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Research article

Projective class rings of a kind of category of Yetter-Drinfeld modules

  • Received: 13 January 2023 Revised: 14 February 2023 Accepted: 20 February 2023 Published: 08 March 2023
  • MSC : 16D70, 16T05, 16T99

  • 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=HN for some simple subcomodule N of HL, 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, m2 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)=1xi+xizm,Δ(z)=(11+x2x2zm)(zz),ε(xi)=0,ε(z)=1,S(xi)=xizm,S(z)=z1.

    for i=1,2. It is easy to see that

    {xs1xt2zi|iZ2m,s,t[0,1]}

    is a basis of H8m, and

    Δ(zl)=zlzl+alx2zlx2zm+l (2.1)

    for lZ2m, where al=1ω2l1ω2. In particular,

    Δ(zm)=zmzm.

    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 iZ2m. The actions are given by

    x1υi=0,x2υi=0,zυi=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(hm)=h(1)m(1)S(h(3))h(2)m(0),mM,hH,

    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 VHHYD, the left dual V is defined by

    hf,v=f,S(h)v,f(1)f(0),v=S1(v(1))f,v(0). (2.2)

    According to Radford's results in [14], we have the following results.

    Proposition 2.1. If V,WHHYD, then VWHHYD. The actions and coactions are as follows:

    h(υω)=(h)h(1)υh(2)ω,ρ(υω)=(υ),(ω)υ(1)ω(1)υ(0)ω(0),

    where υV,ωW,hH.

    Lemma 2.2. Let LHM. Then, we have

    (1) HLHHYD; the module and comodule actions are given by

    g(hl)=(g)g(1)hS(g(3))g(2)l, (2.3)
    ρ(hl)=(h)h(1)h(2)l,h,gH,lL. (2.4)

    (2)If M is a simple Yetter-Drinfeld H-module, then M=HN for some simple subcomodule N of HL 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 H8mSi for all iZ2m.

    Lemma 3.1. Let 0L be a subcomodule of H8mSi, then there exists jZ2m such that zjυiL or (zj+lx2zj)υiL, where 0lK.

    Proof. Let 0L be a subcomodule of H8mSi and 0u=(2m1j=01s,t=0ks,t,ixs1xt2zi)υiL, where ks,t,jK. By (2.4), we have

    ρ(u)=2m1j=0(α0,jzj+α1,jx1zj+α2,jx2zj+α3,jx1x2zj)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,jzjk1,0,jajx1x2zj+k0,0,m+jam+jx2zm+jk1,1,m+jx1zm+j,α3,j=k1,1,jzj+k1,0,m+jam+jx2zm+j.

    Assume that j0,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 aj0. Now, we complete the proof by discussing the cases (i)-(vi).

    (i) k1,1,j0,k1,0,j0.

    (a) If α0,j and α2,j are linearly independent, we have zjυiL. Moreover, x2zm+jυiL since

    ρ(zjυi)=(zjzj+ajx2zjx2zm+j)υi;

    (b) If α0,j and α2,j are linearly dependent, there exist a 0lK such that (zj+lx2zj)υiL. Moreover, (ajx2zm+j+lzm+j)υiL since

    ρ((zj+lx2zj)υi)=(zj(zj+lx2zj)+x2zj(ajx2zm+j+lzm+j))υi;

    (ii) k1,1,j0,k1,0,j=0, we have zjυiL;

    (iii) k1,1,j=0,k1,0,j0, we have x2zjL. Moreover, zm+jυiL since

    ρ(x2zjυi)=(zjx2zj+x2zjzm+j)υi;

    (iv) k1,1,j=0,k1,0,j=0,k0,0,j0,k0,1,j0, similar to (i);

    (v) k1,1,j=0,k1,0,j=0,k0,0,j0,k0,1,j=0, similar to (ii);

    (vi) k1,1,j=0,k1,0,j=0,k0,0,j=0,k0,1,j0, similar to (iii).

    Lemma 3.2. For iZ2m, all simple subcomodules of H8mSi are as follows:

    (1) U(i,j):=K{zjυi}, where j=0,m;

    (2) V(i,j):=K{zjυi,x2zm+jυi}, where jZ2mandj0,m;

    (3) W(i,j,l):=K{(zj+lx2zj)υi,(ajx2zm+j+lzm+j)υi}, where 0lK,jZ2mandj0,m.

    Proof. It is easy to see that U(i,j) is a simple subcomodule of H8mSi for j=0,m. One also see that V(i,j) is a subcomodule of H8mSi by (2.1). We show that V(i,j) is a simple. Indeed, let 0V be a subcomodule of V(i,j) and 0υ=(ki,jzj+li,jx2zm+j)υiV, where ki,j,li,jK and (ki,j,li,j)0. By (2.4), we have

    ρ(υ)=(ki,jzj+li,jx2zm+j)zjυi+(ki,jajx2zj+li,jzm+j)x2zm+jυi.

    One can check that ki,jzj+li,jx2zm+j,ki,jajx2zj+li,jzm+j are linearly independent. It follows that zjυi,x2zm+jυiV and V=V(i,j). Hence, V(i,j) is a simple subcomodule of H8mSi. Similarly, we have W(i,j,l) is a simple subcomodule of H8mSi for 0lK,jZ2m and j0,m. By Lemma3.1, the vector spaces defined in (1)(3) are all simple subcomodules of H8mSi.

    Lemma 3.3. As Yetter-Drinfeld modules, H8mV(i,j)=H8mW(i,j,l), where 0lK,jZ2mandj0,m.

    Proof. By (2.3), we have

    x2((zj+lx2zj)υi)=((1)iωj)x2zm+jυiH8mW(i,j,l).

    By (2.4), we have

    ρ(x2zm+jυi)=x2zm+jzjυi+zm+jx2zm+jυi.

    Then zjυiH8mW(i,j,l). Hence

    x1(zjυi)=((1)i+(1)j+1ωj)x1zm+jυiH8mW(i,j,l),x1(x2zm+jυi)=((1)i+(1)j+1ωj)x1x2zjυiH8mW(i,j,l).

    Therefore, as Yetter-Drinfeld modules, we have H8mV(i,j)=H8mW(i,j,l), where 0lK,jZ2m and j0,m.

    By Lemma 3.2 and Lemma 3.3, we just need to consider the following Yetter-Drinfeld modules:

    H8mU(i,j), where iZ2m,j=0,m;

    H8mV(i,j), where i,jZ2m,j0,m.

    Now, we consider M(i,j):=H8mU(i,j), where iZ2m,j=0,m. There are two cases:

    Case 1: Mi:=M(i,j) for iZ2m, 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=zimυi} and the coactions are given by

    x1νi=0,x2νi=0,zνi=ωiνi,ρ(νi)=zimνi.

    Case 2: M(i,j) for iZ2m, 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=zjυi,νi,j1=x1zm+jυi,νi,j2=x2zm+jυi,νi,j3=x1x2zjυi} and the coaction are given by

    x1νi,j0=(1)i2νi,j1,x2νi,j0=(1)i2νi,j2,zνi,j0=ωiνi,j0,x1νi,j1=0,x2νi,j1=(1)i+12νi,j3,zνi,j1=ωi1νi,j1,x1νi,j2=(1)i2νi,j3,x2νi,j2=0,zνi,j2=ωi1νi,j2,x1νi,j3=0,x2νi,j3=0,zνi,j3=ωi2νi,j3.
    ρ(νi,j0)=zjνi,j0,ρ(νi,j1)=x1zm+jνi,j0+zm+jνi,j1,ρ(νi,j2)=x2zm+jνi,j0+zm+jνi,j2,ρ(νi,j3)=x1x2zjνi,j0+x2zjνi,j1x1zjνi,j2+zjνi,j3.

    For N(i,j):=H8mV(i,j), i,jZ2m,j0,m. N(i,j) are 4-dimension simple Yetter-Drinfeld modules. The action on the basis {νi,j0=zjυi,νi,j1=x1zm+jυi,νi,j2=x2zm+jυi,νi,j3=x1x2zjυi} and the coaction are given by

    x1νi,j0=ai,jνi,j1,x2νi,j0=bi,jνi,j2,zνi,j0=ωiνi,j0,x1νi,j1=0,x2νi,j1=bi,jνi,j3,zνi,j1=ωi1νi,j1,x1νi,j2=ai,jνi,j3,x2νi,j2=0,zνi,j2=ωi1νi,j2,x1νi,j3=0,x2νi,j3=0,zνi,j3=ωi2νi,j3.
    ρ(νi,j0)=zjνi,j0+ajx2zjνi,j2,ρ(νi,j1)=x1zm+jνi,j0+zm+jνi,j1ajx1x2zm+jνi,j2+ajx2zm+jνi,j3,ρ(νi,j2)=x2zm+jνi,j0+zm+jνi,j2,ρ(νi,j3)=x1x2zjνi,j0+x2zjνi,j1x1zjνi,j2+zjνi,j3.

    Here ai,j=(1)i+(1)j+1ωj,bi,j=(1)iωj,aj=1ω2j1ω2.

    Let Λ={(i,j)|i,jZ2m and jim(mod2m)} and

    Ni,j:=K{νi,j0=zjυi,νi,j1=x1zm+jυi,νi,j2=x2zm+jυi,νi,j3=x1x2zjυi},

    where (i,j)Λ. Note that

    {Ni,j|(i,j)Λ}={M(i,j)|iZ2m,j=0,m,jim(mod2m)}{N(i,j)|i,jZ2m,j0,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|iZ2m}{Ni,j|(i,j)Λ}

    forms a complete list of non-isomorphic simple Yetter-Drinfeld modules over H8m.

    Proof. Firstly, we show that

    MiMi if and only if i=i for i,iZ2m.

    Let f:MiMi,νiaνi be a Yetter-Drinfeld module isomorphism, where 0aK. Then, we have

    zf(νi)=azνi=aωiνi=f(zνi)=ωif(νi)=aωiνi,(idf)ρ(νi)=(idf)(zimνi)=azimνi=ρ(f(νi))=ρ(νi)=azimνi.

    Hence i=i.

    Now we show that each Ni,j for (i,j)Λ is simple Yetter-Drinfeld module.

    Let 0V be a Yetter-Drinfeld submodule of N(i,j) and 0υ=3k=0αi,jkνi,jkV, where αi,jkK 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,j0ajx2zjαi,j1ajx1x2zm+j+αi,j2zm+jαi,j3x1zj)νi,j2+(αi,j1ajx2zm+j+αi,j3zj)νi,j3.

    If (αi,j1,αi,j3)=0, it is easy to see that νi,j0V. 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,j0V. It follows that

    νi,j1=1ai,jx1νi,j0V,νi,j2=1bi,jx2νi,j0V,νi,j3=1ai,jbi,jx1x2νi,j0V.

    Hence, V=N(i,j) and N(i,j) is a simple Yetter-Drinfeld module.

    Finally we show that

    Ni,jNi,j if and only if (i,j)=(i,j) for (i,j),(i,j)Λ.

    Let g:Ni,jNi,j,νi,jk3l=0αk,lνi,jl, where k{0,1,2,3},αk,lK and there exists l{0,1,2,3} such that αk,l0 for each k{0,1,2,3}. Then we have

    x1g(νi,j0)=α0,0ai,jνi,j1+α0,2ai,jνi,j3=g(x1νi,j0)=ai,j3l=0α1,lνi,jl,x2g(νi,j0)=α0,0bi,jνi,j2α0,1bi,jνi,j3=g(x2νi,j0)=bi,j3l=0α2,lνi,jl,zg(νi,j0)=α0,0ωiνi,j0α0,1ωi1νi,j1+α0,2ωi1νi,j2α0,3ωi2νi,j3=g(zνi,j0)=ωi3l=0α0,lνi,jl,x1g(νi,j1)=α1,0ai,jνi,j1+α1,2ai,jνi,j3=g(x1νi,j1)=0,x2g(νi,j1)=α1,0bi,jνi,j2α1,1bi,jνi,j3=g(x2νi,j1)=bi,j3l=0α3,lνi,jl,zg(νi,j1)=α1,0ωiνi,j0α1,1ωi1νi,j1+α1,2ωi1νi,j2α1,3ωi2νi,j3=g(zνi,j1)=ωi13l=0α1,lνi,jl,x1g(νi,j2)=α2,0ai,jνi,j1+α2,2ai,jνi,j3=g(x1νi,j2)=ai,j3l=0α3,lνi,jl,x2g(νi,j2)=α2,0bi,jνi,j2α2,1bi,jνi,j3=g(x2νi,j2)=0,zg(νi,j2)=α2,0ωiνi,j0α2,1ωi1νi,j1+α2,2ωi1νi,j2α2,3ωi2νi,j3=g(zνi,j2)=ωi13l=0α2,lνi,jl,x1g(νi,j3)=α3,0ai,jνi,j1+α3,2ai,jνi,j3=g(x1νi,j3)=0,x2g(νi,j3)=α3,0bi,jνi,j2α3,1bi,jνi,j3=g(x2νi,j3)=0,zg(νi,j3)=α3,0ωiνi,j0α3,1ωi1νi,j1+α3,2ωi1νi,j2α3,3ωi2νi,j3=g(zνi,j3)=ωi23l=0α3,lνi,jl.

    Hence i=i and αk,l=0 for kl. Also, we have

    (idg)ρ(νi,j3)=(idg)(x1x2zjνi,j0+x2zjνi,j1x1zjνi,j2+zjνi,j3)=α0,0x1x2zjνi,j0+α1,1x2zjνi,j1α2,2x1zjνi,j2+α3,3zjνi,j3=ρ(g(νi,j3))=α3,3ρ(νi,j3)=α3,3x1x2zjνi,j0+α3,3x2zjνi,j1α3,3x1zjνi,j2+α3,3zjνi,j3.

    Hence j=j and α0,0=α1,1=α2,2=α3,3. Therefore,

    Ni,jNi,j if and only if (i,j)=(i,j) for (i,j),(i,j)Λ.

    The proof is finished.

    Corollary 3.5. Ni,jNm+2i,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

    x1μi,j0=am+2i,jμi,j1,x2μi,j0=bm+2i,jμi,j2,zμi,j0=ω2iμi,j0,x1μi,j1=0,x2μi,j1=bm+2i,jμi,j3,zμi,j1=ω1iμi,j1,x1μi,j2=am+2i,jμi,j3,x2μi,j2=0,zμi,j2=ω1iμi,j2,x1μi,j3=0,x2μi,j3=0,zμi,j3=ωiμi,j3.

    and

    ρ(μi,j0)=zjμi,j0+ajx2zjμi,j2,ρ(μi,j1)=x1zmjμi,j0+zmjμi,j1ajx1x2zmjμi,j2+ajx2zmjμi,j3,ρ(μi,j2)=x2zmjμi,j0+zmjμi,j2,ρ(μi,j3)=x1x2zjμi,j0+x2zjμi,j1x1zjμi,j2+zjμi,j3.

    Hence Ni,jNm+2i,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]=[MN]. The Green ring r(H) is defined to be the quotient ring of F(H) module the relations [MN]=[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, HHYDD(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 VH8mH8mYD. 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

    DVIrr(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),iZ2m and firstly introduce some lemmas.

    Lemma 4.1. (1) MiMjMi+jMjMi and MiNk,lNi+k,im+lNk,lMi for i,jZ2m,(k,l)Λ;

    (2) P(Ni,j)Ni,j, (i,j)Λ;

    (3) MiPjPi+jPjMi and dimPi=16 for i,jZ2m;

    (4) For (i,j),(k,l)Λ,hZ2m,Hom(Ni,jNk,l,Mh)0 if and only if hi+m+k2(mod2m) and j(i+k)ml(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(Nik,jmk)Mk is projective and

    HomD(P(Nik,jmk)Mk,Nik,jmkMk)HomD(P(Nik,jmk),Nik,jmkMkMk)HomD(P(Nik,jmk),Nik,jmk)0,

    we have P(Ni,j)P(Nik,jmkMk)P(Nik,jmk)Mk, which implies that

    dimP(Nik,jmk)dimP(Ni,j)8.

    Let J={(s,t)Λ|(s,t)=(ik,jmk)\; for kZ2m}. It is obvious that

    |J|=2m and |ΛJ|=4m24m.

    Then

    dimD=2m1i=0dimP(Mi)+(s,t)J4dimP(Ns,t)+(s,t)ΛJ4dimP(Ns,t)2m1i=0dimP(Mi)+32|J|+16|ΛJ|>32|J|+16|ΛJ|=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=dimD4|Λ|dimNi,j=32m and hence dimPi=16. Similar to (2), we have PiP0Mi. Then, PiP0Mi. Hence, PjMiP0MjMiP0Mi+jPi+j.

    (4) Since D is quasi-triangular, MKNNKM for M,NDM,

    HomD(Ni,jNk,l,Mh)HomD(Ni,j,HomK(Nk,l,Mh))HomD(Ni,j,Nk,lMh)HomD(Ni,j,MhNk,l)HomD(Ni,j,MhNm+2k,l)HomD(Ni,j,Nh+m+2k,hml).

    Then by Schur's lemma, HomD(Ni,jNk,l,Mh)0 if and only if ih+m+2k(mod2m) and jhml(mod2m), if and only if hi+m+k2(mod2m) and j(i+k)ml(mod2m).

    Corollary 4.2. We have

    DD(2m1i=0Pi)((j,k)ΛN4j,k).

    Now we describe the projective cover Pi of the simple module Mi for iZ2m. For convenience, we let

    A=(0000100000000010),B=(1000010000100001),C=(0000000010000100),D=(10000ω10000ω10000ω2),E=(00000000ω0000100),F=(10000zm0000zm00101),G=(00002x1zm0000000002x10),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]=(C0000C00B0C00B0C),[z]=(ω2D0000ωD00ωE0ωD00E0D),
    ρ(P)=(FGH+K0002x1zm(H+F)zm(F+G+H+K)002x2zm(FG)0zm(F+G+H+K)04x1x2F2x2(F+G)2x1(HF)FGH+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=MN. Let

    κ0=p3+p12+p9p6,κ1=p7p13,κ2=p11p14,κ3=p15,κ4=p1+p4,κ5=p5,κ6=p2+p8,κ7=p10,κ8=p0,κ9=p3p12,κ10=p6+p9,κ11=p11+p14,κ12=p1,κ13=p2,κ14=p7,κ15=p6,

    and K1=0,Kl=li=0Kκi for l{0,,15}. One can check that

    0=K1K0K1K15=P

    is a Yetter-Drinfeld submodules chain of P such that Kl/Kl1 is a one dimensional Yetter-Drinfeld module. And

    Kl/Kl1M1, if l=2,11;Kl/Kl1M0, if l=0,9,10,15;Kl/Kl1M1, if l=4,12;Kl/Kl1Mm2, if l=3;Kl/Kl1Mm1, if l=1,14;Kl/Kl1Mm, if l=5,7;Kl/Kl1Mm+1, if l=6,13;Kl/Kl1Mm+2, if l=8.

    We claim that κ15M and κ15N. If κ15M, then κ11κ2,κ4κ12,κ8,κ13,κ14M since

    x1κ15=κ14,x2κ15=κ11κ22,ρ(κ15)=4x1x2κ8+2x2(κ4κ12)+2x1κ13+1κ15,

    which implies that

    κ0=x1x2κ8M,κ1=x1x2(κ4κ12)M,κ2=x2x1κ13M,κ3=x2κ14M,κ4=x1κ8M,κ5=x1(κ4κ12)M,κ6=x2κ8M,κ7=x2κ13M,κ9=x1κ13+x2(κ4κ12)M,κ10=x1x2κ8+x2(κ4κ12)x1κ13M,κ11=x2x1κ13+2x2κ15M,κ12=x1κ8+κ12κ4M.

    Then M=P. It's a contradiction. Similarly, if κ15N, then N=P and also a contradiction. Hence the claim follows. Therefore, there exist some αiK, for i{0,1,,14}, such that

    κ=14i=0αiκi+κ15M.

    Then

    ρ(κ)=(α0+(2α1+α14)x1zm+2α2x2zm+4α3x1x2)κ0+(α1zm+2α3x2)κ1+(α2zm2α3x1)κ2+α31κ3+(α4zm+2α5x1+2(α10α9+α15)x2+4α14x1x2zm)κ4+(α5+2α14x2zm)κ5+(α6zm+2α7x2+2(α9+α10)x1+8α11x1x2zm)κ6+(α74α11x1zm)κ7+(α82α12x1zm2α13x2zm+4α15x1x2)κ8+(α9+2α11x2zm+α14x1zm)κ9+(α10+2α11x2zmα14x1zm)κ10+α11zmκ11+(α12zm2α15x2)κ12+(α13zm+2α15x1)κ13+α14zmκ14+1κ15.

    It is observe that ˆκ1=α0κ0+α3κ3+α5κ5+α7κ7+α8κ8+α9κ9+α10κ10+κ15M. Hence

    x1ˆκ1=(α9+α10)κ1+α7κ2+α8κ4+κ14M,x2ˆκ1=α5κ1+(α9α1012)κ2+α8κ6+κ112M,x1x2ˆκ1=α8κ0+κ32M,

    and

    ρ(α8κ0+κ32)=(α8+2x1x2)κ0+x2κ1x1κ2+121κ3.

    Then κ0,κ1,κ2,κ3 and α8κ4+κ14,α8κ6+κ112M. Note that

    ρ(α8κ4+κ14)=(α8zm+4x1x2zm)κ4+2x2zmκ5+x1zm(κ0+κ9κ10)+zmκ14,ρ(α8κ6+κ112)=(α8+4x1x2zm)κ62x1zmκ7+x2zm(κ9+κ10)+zm2κ11.

    Then κ4,κ5,κ6,κ7,κ9,κ10,κ11,κ14M. Hence ˆκ2=α8κ8+α12κ12+α13κ13+κ15M, and

    ρ(ˆκ2)=2x2κ4+(α82α12x1zm2α13x2zm+4x1x2)κ8+(α12zm2x2)κ12+(α13zm+2x1)κ13+1κ15.

    Thus κ15M. It's a contradiction. Consequently, P is indecomposable.

    Lemma 4.4. PP0 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 iZ2m and the socle and top of P0 coincide. Therefore P0E(M0). On the other hand, by Lemma 4.3, we know that P is an indecomposable module with SocPM0. Thus, P embeds in E(M0), which implies that PE(M0), since they have the same dimension. Hence, PP0.

    Now we calculate the tensor decompositions of the simple and indecomposable projective Yetter-Drinfeld modules. Denote ±Va±b:=Va+bVab, for a,bZ2m,VH8mH8mYD.

    Lemma 4.5. (1) For i,jZ2m,

    PiPj±(P2i+j±1P2i+j+m±1Pi+j+m±2)P4i+jP2i+j+m.

    (2) For i,jΛ,kZ2m,

    Ni,jPkPkNi,j±(N2i+k±1,j+km+mN2i+k+m±1,j+km+mNi+k+m±2,j+km)N4i+k,j+kmN2i+k+m,j+km.

    Proof. It suffices to prove the lemma for P0P0 and P0Ni,j by Lemma 4.1 (1)(3).

    By the proof of Lemma 4.3,

    [P0]=2[M1]+2[M1]+2[Mm+1]+2[Mm1]+[Mm+2]+[Mm2]+4[M0]+2[Mm]

    in the Grothendieck ring of the category of the Yetter-Drifeld modules over H8m. It follows that

    P0P0±((M±1P0)2(Mm±1P0)2(Mm±2P0))(M0P0)4(MmP0)2±(P2±1P2m±1Pm±2)P40P2m,

    and

    P0Ni,j±((M±1Ni,j)2(Mm±1Ni,j)2(Mm±2Ni,j))(M0Ni,j)4(MmNi,j)2±(N2i±1,j+mN2i+m±1,j+mNi+m±2,j)N4i,jN2i+m,j,

    The proof is finished.

    Lemma 4.6. For (i,j),(k,l)Λ, we have

    Ni,jNk,l{Pi+m+k2,ifj(i+k)ml(mod2m);1t=0(Ni+k+(m1)t,j+l+mtNi+k+(m1)t1,j+l+(t+1)m),otherwise.

    Proof. If j(i+k)ml(mod2m), then by Lemma 4.1(4), Hom(Ni,jNk,l,Mh)0 if and only if hi+m+k2(mod2m). Since Ni,jNk,l is projective and dimPi+m+k2=dim(Ni,jNk,l)=16, we get that Pi+m+k2Ni,jNk,l.

    If j(i+k)ml(mod2m), then by Lemma 4.1(4), Hom(Ni,jNk,l,Mh)=0 for any hZ2m, which implies that Ph can not the direct summand of Ni,jNk,l. Hence Ni,jNk,l has to be the direct sum of four 4-dimensional simple projective modules.

    Let

    t=(1)kbi,jbk,l1ω2,θr,s=μi,jrνk,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(θ1,0tθ3,2)),β0,2=ai+k,j+l(bk,lθ0,2+(1)kbi,jθ2,0),β0,3=ak,lbk,lθ0,3+ai,jbi,jθ3,0+(1)k1ai,jbk,lθ1,2+(1)kak,lbi,jθ2,1,β1,0=bi+k+m1,j+l+m(θ0,1+tθ2,3(1)jωj(θ1,0tθ3,2)),β1,1=bi+k+m1,j+l+m(θ1,1tθ3,3),β1,2=(1)kbi,j((1)jωjθ3,0θ2,1)bk,l(θ0,3+(1)jωjθ1,2),β1,3=bk,lθ1,3(1)kbi,jθ3,1,β2,0=ai+k1,j+l+m(θ0,2ωjθ2,0),β2,1=ak,l(θ0,3ωjθ2,1)(1)kai,j(θ1,3+ωjθ3,0),β2,2=ai+k1,j+l+mθ2,2,β2,3=ak,lθ2,3(1)kai,jθ3,2,β3,0=θ0,3ωjθ2,1+(1)jωj(θ1,2+ωjθ3,0),β3,1=θ1,3+ωjθ3,1,β3,2=θ2,3+(1)jωjθ3,2,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+m1,j+l+m,K{β2,0,β2,1,β2,2,β2,3}Ni+k1,j+l+m,K{β3,0,β3,1,β3,2,β3,3}Ni+k+m2,j+l.

    Hence,

    Ni,jNk,lNi+k,j+lNi+k+m1,j+l+mNi+k1,j+l+mNi+k+m2,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,,2m1}

    Lemma 4.7. The following statements hold in rP(D).

    (1) [Mi]=˙yi0 for iZ2m;

    (2) [Ni,j]=˙yi0˙yj+im for (i,j)Λ;

    (3) [Pi]=˙yi+m+20˙y2m for iZ2m;

    (4) For i,j,k,j+k{1,,2m1},

    ˙y2m0=1,˙yi˙y2mi=˙y2m,˙yj˙yk=(1+˙y2m10+˙ym10+˙ym20)˙yj+k,˙yi˙y2m=(1+2˙y2m10+2˙y2m20+2˙y2m30+2˙ym10+4˙ym20+2˙ym30+˙ym40)˙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|iZ2m,j{1,2,2m1}}.

    Proof. By Lemma 4.7(4), ˙y2m0=1, and for k,l,k+l{1,,2m1}, ˙yk˙y2m,˙yk˙yl can be expressed as a linear combination of {˙yi0,˙yi0˙yj,˙yi0˙y2m|iZ2m,j{1,2,2m1}}. It is easy to check that the set

    {˙yi0,˙yi0˙yj,˙yi0˙y2m|iZ2m,j{1,2,2m1}}

    is a independent set since #{˙yi0,˙yi0˙yj,˙yi0˙y2m|iZ2m,j{1,2,2m1}}=4m2+2m, the number of Z-basis of rP(D).

    Hence, {˙yi0,˙yi0˙yj,˙yi0˙y2m|iZ2m,j{1,2,2m1}} 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,,y2m1] modulo the ideal I generated by the following elements

    y2m01,yiy2miy2m,yjyk(1+y2m10+ym10+ym20)yj+k,yiy2m(1+2y2m10+2y2m20+2y2m30+2ym10+4ym20+2ym30+ym40)yi, (4.1)

    where i,j,k,j+k{1,,2m1}.

    Proof. By Corollary 4.8, there is a unique ring epimorphism

    Φ:Z[y0,y1,,y2m1]rP(H8m)

    such that Φ(yi)=˙yi for iZ2m. By Lemma 4.7(4), we have

    Φ(y2m01)=Φ(y0)2m1=˙y2m01=0,Φ(yiy2miy2m)=Φ(yi)Φ(y2mi)Φ(ym)2=˙yi˙y2mi˙y2m=0,Φ(yjyk(1+y2m10+ym10+ym20)yj+k)=Φ(yj)Φ(yk)(1+Φ(y0)2m1+Φ(y0)m1+Φ(y0)m2)Φ(yj+k)=˙yj˙yk(1+˙y2m10+˙ym10+˙ym20)˙yj+k=0,Φ(yiy2m(1+2y2m10+2y2m20+2y2m30+2ym10+4ym20+2ym30+ym40)yi)=˙yi˙y2m(1+2˙y2m10+2˙y2m20+2˙y2m30+2˙ym10+4˙ym20+2˙ym30+˙ym40)˙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,,y2m1]/IrP(H8m)

    such that ¯Φ(¯ν)=Φ(ν) for all νZ[y0,y1,,y2m1], where ¯ν=ν+I.

    It is straightforward to check that the ring Z[y0,y1,,y2m1]/I is Z-spanned by

    {yi0,yi0yj,yi0y2m|iZ2m,j{1,2,2m1}}.

    This means the Z-rank of Z[y0,y1,,y2m1]/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.



    [1] D. N. Yetter, Quantum groups and representations of monoidal categories, Math. Proc. Cambridge, 108 (1990), 261–290. http://doi.org/10.1017/S0305004100069139 doi: 10.1017/S0305004100069139
    [2] C. Kassel, Quantum groups, New York: Springer, 1995. https://doi.org/10.1007/978-1-4612-0783-2
    [3] N. Andruskiewitsch, H. J. Schneider, Lifting of quantum linear spaces and pointed Hopf algebras of orderp3, J. Algebra, 209 (1998), 658–691. http://doi.org/10.1006/jabr.1998.7643 doi: 10.1006/jabr.1998.7643
    [4] N. Andruskiewitsch, H. J. Schneider, Finite quantum groups and Cartan matrices, Adv. Math., 154 (2000), 1–45. http://doi.org/10.1006/aima.1999.1880 doi: 10.1006/aima.1999.1880
    [5] N. Andruskiewitsch, H. J. Schneider, Pointed Hopf algebras, In: New Directions in Hopf algebras, Cambridge: Cambridge University Press, 2002, 1–68.
    [6] N. Andruskiewitsch, H. J. Schneider, On the classification of finite dimensional pointed Hopf algebras, Ann. Math., 171 (2010), 375–417. http://doi.org/10.4007/annals.2010.171.375 doi: 10.4007/annals.2010.171.375
    [7] N. Andruskiewitsch, G. Carnovale, G. A. García, Finite dimensional pointed Hopf algebras over finite simple groups of Lie type I. Non-semisimple classes in PSLn(q), J. Algebra, 442 (2015), 36–65. http://doi.org/10.1016/j.jalgebra.2014.06.019 doi: 10.1016/j.jalgebra.2014.06.019
    [8] G. A. García, J. M. J. Giraldi, On Hopf algebras over quantum subgroups, J. Pure Appl. Algebra, 223 (2019), 738–768. https://doi.org/10.1016/j.jpaa.2018.04.018 doi: 10.1016/j.jpaa.2018.04.018
    [9] N. Hu, R. Xiong, Some Hopf algebras of dimension 72 without the Chevalley property, 2016, arXiv: 1612.04987. http://doi.org/10.48550/arXiv.1612.04987
    [10] Y. X. Shi, Finite dimensional Nichols algebras over Kac-Paljutkin algebra H8, Rev. Unión Mat. Argent., 60 (2019), 265–298. http://doi.org/10.33044/revuma.v60n1a17 doi: 10.33044/revuma.v60n1a17
    [11] R. C. Xiong, On Hopf algebras over the unique 12-dimensional Hopf algebra without the dual Chevalley property, Commun. Algebra, 47 (2019) 1516–1540. http://doi.org/10.1080/00927872.2018.1508582
    [12] Y. Zheng, Y. Gao, N. H. Hu, Finite dimensional Hopf algebras over the Hopf algebra Hb:1 of Kashina, J. Algebra, 567 (2021), 613–659. http://doi.org/10.1016/j.jalgebra.2020.09.035 doi: 10.1016/j.jalgebra.2020.09.035
    [13] Y. Zheng, Y. Gao, N. H. Hu, Finite dimensional Hopf algebras over the Hopf algebra Hd:1,1 of Kashina, J. Pure Appl. Algebra, 225 (2021), 106527. https://doi.org/10.1016/j.jpaa.2020.106527 doi: 10.1016/j.jpaa.2020.106527
    [14] D. E. Radford, On oriented quantum algebras derived from representations of the quantum double of a finite dimensional Hopf algebra, J. Algebra, 270 (2003), 670–695. http://doi.org/10.1016/j.jalgebra.2003.07.006 doi: 10.1016/j.jalgebra.2003.07.006
    [15] H. Zhu, H. X. Chen, Yetter-Drinfeld modules over the Hopf-Ore extension of the group algebra of dihedral group, Acta Math. Sin., 28 (2012), 487–502. http://doi.org/10.1007/s10114-011-9777-4 doi: 10.1007/s10114-011-9777-4
    [16] S. L. Yang, Y. F. Zhang, Ore extensions for the Sweedler's Hopf algebra H4, Mathematics, 8 (2020), 1293. http://doi.org/10.3390/math8081293 doi: 10.3390/math8081293
    [17] J. Chen, S. Yang, D. Wang, Y. Xu, On 4n-dimension neither pointed nor semisimple Hopf algebras and the associated weak Hopf algebras, 2018, arXiv: 1809.00514. http://doi.org/10.48550/arXiv.1809.00514
    [18] R. C. Xiong, Some classification results on finite dimensional Hopf algebras, PhD thesis, East China Normal University, 2019.
    [19] Y. Zhang, The Ore extensions of Hopf algebras and their related topics, PhD thesis, Beijing University of Technology, 2020.
    [20] Y. Guo, S. Yang, The Grothendieck ring of Yetter-Drinfeld modules over a class of 2n2-dimension Kac-Paljutkin Hopf algebras, submmited for publication.
    [21] Y. Guo, S. Yang, Representations of a family of 8m-dimension Hopf algebras of rank two, submmited for publication.
    [22] G. Liu, Basic Hopf algebras of tame type, Algebr. Represent. Theory, 16 (2013), 771–791. https://doi.org/10.1007/s10468-011-9331-1 doi: 10.1007/s10468-011-9331-1
    [23] B. Wald, J. Waschbüsch, Tame biserial algebras, J. Algebra, 95 (1985), 480–500. http://doi.org/10.1016/0021-8693(85)90119-X doi: 10.1016/0021-8693(85)90119-X
    [24] S. Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, 1993. http://doi.org/10.1090/cbms/082
    [25] M. Auslander, I. Reiten, S. O. Smalø, Representation theory of Artin algebras, Cambridge: Cambridge University Press, 1995. http://doi.org/10.1017/CBO9780511623608
    [26] M. Lorenz, Representations of finite dimensional Hopf algebra, J. Algebra, 188 (1997), 476–505. https://doi.org/10.1006/jabr.1996.6827 doi: 10.1006/jabr.1996.6827
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