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Research article Special Issues

Representations of a non-pointed Hopf algebra

  • Received: 21 May 2021 Accepted: 13 July 2021 Published: 20 July 2021
  • MSC : 16D70, 16G60, 16T05

  • In this paper, we construct all the indecomposable modules of a class of non-pointed Hopf algebras, which are quotient Hopf algebras of a class of prime Hopf algebras of GK-dimension one. Then the decomposition formulas of the tensor product of any two indecomposable modules are established. Based on these results, the representation ring of the Hopf algebras is characterized by generators and some relations.

    Citation: Ruifang Yang, Shilin Yang. Representations of a non-pointed Hopf algebra[J]. AIMS Mathematics, 2021, 6(10): 10523-10539. doi: 10.3934/math.2021611

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  • In this paper, we construct all the indecomposable modules of a class of non-pointed Hopf algebras, which are quotient Hopf algebras of a class of prime Hopf algebras of GK-dimension one. Then the decomposition formulas of the tensor product of any two indecomposable modules are established. Based on these results, the representation ring of the Hopf algebras is characterized by generators and some relations.



    The radial addition K˜+L of star sets K and L can be defined by

    ρ(K˜+L,)=ρ(K,)+ρ(L,),

    where a star set is a compact set that is star-shaped at o and contains o and ρ(K,) denotes the radial function of star set K. The radial function is defined by

    ρ(K,u)=max{c0:cuK}, (1.1)

    for uSn1, where Sn1 denotes the surface of the unit ball centered at the origin. The initial study of the radial addition can be found in [1, p. 235]. K is called a star body if ρ(K,) is positive and continuous, and let Sn denote the set of star bodies. The radial addition and volume are the basis and core of the dual Brunn-Minkowski theory (see, e.g., [2,3,4,5,6,7,8,9,10]). It is important that the dual Brunn-Minkowski theory can count among its successes the solution of the Busemann-Petty problem in [3,11,12,13,14]. Recently, it has turned to a study extending from Lp-dual Brunn-Minkowski theory to Orlicz dual Brunn-Minkowski theory. The Orlicz dual Brunn-Minkowski theory and its dual have attracted people's attention [15,16,17,18,19,20,21,22,23,24,25,26,27,28].

    For KSn and uSn1, the half chord of K in the direction u is defined by

    d(K,u)=12(ρ(K,u)+ρ(K,u)).

    If there exists a constant λ>0 such that d(K,u)=λd(L,u), for all uSn1, then star bodies K,L are said to have similar chord (see Gardner [1] or Schneider [29]). Lu [30] introduced the i-th chord integral of star bodies: For KSn and 0i<n, the i-th chord integral of K, is denoted by Bi(K), is defined by

    Bi(K)=1nSn1d(K,u)nidS(u). (1.2)

    Obviously, for i=0, Bi(K) becomes the chord integral B(K).

    The main aim of the present article is to generalize the chord integrals to Orlicz space. We introduce a new affine geometric quantity which we shall call Orlicz mixed chord integrals. The fundamental notions and conclusions of the chord integral and related isoperimetric inequalities for the chord integral are extended to an Orlicz setting. The new inequalities in special cases yield the Lp-dual Minkowski and Lp-dual Brunn-Minkowski inequalities for the Lp-mixed chord integrals. The related concepts and inequalities of Lp-mixed chord integrals are derived. As extensions, Orlicz multiple mixed chord integrals and Orlicz-Aleksandrov-Fenchel inequality for the Orlicz multiple mixed chord integrals are also derived.

    In Section 3, we introduce the following new notion of Orlicz chord addition of star bodies.

    Orlicz chord addition Let K and L be star bodies, the Orlicz chord addition of K and L, is denoted by Kˇ+ϕL, is defined by

    ϕ(d(K,u)d(Kˇ+ϕL,u),d(L,u)d(Kˇ+ϕL,u))=1, (1.3)

    where uSn1, and ϕΦ2, which is the set of convex functions ϕ:[0,)2(0,) that are decreasing in each variable and satisfy ϕ(0,0)= and ϕ(,1)=ϕ(1,)=1.

    The particular instance of interest corresponds to using (1.3) with ϕ(x1,x2)=ϕ1(x1)+εϕ2(x2) for ε>0 and some ϕ1,ϕ2Φ, which are the sets of convex functions ϕ1,ϕ2:[0,)(0,) that are decreasing and satisfy ϕ1(0)=ϕ2(0)=, ϕ1()=ϕ2()=0 and ϕ1(1)=ϕ2(1)=1.

    In accordance with the spirit of Aleksandrov [31], Fenchel and Jessen's [32] introduction of mixed quermassintegrals, and introduction of Lutwak's [33] Lp-mixed quermassintegrals, we are based on the study of first-order variations of the chord integrals. In Section 4, we prove that the first order Orlicz variation of the mixed chord integral can be expressed as: For K,LSn, ϕ1,ϕ2Φ, 0i<n and ε>0,

    ddε|ε=0+Bi(Kˇ+ϕεL)=(ni)1(ϕ1)r(1)Bϕ2,i(K,L), (1.4)

    where (ϕ1)r(1) denotes the value of the right derivative of convex function ϕ1 at point 1. In this first order variational equation (1.4), we find a new geometric quantity. Based on this, we extract the required geometric quantity, denoted by Bϕ,i(K,L) which we shall call Orlicz mixed chord integrals of K and L, as follows

    Bϕ2,i(K,L)=1ni(ϕ1)r(1)ddε|ε=0+Bi(Kˇ+ϕεL). (1.5)

    We show also that the new affine geometric quantity has an integral representation as follows:

    Bϕ,i(K,L)=1nSn1ϕ(d(L,u)d(K,u))d(K,u)nidS(u). (1.6)

    When ϕ(t)=tp and p1, the new affine geometric quantity becomes a new Lp-mixed chord integrals of K and L, denoted by Bp,i(K,L), which as is in (2.7).

    In Section 5, we establish an Orlicz Minkowski inequality for the mixed chord and Orlicz mixed chord integrals.

    Orlicz Minkowski inequality for the Orlicz mixed chord integrals If K,LSn, 0i<n and ϕΦ, then

    Bϕ,i(K,L)Bi(K)ϕ((Bi(L)Bi(K))1/(ni)). (1.7)

    If ϕ is strictly convex, the equality holds if and only if K and L are similar chord.

    When ϕ(t)=tp and p1, (1.7) becomes a new Lp-Minkowski inequality (2.8) for the Lp-mixed chord integrals.

    In Section 6, as an application, we establish an Orlicz Brunn-Minkowski inequality for the Orlicz chord additions and the mixed chord integrals:

    Orlicz Brunn-Minkowski inequality for the Orlicz chord additions If K,LSn, 0i<n and ϕΦ2, then

    1ϕ((Bi(K)Bi(Kˇ+ϕL))1/(ni),(Bi(L)Bi(Kˇ+ϕL))1/(ni)). (1.8)

    If ϕ is strictly convex, the equality holds if and only if K and L are similar chord.

    When ϕ(t)=tp and p1, (1.8) becomes a new Lp-Brunn-Minkowski inequality (2.9) for the mixed chord integrals.

    A new isoperimetric inequality for the chord integrals is given in Section 7. In Section 8, Orlicz multiple mixed chord integrals is introduced and Orlicz-Aleksandrov-Fenchel inequality for the Orlicz multiple mixed chord integrals is established.

    The setting for this paper is n-dimensional Euclidean space Rn. A body in Rn is a compact set equal to the closure of its interior. For a compact set KRn, we write V(K) for the (n-dimensional) Lebesgue measure of K and call this the volume of K. Associated with a compact subset K of Rn which is star-shaped with respect to the origin and contains the origin, its radial function is ρ(K,):Sn1[0,) is defined by

    ρ(K,u)=max{λ0:λuK}.

    Note that the class (star sets) is closed under union, intersection, and intersection with subspace. The radial function is homogeneous of degree 1, that is (see e.g. [1]),

    ρ(K,ru)=r1ρ(K,u),

    for all uSn1 and r>0. Let ˜δ denote the radial Hausdorff metric, as follows: if K,LSn, then

    ˜δ(K,L)=|ρ(K,u)ρ(L,u)|.

    From the definition of the radial function, it follows immediately that for AGL(n) the radial function of the image AK={Ay:yK} of K is given by (see e.g. [29])

    ρ(AK,x)=ρ(K,A1x), (2.1)

    for all xRn.

    For KiSn,i=1,,m, define the real numbers RKi and rKi by

    RKi=maxuSn1d(Ki,u),andrKi=minuSn1d(Ki,u). (2.2)

    Obviously, 0<rKi<RKi, for all KiSn. Writing R=max{RKi} and r=min{rKi}, where i=1,,m.

    If K1,,KnSn, the mixed chord integral of K1,,Kn, is denoted by B(K1,,Kn), is defined by (see [30])

    B(K1,,Kn)=1nSn1d(K1,u)d(Kn,u)dS(u).

    If K1==Kni=K, Kni+1==Kn=L, the mixed chord integral B(K1,,Kn) is written as Bi(K,L). If L=B (B is the unit ball centered at the origin), the mixed chord integral Bi(K,L)=Bi(K,B) is written as Bi(K) and called the i-th chord integral of K. Obviously, For KSn and 0i<n, we have

    Bi(K)=1nSn1d(K,u)nidS(u). (2.3)

    If K1==Kni1=K, Kni==Kn1=B and Kn=L, the mixed chord integral B(K,,Kni1,B,,Bi,L) is written as Bi(K,L) and called the i-th mixed chord integral of K and L. For K,LSn and 0i<n, it is easy to see that

    Bi(K,L)=1nSn1d(K,u)ni1d(L,u)dS(u). (2.4)

    This integral representation (2.4), together with the Hölder inequality, immediately give the Minkowski inequality for the i-th mixed chord integral: If K,LSn and 0i<n, then

    Bi(K,L)niBi(K)ni1Bi(L), (2.5)

    with equality if and only if K and L are similar chord.

    Definition 2.1 (The Lp-chord addition) Let K,LSn and p1, the Lp chord addition ˇ+p of star bodies K and L, is defined by

    d(Kˇ+pL,u)p=d(K,u)p+d(L,u)p, (2.6)

    for uSn1.

    Obviously, putting ϕ(x1,x2)=xp1+xp2 and p1 in (1.3), (1.3) becomes (2.6). The following result follows immediately from (2.6) with p1.

    npnilimε0+Bi(Kˇ+pεL)Bi(L)ε=1nSn1d(K,u)ni+pd(L,u)pdS(u).

    Definition 2.2 (The Lp-mixed chord integrals) Let K,LSn, 0i<n and p1, the Lp-mixed chord integral of star K and L, denoted by Bp,i(K,L), is defined by

    Bp,i(K,L)=1nSn1d(K,u)ni+pd(L,u)pdS(u). (2.7)

    Obviously, when K=L, the Lp-mixed chord integral Bp,i(K,K) becomes the i-th chord integral Bi(K). This integral representation (2.7), together with the Hölder inequality, immediately gives:

    Proposition 2.3 If K,LSn, 0i<n and p1, then

    Bp,i(K,L)niBi(K)ni+pBi(L)p, (2.8)

    with equality if and only if K and L are similar chord.

    Proposition 2.4 If K,LSn, 0i<n and p1, then

    Bi(Kˇ+pL)p/(ni)Bi(K)p/(ni)+Bi(L)p/(ni), (2.9)

    with equality if and only if K and L are similar chord.

    Proof From (2.6) and (2.7), it is easily seen that the Lp-chord integrals is linear with respect to the Lp-chord addition, and together with inequality (2.8), we have for p1

    Bp,i(Q,Kˇ+pL)=Bp,i(Q,K)+Bp,i(Q,L)Bi(Q)(ni+p)/(ni)(Bi(K)p/(ni)+Bi(L)p/(ni)),

    with equality if and only if K and L are similar chord.

    Take Kˇ+pL for Q, recall that Bp,i(Q,Q)=Bi(Q), inequality (2.9) follows easily.

    Throughout this paper, the standard orthonormal basis for Rn will be {e1,,en}. Let Φn, nN, denote the set of convex functions ϕ:[0,)n(0,) that are strictly decreasing in each variable and satisfy ϕ(0)= and ϕ(ej)=1, j=1,,n. When n=1, we shall write Φ instead of Φ1. The left derivative and right derivative of a real-valued function f are denoted by (f)l and (f)r, respectively. We first define the Orlicz chord addition.

    Definition 3.1 (The Orlicz chord addition) Let m2,ϕΦm, KjSn and j=1,,m, the Orlicz chord addition of K1,,Km, is denoted by ˇ+ϕ(K1,,Km), is defined by

    d(ˇ+ϕ(K1,,Km),u)=sup{λ>0:ϕ(d(K1,u)λ,,d(Km,u)λ)1}, (3.1)

    for uSn1. Equivalently, the Orlicz chord addition ˇ+ϕ(K1,,Km) can be defined implicitly by

    ϕ(d(K1,u)d(ˇ+ϕ(K1,,Km),u),,d(Km,u)d(ˇ+ϕ(K1,,Km),u))=1, (3.2)

    for all uSn1.

    An important special case is obtained when

    ϕ(x1,,xm)=mj=1ϕj(xj),

    for some fixed ϕjΦ such that ϕ1(1)==ϕm(1)=1. We then write ˇ+ϕ(K1,,Km)=K1ˇ+ϕˇ+ϕKm. This means that K1ˇ+ϕˇ+ϕKm is defined either by

    d(K1ˇ+ϕˇ+ϕKm,u)=sup{λ>0:mj=1ϕj(d(Kj,u)λ)1}, (3.3)

    for all uSn1, or by the corresponding special case of (3.2).

    Lemma 3.2 The Orlicz chord addition ˇ+ϕ:(Sn)mSn is monotonic.

    Proof This follows immediately from (3.1).

    Lemma 3.3 The Orlicz chord addition ˇ+ϕ:(Sn)mSn is GL(n) covariant.

    Proof From (2.1), (3.1) and let AGL(n), we obtain

    d(ˇ+ϕ(AK1,AK2,AKm),u)
    =sup{λ>0:ϕ(d(AK1,u)λ,d(AK2,u)λ,,d(AKm,u)λ)1}=sup{λ>0:ϕ(d(K1,A1u)λ,d(K2,A1u)λ,,d(Km,A1u)λ)1}=d(ˇ+ϕ(K1,,Km),A1u)=d(ˇ+ϕ(K1,,Km),u).

    This shows Orlicz chord addition ˇ+ϕ is GL(n) covariant.

    Lemma 3.4 Suppose K1,,KmSn. If ϕΦ, then

    ϕ(d(K1,u)t)++ϕ(d(Km,u)t)=1

    if and only if

    d(ˇ+ϕ(K1,,Km),u)=t

    Proof This follows immediately from Definition 3.1.

    Lemma 3.5 Suppose Km,,KmSn. If ϕΦ, then

    rϕ1(1m)d(ˇ+ϕ(K1,,Km),u)Rϕ1(1m).

    Proof Suppose d(ˇ+ϕ(K1,,Km),u)=t, from Lemma 3.4 and noting that ϕ is strictly deceasing on (0,), we have

    1=ϕ(d(K1,u)t)++ϕ(d(Km,u)t)ϕ(rK1t)++ϕ(rKmt)=mϕ(rt).

    Noting that the inverse ϕ1 is strictly deceasing on (0,), we obtain the lower bound for d(ˇ+ϕ(K1,,Km),u):

    trϕ1(1m).

    To obtain the upper estimate, observe the fact from the Lemma 3.4, together with the convexity and the fact ϕ is strictly deceasing on (0,), we have

    1=ϕ(d(K1,u)t)++ϕ(d(Km,u)t)mϕ(d(K1,u)++d(Km,u)mt)mϕ(Rt).

    Then we obtain the upper estimate:

    tRϕ1(1m).

    Lemma 3.6 The Orlicz chord addition ˇ+ϕ:(Sn)mSn is continuous.

    Proof To see this, indeed, let KijSn, iN{0}, j=1,,m, be such that KijK0j as i. Let

    d(ˇ+ϕ(Ki1,,Kim),u)=ti.

    Then Lemma 3.5 shows

    rijϕ1(1m)tiRijϕ1(1m),

    where rij=min{rKij} and Rij=max{RKij}. Since KijK0j, we have RKijRK0j< and rKijrK0j>0, and thus there exist a,b such that 0<atib< for all i. To show that the bounded sequence {ti} converges to d(ˇ+ϕ(K01,,K0m),u), we show that every convergent subsequence of {ti} converges to d(ˇ+ϕ(K01,,K0m),u). Denote any subsequence of {ti} by {ti} as well, and suppose that for this subsequence, we have

    tit.

    Obviously atb. Noting that ϕ is a continuous function, we obtain

    tsup{t>0:ϕ(d(K01,u)t,,d(K0m,u)t)1}
    =d(ˇ+ϕ(K01,,K0m),u).

    Hence

    d(ˇ+ϕ(Ki1,,Kim),u)d(ˇ+ϕ(K01,,K0m),u)

    as i.

    This shows that the Orlicz chord addition ˇ+ϕ:(Sn)mSn is continuous.

    Next, we define the Orlicz chord linear combination for the case m=2.

    Definition 3.7 (The Orlicz chord linear combination) The Orlicz chord linear combination, denoted by ˇ+ϕ(K,L,α,β) for K,LSn, and α,β0 (not both zero), is defined by

    αϕ1(d(K,u)d(ˇ+ϕ(K,L,α,β),u))+βϕ2(d(L,u)d(ˇ+ϕ(K,L,α,β),u))=1, (3.4)

    for ϕ1,ϕ2Φ and all uSn1.

    We shall write Kˇ+ϕεL instead of ˇ+ϕ(K,L,1,ε), for ε0 and assume throughout that this is defined by (3.1), if α=1,β=ε and ϕΦ. We shall write Kˇ+ϕL instead of ˇ+ϕ(K,L,1,1) and call it the Orlicz chord addition of K and L.

    In order to define Orlicz mixed chord integrals, we need the following Lemmas 4.1-4.4.

    Lemma 4.1 Let ϕΦ and ε>0. If K,LSn, then Kˇ+ϕεLSn.

    Proof Let u0Sn1, and {ui}Sn1 be any subsequence such that uiu0 as i.

    Let

    d(Kˇ+ϕL,ui)=λi.

    Then Lemma 3.5 shows

    rϕ1(12)λiRϕ1(12),

    where R=max{RK,RL} and r=min{rK,rL}.

    Since K,LSn, we have 0<rKRK< and 0<rLRL<, and thus there exist a,b such that 0<aλib< for all i. To show that the bounded sequence {λi} converges to d(Kˇ+ϕεL,u0), we show that every convergent subsequence of {λi} converges to d(Kˇ+ϕεL,u0). Denote any subsequence of {λi} by {λi} as well, and suppose that for this subsequence, we have

    λiλ0.

    Obviously aλ0b. From (3.4) and note that ϕ1,ϕ2 are continuous functions, so ϕ11 is continuous, we obtain

    λid(K,u0)ϕ11(1εϕ2(d(L,u0)λ0))

    as i. Hence

    ϕ1(d(K,u0)λ0)+εϕ2(d(L,u0)λ0)=1.

    Therefore

    λ0=d(Kˇ+ϕεL,u0).

    That is

    d(Kˇ+ϕεL,ui)d(Kˇ+ϕεL,u0).

    as i.

    This shows that Kˇ+ϕεLSn.

    Lemma 4.2 If K,LSn, ε>0 and ϕΦ, then

    Kˇ+ϕεLK (4.1)

    as ε0+.

    Proof This follows immediately from (3.4).

    Lemma 4.3 If K,LSn, 0i<n and ϕ1,ϕ2Φ, then

    ddε|ε=0+d(Kˇ+ϕεL,u)ni=ni(ϕ1)r(1)ϕ2(d(L,u)d(K,u))d(K,u)ni. (4.2)

    Proof From (3.4), Lemma 4.2 and notice that ϕ11, ϕ2 are continuous functions, we obtain for 0i<n

    ddε|ε=0+d(Kˇ+ϕεL,u)ni
    =limε0+(ni)d(K,u)ni1(d(K,u)ϕ2(d(L,u)d(Kˇ+ϕεL,u)))×limy1ϕ11(y)ϕ11(1)y1=ni(ϕ1)r(1)ϕ2(d(L,u)d(K,u))d(K,u)ni,

    where

    y=1εϕ2(d(L,u)d(Kˇ+ϕεL,u)),

    and note that y1 as ε0+.

    Lemma 4.4 If ϕΦ2, 0i<n and K,LSn, then

    (ϕ1)r(1)niddε|ε=0+Bi(Kˇ+ϕεL)=1nSn1ϕ2(d(L,u)d(K,u))d(K,u)nidS(u). (4.3)

    Proof This follows immediately from (2.1) and Lemma 4.3.

    Denoting by Bϕ,i(K,L), for any ϕΦ and 0i<n, the integral on the right-hand side of (4.3) with ϕ2 replaced by ϕ, we see that either side of the equation (4.3) is equal to Bϕ2,i(K,L) and hence this new Orlicz mixed chord integrals Bϕ,i(K,L) has been born.

    Definition 4.5 (The Orlicz mixed chord integral) For ϕΦ and 0i<n, Orlicz mixed chord integral of star bodies K and L, Bϕ,i(K,L), is defined by

    Bϕ,i(K,L)=:1nSn1ϕ(d(L,u)d(K,u))d(K,u)nidS(u). (4.4)

    Lemma 4.6 If ϕ1,ϕ2Φ, 0i<n and K,LSn, then

    Bϕ2,i(K,L)=(ϕ1)r(1)nilimε0+Bi(Kˇ+ϕεL)Bi(K)ε. (4.5)

    Proof This follows immediately from Lemma 4.4 and (4.4).

    Lemma 4.7 If K,LSn, ϕΦ and any ASL(n), then for ε>0

    A(Kˇ+ϕεL)=(AK)ˇ+ϕε(AL). (4.6)

    Proof This follows immediately from (2.1) and (3.3).

    We find easily that Bϕ,i(K,L) is invariant under simultaneous unimodular centro-affine transformation.

    Lemma 4.8 If ϕΦ, 0i<n and K,LSn, then for ASL(n),

    Bϕ,i(AK,AL)=Bϕ,i(K,L). (4.7)

    Proof This follows immediately from Lemmas 4.6 and 4.7.

    In this section, we will define a Borel measure in Sn1, denoted by Bn,i(K,υ), which we shall call the chord measure of star body K.

    Definition 5.1 (The chord measure) Let KSn and 0i<n, the chord measure of star body K, denoted by Bn,i(K,υ), is defined by

    dBn,i(K,υ)=1nd(K,υ)niBi(K)dS(υ). (5.1)

    Lemma 5.2 (Jensen's inequality) Let μ be a probability measure on a space X and g:XIR be a μ-integrable function, where I is a possibly infinite interval. If ψ:IR is a convex function, then

    Xψ(g(x))dμ(x)ψ(Xg(x)dμ(x)). (5.2)

    If ψ is strictly convex, the equality holds if and only if g(x) is constant for μ-almost all xX (see [34, p. 165]).

    Lemma 5.3 Suppose that ϕ:[0,)(0,) is decreasing and convex with ϕ(0)=. If K,LSn and 0i<n, then

    1nBi(K)Sn1ϕ(d(L,u)d(K,u))d(K,u)nidS(u)ϕ((Bi(L)Bi(K))1/(ni)). (5.3)

    If ϕ is strictly convex, the equality holds if and only if K and L are similar chord.

    Proof For KSn1, 0i<n and any uSn1, the chord measure d(K,u)ninBi(K)dS(u) is a probability measure on Sn1. Hence, from (2.4), (2.5), (5.1) and by using Jensen's inequality, and in view of ϕ is decreasing, we obtain

    1nBi(K)Sn1ϕ(d(L,u)d(K,u))d(K,u)nidS(u)
    =Sn1ϕ(d(L,u)d(K,u))dBn,i(K,u)ϕ(Bi(K,L)Bi(K))ϕ((Bi(L)Bi(K))1/(ni)).

    Next, we discuss the equality in (5.3). If ϕ is strictly convex, suppose the equality holds in (5.3), form the equality necessary conditions of Jensen's inequality and (2.5), it follows that d(L,u)/d(K,u) is constant, and K and L are similar chord, respectively. This yields that there exists r>0 such that d(L,u)=rd(K,u), for all uSn1. On the other hand, suppose that K and L are similar chord, i.e. there exists λ>0 such that d(L,u)=λd(K,u) for all uSn1. Hence

    1nBi(K)Sn1ϕ(d(L,u)d(K,u))d(K,u)nidS(u)
    =1nBi(K)Sn1ϕ((Bi(L)Bi(K))1/(ni))d(K,u)nidS(u)=ϕ((Bi(L)Bi(K))1/(ni)).

    This implies the equality in (5.3) holds.

    Theorem 5.4 (Orlicz chord Minkowski inequality) If K,LSn, 0i<n and ϕΦ, then

    Bϕ,i(K,L)Bi(K)ϕ((Bi(L)Bi(K))1/(ni)). (5.4)

    If ϕ is strictly convex, the equality holds if and only if K and L are similar chord.

    Proof This follows immediately from (4.4) and Lemma 5.3.

    Corollary 5.5 If K,LSn, 0i<n and p1, then

    Bp,i(K,L)niBi(K)ni+pBi(L)p, (5.5)

    with equality if and only if K and L are similar chord.

    Proof This follows immediately from Theorem 5.4 with ϕ1(t)=ϕ2(t)=tp and p1.

    Taking i=0 in (5.5), this yields Lp-Minkowski inequality: If K,LSn and p1, then

    Bp(K,L)nB(K)n+pB(L)p,

    with equality if and only if K and L are similar chord.

    Corollary 5.6 Let K,LMSn, 0i<n and ϕΦ, and if either

    Bϕ,i(Q,K)=Bϕ,i(Q,L),forallQM (5.6)

    or

    Bϕ,i(K,Q)Bi(K)=Bϕ,i(L,Q)Bi(L),forallQM, (5.7)

    then K=L.

    Proof Suppose (5.6) holds. Taking K for Q, then from (2.3), (4.4) and (5.3), we obtain

    Bi(K)=Bϕ,i(K,L)Bi(K)ϕ((Bi(L)Bi(K))1/(ni))

    with equality if and only if K and L are similar chord. Hence

    Bi(K)Bi(L),

    with equality if and only if K and L are similar chord. On the other hand, if taking L for Q, by similar arguments, we get Bi(K)Bi(L), with equality if and only if K and L are similar chord. Hence Bi(K)=Bi(L), and K and L are similar chord, it follows that K and L must be equal.

    Suppose (5.7) holds. Taking L for Q, then from from (2.3), (4.4) and (5.3), we obtain

    1=Bϕ,i(K,L)Bi(K)ϕ((Bi(L)Bi(K))1/(ni)),

    with equality if and only if K and L are similar chord. Hence

    Bi(K)Bi(L),

    with equality if and only if K and L are similar chord. On the other hand, if taking K for Q, by similar arguments, we get Bi(K)Bi(L), with equality if and only if K and L are similar chord. Hence Bi(K)=Bi(L), and K and L have similar chord, it follows that K and L must be equal.

    When ϕ1(t)=ϕ2(t)=tp and p1, Corollary 5.6 becomes the following result.

    Corollary 5.7 Let K,LMSn, 0i<n and p1, and if either

    Bp,i(K,Q)=Bp,i(L,Q),forallQM

    or

    Bp,i(K,Q)Bi(K)=Bp,i(L,Q)Bi(L),forallQM,

    then K=L.

    Lemma 6.1 If K,LSn, 0i<n, and ϕ1,ϕ2Φ, then

    Bi(Kˇ+ϕL)=Bϕ1,i(Kˇ+ϕL,K)+Bϕ2,i(Kˇ+ϕL,L). (6.1)

    Proof From (3.1), (3.4) and (4.4), we have for Kˇ+ϕL=QSn

    Bϕ1,i(Q,K)+Bϕ2,i(Q,L)
    =1nSn1ϕ(d(K,u)d(Q,u),d(L,u)d(Q,u))d(Q,u)nidS(u)
    =Bi(Q). (6.2)

    The completes the proof.

    Lemma 6.2 Let K,LSn, ε>0 and ϕΦ.

    (1) If K and L are similar chord, then K and Kˇ+ϕεL are similar chord.

    (2) If K and Kˇ+ϕεL are similar chord, then K and L are similar chord.

    Proof Suppose exist a constant λ>0 such that d(L,u)=λd(K,u), we have

    ϕ(d(K,u)d(Kˇ+ϕεL,u))+εϕ(λd(K,u)d(Kˇ+ϕεL,u))=1.

    On the other hand, the exist unique constant δ>0 such that

    ϕ(d(K,u)d(δK,u))+εϕ(λd(K,u)d(δK,u))=1,

    where δ satisfies that

    ϕ(1δ)+εϕ(λδ)=1.

    This shows that d(Kˇ+ϕεL,u)=δd(K,u).

    Suppose exist a constant λ>0 such that d(Kˇ+ϕεL,u)=λd(K,u). Then

    ϕ(1λ)+εϕ(d(L,u)d(Kˇ+ϕεL,u))=1.

    This shows that

    d(L,u)d(Kˇ+ϕεL,u)

    is a constant. This yields that Kˇ+ϕεL and L are similar chord. Namely K and L are similar chord.

    Theorem 6.3 (Orlicz chord Brunn-Minkowski inequality) If K,LSn, 0i<n and ϕΦ2, then

    1ϕ((Bi(K)Bi(Kˇ+ϕL))1/(ni),(Bi(L)Bi(Kˇ+ϕL))1/(ni)). (6.3)

    If ϕ is strictly convex, the equality holds if and only if K and L are similar chord.

    Proof From (5.4) and Lemma 6.1, we have

    Bi(Kˇ+ϕL)=Bϕ1,i(Kˇ+ϕL,K)+Bϕ2,i(Kˇ+ϕL,L)Bi(Kˇ+ϕL)(ϕ1((Bi(K)Bi(Kˇ+ϕL))1/(ni))+ϕ2((Bi(L)kBi(Kˇ+ϕL))1/(ni)))=Bi(Kˇ+ϕL)ϕ((Bi(K)Bi(Kˇ+ϕL))1/(ni),(Bi(L)Bi(Kˇ+ϕL))1/(ni)).

    This is just inequality (6.3). From the equality condition of (5.4) and Lemma 6.3, it yields that if ϕ is strictly convex, equality in (6.3) holds if and only if K and L are similar chord.

    Corollary 6.4 If K,LSn, 0i<n and p1, then

    Bi(Kˇ+pL)p/(ni)Bi(K)p/(ni)+Bi(L)p/(ni), (6.4)

    with equality if and only if K and L are similar chord.

    Proof This follows immediately from Theorem 6.2 with ϕ(x1,x2)=xp1+xp2 and p1.

    Taking i=0 in (6.4), this yields the Lp-Brunn-Minkowski inequality for the chord integrals. If K,LSn and p1, then

    B(Kˇ+pL)p/nB(K)p/n+B(L)p/n,

    with equality if and only if K and L are similar chord.

    As a application, in the section, we give a new isoperimetric inequality for chord integrals. As we all know, the isoperimetric inequality for convex bodies can be stated below (see e.g. [26], p. 318).

    The isoperimetric inequality If K is convex body in Rn, then

    (V(K)V(B))n1(S(K)S(B))n, (7.1)

    with equality if and only if K is an n-ball.

    Here B is the unit ball centered at the origin, V(K) denotes the volume of K and S(K) is the surface area of K, defined by (see [26], p. 318)

    S(K)=limε0V(K+εB)V(K)ε=nV1(K,B),

    where + the usual Minkowski sum. Here, the mixed volume of convex bodies K and L, V1(K,L), defined by (see e.g. [1])

    V1(K,L)=1nSn1h(L,u)dS(K,u). (7.2)

    Next, we give some new isoperimetric inequalities for mixed chord integrals by using the Orlicz chord Minkowski inequality established in Section 5.

    Theorem 7.1 (The Lp-isoperimetric inequality for mixed chord integrals) If KSn, 0i<n and p1, then

    (˜Bp,i(K)S(B))ni(Bi(K)V(B))ni+p, (7.3)

    with equality if and only if K is an n-ball, where ˜Bp,i(K)=nBp,i(K,B).

    Proof Putting L=B, ϕ(t)=tp and p1 in Orlicz chord Minkowski inequality (5.4)

    Bp,i(K,B)Bi(K)(Bi(B)Bi(K))p/(ni).

    That is

    (Bp,i(K,B)Bi(K))ni(Bi(K)V(B))p.

    Hence

    (nBp,i(K,B)S(B))ni(Bi(K)V(B))ni+p.

    From the equality of (5.4), we find that the equality in (7.3) holds if and only if K and B are similar chord. This yields that the equality in (7.3) holds if and only if K is an n-ball.

    Theorem 7.2 (The isoperimetric inequality for the chord integrals) If KSn, then

    (˜B(K)S(B))n(B(K)V(B))n+1, (7.4)

    with equality if and only if K is an n-ball, where ˜B(K)=nB1(K,B).

    Proof This follows immediately from (7.3) with p=1 and i=0.

    This is just a similar form of the classical isoperimetric inequality (7.1).

    As extensions, in the Section, the Orlicz mixed chord integral of K and L, Bϕ(K,L), is generalized into Orlicz multiple mixed chord integral of (n+1) star bodies L1,K1,,Kn. Further, we generalize the Orlicz-Minklowski inequality into Orlicz-Aleksandrov-Fenchel inequality for the Orlicz multiple mixed chord integrals.

    Theorem 8.1 If L1,K1,,KnSn and ϕ1,ϕ2Φ, then

    ddε|ε=0+B(L1ˇ+ϕεK1,K2,,Kn)=1n(ϕ1)r(1)
    ×Sn1ϕ2(d(K1,u)d(L1,u))d(L1,u)d(K2,u)d(Kn,u)dS(u). (8.1)

    Proof This may yield by using a generalized idea and method of proving Lemma 4.4. Here, we omit the details.

    Obviously, (4.3) is a special case of (8.1). Moreover, from Theorem 8.1, we can find the following definition:

    Definition 8.2 (Orlicz multiple mixed chord integrals) Let L1,K1,,KnSn and ϕΦ, the Orlicz multiple mixed chord integral of (n+1) star bodies L1,K1,,Kn, is denoted by Bϕ(L1,K1,,Kn), is defined by

    Bϕ(L1,K1,,Kn)=1nSn1ϕ(d(K1,u)d(L1,u))d(L1,u)d(K2,u)d(Kn,u)dS(u). (8.2)

    When L1=K1, Bϕ(L1,K1,,Kn) becomes the well-known mixed chord integral B(K1,,Kn). Obviously, for 0i<n, Bϕ,i(K,L) is also a special case of Bϕ(L1,K1,,Kn).

    Corollary 8.3 If L1,K1,,KnSn and ϕ1,ϕ2Φ, then

    Bϕ2(L1,K1,,Kn)=(ϕ1)r(1)ddε|ε=0B(L1+ϕεK1,K2,,Kn). (8.3)

    Proof This yields immediately from (8.1) and (8.2).

    Similar to the proof of Theorem 5.4, we may establish an Orlicz-Aleksandrov-Fenchel inequality as follows:

    Theorem 8.4 (Orlicz-Aleksandrov-Fenchel inequality for the Orlicz multiple mixed chord integrals) If L1,K1,,KnSn, ϕΦ and 1rn, then

    Bϕ(L1,K1,K2,,Kn)B(L1,K2,,Kn)ϕ(ri=1B(Ki,Ki,Kr+1,,Kn)1rB(L1,K2,Kn)). (8.4)

    If ϕ is strictly convex, equality holds if and only if L1,K1,,Kr are all of similar chord.

    Proof This yields immediately by using a generalized idea and method of proving Theorem 5.4. Here, we omit the details.

    Obviously, the Orlicz-Minklowski inequality (5.4) is a special case of the Orlicz-Aleksandrov-Fenchel inequality (8.4). Moreover, when L1=K1, (8.4) becomes the following Aleksandrov-Fenchel inequality for the mixed chords.

    Corollary 8.5 (Aleksandrov-Fenchel inequality for the mixed chord integrals) If K1,,KnSn and 1rn, then

    B(K1,,Kn)ri=1B(Ki,Ki,Kr+1,,Kn)1r. (8.5)

    with equality if and only if K1,,Kr are all of similar chord.

    Finally, it is worth mentioning: when ϕ(t)=tp and p1, Bϕ(L1,K1,,Kn) written as Bp(L1,K1,,Kn) and call it Lp-multiple mixed chord integrals of (n+1) star bodies L1,K1,,Kn. So, the new concept of Lp-multiple mixed chord integrals and Lp-Aleksandrov-Fenchel inequality for the Lp-multiple mixed chord integrals may be also derived. Here, we omit the details of all derivations.

    Research is supported by National Natural Science Foundation of China (11371334, 10971205).

    The author declares that no conflicts of interest in this paper.



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