−2.
Let Λ={(i,j)|i,j∈Z2m and j≢im(mod2m)} and
Ni,j:=K{νi,j0=zj⊗υi,νi,j1=x1zm+j⊗υi,νi,j2=x2zm+j⊗υi,νi,j3=x1x2zj⊗υi}, |
where (i,j)∈Λ. Note that
{Ni,j|(i,j)∈Λ}={M(i,j)|i∈Z2m,j=0,m,j≢im(mod2m)}∪{N(i,j)|i,j∈Z2m,j≠0,m}. |
In the sequel the operations of subscripts are in Z2m.
We get the first result of this paper.
Theorem 3.4. The set
{Mi|i∈Z2m}⋃{Ni,j|(i,j)∈Λ} |
forms a complete list of non-isomorphic simple Yetter-Drinfeld modules over H8m.
Proof. Firstly, we show that
Mi≅Mi′ if and only if i=i′ for i,i′∈Z2m. |
Let f:Mi→Mi′,νi↦aνi′ be a Yetter-Drinfeld module isomorphism, where 0≠a∈K. Then, we have
z⋅f(νi)=az⋅νi′=aωi′νi′=f(z⋅νi)=ωif(νi)=aωiνi′,(id⊗f)ρ(νi)=(id⊗f)(zim⊗νi)=azim⊗νi′=ρ(f(νi))=ρ(νi′)=azi′m⊗νi′. |
Hence i=i′.
Now we show that each Ni,j for (i,j)∈Λ is simple Yetter-Drinfeld module.
Let 0≠V be a Yetter-Drinfeld submodule of N(i,j) and 0≠υ=3∑k=0αi,jkνi,jk∈V, where αi,jk∈K and (αi,j0,αi,j1,αi,j2,αi,j3)≠0. By (2.4), we have
ρ(υ)=(αi,j0zj+αi,j1x1zm+j+αi,j2x2zm+j+αi,j3x1x2zj)⊗νi,j0+(αi,j1zm+j+αi,j3x2zj)⊗νi,j1+(αi,j0ajx2zj−αi,j1ajx1x2zm+j+αi,j2zm+j−αi,j3x1zj)⊗νi,j2+(αi,j1ajx2zm+j+αi,j3zj)⊗νi,j3. |
If (αi,j1,αi,j3)=0, it is easy to see that νi,j0∈V. If (αi,j1,αi,j3)≠0, αi,j0zj+αi,j1x1zm+j+αi,j2x2zm+j+αi,j3x1x2zj and αi,j1ajx2zm+j+αi,j3zj are linearly independent. We also get that νi,j0∈V. It follows that
νi,j1=1ai,jx1⋅νi,j0∈V,νi,j2=1bi,jx2⋅νi,j0∈V,νi,j3=1ai,jbi,jx1x2⋅νi,j0∈V. |
Hence, V=N(i,j) and N(i,j) is a simple Yetter-Drinfeld module.
Finally we show that
Ni,j≅Ni′,j′ if and only if (i,j)=(i′,j′) for (i,j),(i′,j′)∈Λ. |
Let g:Ni,j→Ni′,j′,νi,jk↦3∑l=0αk,lνi′,j′l, where k∈{0,1,2,3},αk,l∈K and there exists l∈{0,1,2,3} such that αk,l≠0 for each k∈{0,1,2,3}. Then we have
x1⋅g(νi,j0)=α0,0ai′,j′νi′,j′1+α0,2ai′,j′νi,j3=g(x1⋅νi,j0)=ai,j3∑l=0α1,lνi′,j′l,x2⋅g(νi,j0)=α0,0bi′,j′νi′,j′2−α0,1bi′,j′νi,j3=g(x2⋅νi,j0)=bi,j3∑l=0α2,lνi′,j′l,z⋅g(νi,j0)=α0,0ωi′νi′,j′0−α0,1ωi′−1νi,j1+α0,2ωi′−1νi′,j′2−α0,3ωi′−2νi,j3=g(z⋅νi,j0)=ωi3∑l=0α0,lνi′,j′l,x1⋅g(νi,j1)=α1,0ai′,j′νi′,j′1+α1,2ai′,j′νi,j3=g(x1⋅νi,j1)=0,x2⋅g(νi,j1)=α1,0bi′,j′νi′,j′2−α1,1bi′,j′νi,j3=g(x2⋅νi,j1)=−bi,j3∑l=0α3,lνi′,j′l,z⋅g(νi,j1)=α1,0ωi′νi′,j′0−α1,1ωi′−1νi,j1+α1,2ωi′−1νi′,j′2−α1,3ωi′−2νi,j3=g(z⋅νi,j1)=−ωi−13∑l=0α1,lνi′,j′l,x1⋅g(νi,j2)=α2,0ai′,j′νi′,j′1+α2,2ai′,j′νi,j3=g(x1⋅νi,j2)=ai,j3∑l=0α3,lνi′,j′l,x2⋅g(νi,j2)=α2,0bi′,j′νi′,j′2−α2,1bi′,j′νi,j3=g(x2⋅νi,j2)=0,z⋅g(νi,j2)=α2,0ωi′νi′,j′0−α2,1ωi′−1νi,j1+α2,2ωi′−1νi′,j′2−α2,3ωi′−2νi,j3=g(z⋅νi,j2)=ωi−13∑l=0α2,lνi′,j′l,x1⋅g(νi,j3)=α3,0ai′,j′νi′,j′1+α3,2ai′,j′νi,j3=g(x1⋅νi,j3)=0,x2⋅g(νi,j3)=α3,0bi′,j′νi′,j′2−α3,1bi′,j′νi,j3=g(x2⋅νi,j3)=0,z⋅g(νi,j3)=α3,0ωi′νi′,j′0−α3,1ωi′−1νi,j1+α3,2ωi′−1νi′,j′2−α3,3ωi′−2νi,j3=g(z⋅νi,j3)=−ωi−23∑l=0α3,lνi′,j′l. |
Hence i=i′ and αk,l=0 for k≠l. Also, we have
(id⊗g)ρ(νi,j3)=(id⊗g)(x1x2zj⊗νi,j0+x2zj⊗νi,j1−x1zj⊗νi,j2+zj⊗νi,j3)=α0,0x1x2zj⊗νi′,j′0+α1,1x2zj⊗νi′,j′1−α2,2x1zj⊗νi′,j′2+α3,3zj⊗νi′,j′3=ρ(g(νi,j3))=α3,3ρ(νi′,j′3)=α3,3x1x2zj′⊗νi′,j′0+α3,3x2zj′⊗νi′,j′1−α3,3x1zj′⊗νi′,j′2+α3,3zj′⊗νi′,j′3. |
Hence j=j′ and α0,0=α1,1=α2,2=α3,3. Therefore,
Ni,j≅Ni′,j′ if and only if (i,j)=(i′,j′) for (i,j),(i′,j′)∈Λ. |
The proof is finished.
Corollary 3.5. N∗i,j≅Nm+2−i,−j for (i,j)∈Λ.
Proof. Let {μi,jk|k∈{0,1,2,3}} be the dual basis of {νi,jk|k∈{0,1,2,3}} such that
μi,j0(νi,j0)=0,μi,j0(νi,j1)=0,μi,j0(νi,j2)=0,μi,j0(νi,j3)=(−1)jω2j,μi,j1(νi,j0)=0,μi,j1(νi,j1)=0,μi,j1(νi,j2)=−ωj,μi,j0(νi,j3)=0,μi,j2(νi,j0)=0,μi,j2(νi,j1)=(−1)jωj,μi,j2(νi,j2)=0,μi,j2(νi,j3)=0,μi,j3(νi,j0)=1,μi,j2(νi,j1)=0,μi,j3(νi,j2)=0,μi,j3(νi,j3)=0. |
By (2.2), we have
x1⋅μi,j0=am+2−i,−jμi,j1,x2⋅μi,j0=bm+2−i,−jμi,j2,z⋅μi,j0=−ω2−iμi,j0,x1⋅μi,j1=0,x2⋅μi,j1=−bm+2−i,−jμi,j3,z⋅μi,j1=ω1−iμi,j1,x1⋅μi,j2=am+2−i,−jμi,j3,x2⋅μi,j2=0,z⋅μi,j2=−ω1−iμi,j2,x1⋅μi,j3=0,x2⋅μi,j3=0,z⋅μi,j3=ω−iμi,j3. |
and
ρ(μi,j0)=z−j⊗μi,j0+a−jx2z−j⊗μi,j2,ρ(μi,j1)=x1zm−j⊗μi,j0+zm−j⊗μi,j1−a−jx1x2zm−j⊗μi,j2+a−jx2zm−j⊗μi,j3,ρ(μi,j2)=x2zm−j⊗μi,j0+zm−j⊗μi,j2,ρ(μi,j3)=x1x2z−j⊗μi,j0+x2z−j⊗μi,j1−x1z−j⊗μi,j2+z−j⊗μi,j3. |
Hence N∗i,j≅Nm+2−i,−j.
Assume that H is a finite dimensional Hopf algebra. Let F(H) be the free abelian group generated by the isomorphic classes [M] of H-modules M. Then the abelian group F(H) becomes a ring equipped with a multiplication given by the tensor product [M][N]=[M⊗N]. The Green ring r(H) is defined to be the quotient ring of F(H) module the relations [M⊕N]=[M]+[N]. The projective class ring of H is the subring of r(H) generated by their projective and simple representations of H. As is known, HHYD≅D(Hcop)M. where D(Hcop)M is the category of the left modules of Drinfeld double D(Hcop). The projective class ring of D(Hcop), or equivalently, the projective class ring of HHYD is denoted by rP(D(Hcop)).
In this section, we denote D:=D(Hcop8m). Let P(V) be the projective cover of a simple D-module V, or equivalently, a simple Yetter-Drinfeld module V∈H8mH8mYD. It is well-known that P(V) is unique (up to isomorphism) indecomposable projective D-module which maps onto V. Let Irr(D) be the set of isomorphism classes of simple D-modules. One sees that
D≅⨁V∈Irr(D)P(V)⊕dimV, |
and D is unimodular and quasi-triangular (see [24,25]).
Let us determine the projective class ring rP(D). For this purpose, we denote Pi:=P(Mi),i∈Z2m and firstly introduce some lemmas.
Lemma 4.1. (1) Mi⊗Mj≅Mi+j≅Mj⊗Mi and Mi⊗Nk,l≅Ni+k,im+l≅Nk,l⊗Mi for i,j∈Z2m,(k,l)∈Λ;
(2) P(Ni,j)≅Ni,j, (i,j)∈Λ;
(3) Mi⊗Pj≅Pi+j≅Pj⊗Mi and dimPi=16 for i,j∈Z2m;
(4) For (i,j),(k,l)∈Λ,h∈Z2m,Hom(Ni,j⊗Nk,l,Mh)≠0 if and only if h≡i+m+k−2(mod2m) and j≡(i+k)m−l(mod2m).
Proof. (1) It follows from a direct computation.
(2) Suppose that P(Ni,j)≆Ni,j for some (i,j)∈Λ. Since D is unimodular, SocP(Ni,j)≅Ni,j, and dimP(Ni,j)≥2dimNi,j=8. Since P(Ni−k,j−mk)⊗Mk is projective and
HomD(P(Ni−k,j−mk)⊗Mk,Ni−k,j−mk⊗Mk)≅HomD(P(Ni−k,j−mk),Ni−k,j−mk⊗Mk⊗M∗k)≅HomD(P(Ni−k,j−mk),Ni−k,j−mk)≠0, |
we have P(Ni,j)≅P(Ni−k,j−mk⊗Mk)⊂P(Ni−k,j−mk)⊗Mk, which implies that
dimP(Ni−k,j−mk)≥dimP(Ni,j)≥8. |
Let J={(s,t)∈Λ|(s,t)=(i−k,j−mk)\; for k∈Z2m}. It is obvious that
|J|=2m and |Λ−J|=4m2−4m. |
Then
dimD=2m−1∑i=0dimP(Mi)+∑(s,t)∈J4dimP(Ns,t)+∑(s,t)∈Λ−J4dimP(Ns,t)≥2m−1∑i=0dimP(Mi)+32|J|+16|Λ−J|>32|J|+16|Λ−J|=64m2. |
It is a contradiction. Hence P(Ni,j)≅Ni,j for (i,j)∈Λ.
(3) Since P(Ni,j)≅Ni,j for any fixed (i,j)∈Λ, it follows that 2mdimPi=dimD−4|Λ|dimNi,j=32m and hence dimPi=16. Similar to (2), we have Pi⊆P0⊗Mi. Then, Pi≅P0⊗Mi. Hence, Pj⊗Mi≅P0⊗Mj⊗Mi≅P0⊗Mi+j≅Pi+j.
(4) Since D is quasi-triangular, M⊗KN≅N⊗KM for M,N∈DM,
HomD(Ni,j⊗Nk,l,Mh)≅HomD(Ni,j,HomK(Nk,l,Mh))≅HomD(Ni,j,N∗k,l⊗Mh)≅HomD(Ni,j,Mh⊗N∗k,l)≅HomD(Ni,j,Mh⊗Nm+2−k,−l)≅HomD(Ni,j,Nh+m+2−k,hm−l). |
Then by Schur's lemma, HomD(Ni,j⊗Nk,l,Mh)≠0 if and only if i≡h+m+2−k(mod2m) and j≡hm−l(mod2m), if and only if h≡i+m+k−2(mod2m) and j≡(i+k)m−l(mod2m).
Corollary 4.2. We have
DD≅(2m−1⨁i=0Pi)⨁(⨁(j,k)∈ΛN⊕4j,k). |
Now we describe the projective cover Pi of the simple module Mi for i∈Z2m. For convenience, we let
A=(0000100000000010),B=(10000−10000−100001),C=(0000000010000−100),D=(10000−ω−10000ω−10000−ω−2),E=(00000000−ω0000100),F=(10000zm0000zm00101),G=(00002x1zm000000000−2x10),H=(000000002x2zm00002x200),K=(0000000000004x1x2000). |
Let P be a vector space with a basis {p0,p1,⋯,p15} and the action and coaction of H8m on P be the form:
[x1]=(A000BA0000A000BA),[x2]=(C0000C00B0C00−B0C),[z]=(−ω2D0000ωD00ωE0−ωD00E0D), |
ρ(P)=(F−G−H+K0002x1zm(H+F)zm(F+G+H+K)002x2zm(F−G)0zm(F+G+H+K)04x1x2F2x2(F+G)2x1(H−F)F−G−H+K)⊗P, |
where P=(p0,p1,⋯,p15)T. By a complex computation, we know that P is a left Yetter-Drinfeld module.
Lemma 4.3. P is an indecomposable Yetter-Drinfeld module over H8m.
Proof. Suppose that P is not indecomposable. Then there exist two non-trivial submodules M and N such that P=M⊕N. Let
κ0=p3+p12+p9−p6,κ1=p7−p13,κ2=p11−p14,κ3=p15,κ4=p1+p4,κ5=p5,κ6=p2+p8,κ7=p10,κ8=p0,κ9=p3−p12,κ10=p6+p9,κ11=p11+p14,κ12=p1,κ13=p2,κ14=p7,κ15=p6, |
and K−1=0,Kl=∑li=0Kκi for l∈{0,⋯,15}. One can check that
0=K−1⊂K0⊂K1⊂⋯⊂K15=P |
is a Yetter-Drinfeld submodules chain of P such that Kl/Kl−1 is a one dimensional Yetter-Drinfeld module. And
Kl/Kl−1≅M−1, if l=2,11;Kl/Kl−1≅M0, if l=0,9,10,15;Kl/Kl−1≅M1, if l=4,12;Kl/Kl−1≅Mm−2, if l=3;Kl/Kl−1≅Mm−1, if l=1,14;Kl/Kl−1≅Mm, if l=5,7;Kl/Kl−1≅Mm+1, if l=6,13;Kl/Kl−1≅Mm+2, if l=8. |
We claim that κ15∉M and κ15∉N. If κ15∈M, then κ11−κ2,κ4−κ12,κ8,κ13,κ14∈M since
x1⋅κ15=κ14,x2⋅κ15=κ11−κ22,ρ(κ15)=4x1x2⊗κ8+2x2⊗(κ4−κ12)+2x1⊗κ13+1⊗κ15, |
which implies that
κ0=x1x2⋅κ8∈M,κ1=x1x2⋅(κ4−κ12)∈M,κ2=x2x1⋅κ13∈M,κ3=−x2⋅κ14∈M,κ4=x1⋅κ8∈M,κ5=x1⋅(κ4−κ12)∈M,κ6=x2⋅κ8∈M,κ7=−x2⋅κ13∈M,κ9=x1⋅κ13+x2⋅(κ4−κ12)∈M,κ10=x1x2⋅κ8+x2⋅(κ4−κ12)−x1⋅κ13∈M,κ11=x2x1⋅κ13+2x2⋅κ15∈M,κ12=x1⋅κ8+κ12−κ4∈M. |
Then M=P. It's a contradiction. Similarly, if κ15∈N, then N=P and also a contradiction. Hence the claim follows. Therefore, there exist some αi∈K, for i∈{0,1,⋯,14}, such that
κ=14∑i=0αiκi+κ15∈M. |
Then
ρ(κ)=(α0+(2α1+α14)x1zm+2α2x2zm+4α3x1x2)⊗κ0+(α1zm+2α3x2)⊗κ1+(α2zm−2α3x1)⊗κ2+α31⊗κ3+(α4zm+2α5x1+2(α10−α9+α15)x2+4α14x1x2zm)⊗κ4+(α5+2α14x2zm)⊗κ5+(α6zm+2α7x2+2(α9+α10)x1+8α11x1x2zm)⊗κ6+(α7−4α11x1zm)⊗κ7+(α8−2α12x1zm−2α13x2zm+4α15x1x2)⊗κ8+(α9+2α11x2zm+α14x1zm)⊗κ9+(α10+2α11x2zm−α14x1zm)⊗κ10+α11zm⊗κ11+(α12zm−2α15x2)⊗κ12+(α13zm+2α15x1)⊗κ13+α14zm⊗κ14+1⊗κ15. |
It is observe that ˆκ1=α0κ0+α3κ3+α5κ5+α7κ7+α8κ8+α9κ9+α10κ10+κ15∈M. Hence
x1⋅ˆκ1=(α9+α10)κ1+α7κ2+α8κ4+κ14∈M,x2⋅ˆκ1=−α5κ1+(α9−α10−12)⊗κ2+α8κ6+κ112∈M,x1x2⋅ˆκ1=α8κ0+κ32∈M, |
and
ρ(α8κ0+κ32)=(α8+2x1x2)⊗κ0+x2⊗κ1−x1⊗κ2+121⊗κ3. |
Then κ0,κ1,κ2,κ3 and α8κ4+κ14,α8κ6+κ112∈M. Note that
ρ(α8κ4+κ14)=(α8zm+4x1x2zm)⊗κ4+2x2zm⊗κ5+x1zm⊗(κ0+κ9−κ10)+zm⊗κ14,ρ(α8κ6+κ112)=(α8+4x1x2zm)⊗κ6−2x1zm⊗κ7+x2zm⊗(κ9+κ10)+zm2⊗κ11. |
Then κ4,κ5,κ6,κ7,κ9,κ10,κ11,κ14∈M. Hence ˆκ2=α8κ8+α12κ12+α13κ13+κ15∈M, and
ρ(ˆκ2)=2x2⊗κ4+(α8−2α12x1zm−2α13x2zm+4x1x2)⊗κ8+(α12zm−2x2)⊗κ12+(α13zm+2x1)⊗κ13+1⊗κ15. |
Thus κ15∈M. It's a contradiction. Consequently, P is indecomposable.
Lemma 4.4. P≅P0 as D-modules.
Proof. It is well known that D is a symmetric algebra [26] and every projective module is injective. In particular, P0=E(Mi) for some i∈Z2m and the socle and top of P0 coincide. Therefore P0≅E(M0). On the other hand, by Lemma 4.3, we know that P is an indecomposable module with SocP≅M0. Thus, P embeds in E(M0), which implies that P≅E(M0), since they have the same dimension. Hence, P≅P0.
Now we calculate the tensor decompositions of the simple and indecomposable projective Yetter-Drinfeld modules. Denote ⨁±Va±b:=Va+b⊕Va−b, for a,b∈Z2m,V∈H8mH8mYD.
Lemma 4.5. (1) For i,j∈Z2m,
Pi⊗Pj≅⨁±(P⊕2i+j±1⊕P⊕2i+j+m±1⊕Pi+j+m±2)⊕P⊕4i+j⊕P⊕2i+j+m. |
(2) For i,j∈Λ,k∈Z2m,
Ni,j⊗Pk≅Pk⊗Ni,j≅⨁±(N⊕2i+k±1,j+km+m⊕N⊕2i+k+m±1,j+km+m⊕Ni+k+m±2,j+km)⊕N⊕4i+k,j+km⊕N⊕2i+k+m,j+km. |
Proof. It suffices to prove the lemma for P0⊗P0 and P0⊗Ni,j by Lemma 4.1 (1)(3).
By the proof of Lemma 4.3,
[P0]=2[M1]+2[M−1]+2[Mm+1]+2[Mm−1]+[Mm+2]+[Mm−2]+4[M0]+2[Mm] |
in the Grothendieck ring of the category of the Yetter-Drifeld modules over H8m. It follows that
P0⊗P0≅⨁±((M±1⊗P0)⊕2⊕(Mm±1⊗P0)⊕2⊕(Mm±2⊗P0))⊕(M0⊗P0)⊕4⊕(Mm⊗P0)⊕2≅⨁±(P⊕2±1⊕P⊕2m±1⊕Pm±2)⊕P⊕40⊕P⊕2m, |
and
P0⊗Ni,j≅⨁±((M±1⊗Ni,j)⊕2⊕(Mm±1⊗Ni,j)⊕2⊕(Mm±2⊗Ni,j))⊕(M0⊗Ni,j)⊕4⊕(Mm⊗Ni,j)⊕2≅⨁±(N⊕2i±1,j+m⊕N⊕2i+m±1,j+m⊕Ni+m±2,j)⊕N⊕4i,j⊕N⊕2i+m,j, |
The proof is finished.
Lemma 4.6. For (i,j),(k,l)∈Λ, we have
Ni,j⊗Nk,l≅{Pi+m+k−2,ifj≡(i+k)m−l(mod2m);1⨁t=0(Ni+k+(m−1)t,j+l+mt⊕Ni+k+(m−1)t−1,j+l+(t+1)m),otherwise. |
Proof. If j≡(i+k)m−l(mod2m), then by Lemma 4.1(4), Hom(Ni,j⊗Nk,l,Mh)≠0 if and only if h≡i+m+k−2(mod2m). Since Ni,j⊗Nk,l is projective and dimPi+m+k−2=dim(Ni,j⊗Nk,l)=16, we get that Pi+m+k−2≅Ni,j⊗Nk,l.
If j≢(i+k)m−l(mod2m), then by Lemma 4.1(4), Hom(Ni,j⊗Nk,l,Mh)=0 for any h∈Z2m, which implies that Ph can not the direct summand of Ni,j⊗Nk,l. Hence Ni,j⊗Nk,l has to be the direct sum of four 4-dimensional simple projective modules.
Let
t=(−1)kbi,jbk,l1−ω−2,θr,s=μi,jr⊗νk,ls |
for r,s∈{0,1,2,3} and
β0,0=ai+k,j+lbi+k,j+l(θ0,0+tθ2,2),β0,1=bi+k,j+l(ak,l(θ0,1+tθ2,3)+(−1)kai,j(θ1,0−tθ3,2)),β0,2=ai+k,j+l(bk,lθ0,2+(−1)kbi,jθ2,0),β0,3=ak,lbk,lθ0,3+ai,jbi,jθ3,0+(−1)k−1ai,jbk,lθ1,2+(−1)kak,lbi,jθ2,1,β1,0=bi+k+m−1,j+l+m(θ0,1+tθ2,3−(−1)jω−j(θ1,0−tθ3,2)),β1,1=bi+k+m−1,j+l+m(θ1,1−tθ3,3),β1,2=(−1)kbi,j((−1)jω−jθ3,0−θ2,1)−bk,l(θ0,3+(−1)jω−jθ1,2),β1,3=bk,lθ1,3−(−1)kbi,jθ3,1,β2,0=ai+k−1,j+l+m(θ0,2−ω−jθ2,0),β2,1=ak,l(θ0,3−ω−jθ2,1)−(−1)kai,j(θ1,3+ω−jθ3,0),β2,2=ai+k−1,j+l+mθ2,2,β2,3=ak,lθ2,3−(−1)kai,jθ3,2,β3,0=θ0,3−ω−jθ2,1+(−1)jω−j(θ1,2+ω−jθ3,0),β3,1=θ1,3+ω−jθ3,1,β3,2=θ2,3+(−1)jω−jθ3,2,β3,3=θ3,3. |
A direct computation shows that
K{β0,0,β0,1,β0,2,β0,3}≅Ni+k,j+l,K{β1,0,β1,1,β1,2,β1,3}≅Ni+k+m−1,j+l+m,K{β2,0,β2,1,β2,2,β2,3}≅Ni+k−1,j+l+m,K{β3,0,β3,1,β3,2,β3,3}≅Ni+k+m−2,j+l. |
Hence,
Ni,j⊗Nk,l≅Ni+k,j+l⊕Ni+k+m−1,j+l+m⊕Ni+k−1,j+l+m⊕Ni+k+m−2,j+l. |
By Lemma 4.1, Lemma 4.5, Lemma 4.6, the projective class ring rP(D) is a commutative ring. Let ˙y0=[M1],˙yi=[N0,i],i∈{1,⋯,2m−1}
Lemma 4.7. The following statements hold in rP(D).
(1) [Mi]=˙yi0 for i∈Z2m;
(2) [Ni,j]=˙yi0˙yj+im for (i,j)∈Λ;
(3) [Pi]=˙yi+m+20˙y2m for i∈Z2m;
(4) For i,j,k,j+k∈{1,⋯,2m−1},
˙y2m0=1,˙yi˙y2m−i=˙y2m,˙yj˙yk=(1+˙y2m−10+˙ym−10+˙ym−20)˙yj+k,˙yi˙y2m=(1+2˙y2m−10+2˙y2m−20+2˙y2m−30+2˙ym−10+4˙ym−20+2˙ym−30+˙ym−40)˙yi. |
Proof. The results are easy to get from Lemma 4.1, Lemma 4.5, Lemma 4.6.
Corollary 4.8. The following set is a Z-basis of rP(D):
{˙yi0,˙yi0˙yj,˙yi0˙y2m|i∈Z2m,j∈{1,2,⋯2m−1}}. |
Proof. By Lemma 4.7(4), ˙y2m0=1, and for k,l,k+l∈{1,⋯,2m−1}, ˙yk˙y2m,˙yk˙yl can be expressed as a linear combination of {˙yi0,˙yi0˙yj,˙yi0˙y2m|i∈Z2m,j∈{1,2,⋯2m−1}}. It is easy to check that the set
{˙yi0,˙yi0˙yj,˙yi0˙y2m|i∈Z2m,j∈{1,2,⋯2m−1}} |
is a independent set since #{˙yi0,˙yi0˙yj,˙yi0˙y2m|i∈Z2m,j∈{1,2,⋯2m−1}}=4m2+2m, the number of Z-basis of rP(D).
Hence, {˙yi0,˙yi0˙yj,˙yi0˙y2m|i∈Z2m,j∈{1,2,⋯2m−1}} is a Z-basis of rP(D).
The results of this section is as follows.
Theorem 4.9. The projective class ring rP(D) is isomorphic to the quotient ring of the ring Z[y0,y1,⋯,y2m−1] modulo the ideal I generated by the following elements
y2m0−1,yiy2m−i−y2m,yjyk−(1+y2m−10+ym−10+ym−20)yj+k,yiy2m−(1+2y2m−10+2y2m−20+2y2m−30+2ym−10+4ym−20+2ym−30+ym−40)yi, | (4.1) |
where i,j,k,j+k∈{1,⋯,2m−1}.
Proof. By Corollary 4.8, there is a unique ring epimorphism
Φ:Z[y0,y1,⋯,y2m−1]→rP(H8m) |
such that Φ(yi)=˙yi for i∈Z2m. By Lemma 4.7(4), we have
Φ(y2m0−1)=Φ(y0)2m−1=˙y2m0−1=0,Φ(yiy2m−i−y2m)=Φ(yi)Φ(y2m−i)−Φ(ym)2=˙yi˙y2m−i−˙y2m=0,Φ(yjyk−(1+y2m−10+ym−10+ym−20)yj+k)=Φ(yj)Φ(yk)−(1+Φ(y0)2m−1+Φ(y0)m−1+Φ(y0)m−2)Φ(yj+k)=˙yj˙yk−(1+˙y2m−10+˙ym−10+˙ym−20)˙yj+k=0,Φ(yiy2m−(1+2y2m−10+2y2m−20+2y2m−30+2ym−10+4ym−20+2ym−30+ym−40)yi)=˙yi˙y2m−(1+2˙y2m−10+2˙y2m−20+2˙y2m−30+2˙ym−10+4˙ym−20+2˙ym−30+˙ym−40)˙yi=0. |
Let I is the ideal generated by the elements (4.1). It follows that, Φ(I)=0 and Φ induces a natural ring epimorphism
¯Φ:Z[y0,y1,⋯,y2m−1]/I→rP(H8m) |
such that ¯Φ(¯ν)=Φ(ν) for all ν∈Z[y0,y1,⋯,y2m−1], where ¯ν=ν+I.
It is straightforward to check that the ring Z[y0,y1,⋯,y2m−1]/I is Z-spanned by
{yi0,yi0yj,yi0y2m|i∈Z2m,j∈{1,2,⋯2m−1}}. |
This means the Z-rank of Z[y0,y1,⋯,y2m−1]/I is 4m2+2m. Hence we get the ring isomorphism ¯Φ.
It's a challenging work to classify all indecomposable modules or indecomposable Yetter-Drinfeld modules over the finite dimensional Hopf algebras over an algebraically closed field of characteristic p>0, and to classify all finite dimensional (Nichols) Hopf algebras over H8m over any field. This is our future attempt.
The work is supported by National Natural Science Foundation of China (Grant No. 11671024). The authors are grateful to thank the referee for careful reading and helpful suggestions.
The authors declare that they have no competing interests.
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