亚洲免费av电影一区二区三区,日韩爱爱视频,51精品视频一区二区三区,91视频爱爱,日韩欧美在线播放视频,中文字幕少妇AV,亚洲电影中文字幕,久久久久亚洲av成人网址,久久综合视频网站,国产在线不卡免费播放

        ?

        Construction of semisimple categoryover generalized Yetter-Drinfeld modules

        2013-01-02 01:25:42ZhangXiaohuiWangShuanhong

        Zhang Xiaohui Wang Shuanhong

        (Department of Mathematics, Southeast University, Nanjing 211189, China)

        In 2007, Panaite and Staic[1]introduced the notion of generalized Yetter-Drinfeld modules which covered both Yetter-Drinfeld modules and anti-Yetter-Drinfeld modules. Liu and Wang[2]studied the notion of generalized weak Yetter-Drinfeld modules and made the category ofHWYDH(α,β) into a braided T-category[3]. The fusion category[4-6]plays an important role in classifying the semisimple Hopf algebra. The semisimple category is the first step to construct a fusion category. In this paper, we discuss the following question: how to make the category of generalized Yetter-Drinfeld modulesHYDH(α,β) into a semisimple category.

        Throughout this paper, we assume thatHis a Hopf algebra[7-8]with a bijective antipode over a fieldk. Denote the set of all the automorphisms ofHby AutHopf(H). Letα,β∈AutHopf(H).

        1 Preliminaries

        Definition1For anyα,β∈AutHopf(H), a (α,β)-Yetter-Drinfeld module is ak-moduleM, such thatMis a leftH-module (with notationh?m|→h·m) and a rightH-comodule (with notationm|→m(0)?m(1)) with the following compatibility condition:

        ρ(h·m)=h2·m(0)?β(h3)m(1)α(S-1(h1))

        for allh∈Handm∈M. The category of (α,β)-Yetter-Drinfeld modules andH-linearH-colinear maps is denoted byHYDH(α,β).

        Define the category of the generalized Yetter-Drinfeld module YD(H) as the disjoint union of allHYDH(α,β).

        Definition2Suppose thatM∈YD(H), thenMis called simple if it has no proper subobjects. A direct sum of simple objects is called semisimple. If every objectM∈YD(H) is semisimple, we call the category YD(H) semisimple.

        2 Making YD(H) into semisimple

        Lemma1[1]Suppose thatM∈HYDH(α,β) andN∈HYDH(γ,δ), thenM?N∈HYDH(αγ,δγ-1βγ) with the following structures

        h·(m?n)=γ(h1)·m?γ-1βγ(h2)·n

        m?n|→(m?n)(0)?(m?n)(1)=(m(0)?n(0))?n(1)m(1)

        Lemma2Ifkis a commutative ring,His a commutative Hopf algebra overk,M∈HYDH(α,β),N∈HYDH(γ,δ), andMis a finitely generated projectiveH-module, then

        1)Hhom(M,N) is anH-comodule, andHhomH(M,N)=Hhom(M,N)coH, where theH-coaction is given byρ(f)(m)=f0(m)?f1f(m0)0?f(m0)1S(m1).

        2)Hhom(M,N)∈HYDH(α,β), where theH-action is given by (h·f)(m)hf(m)=f(h·m).

        Proof1) Define a mapπ:Hhom(M,N)→Hhom(M,N?H) byπ(f)(m)=f(m0)0?f(m0)1S(m1).

        For anym∈M,h∈H, we have

        π(f)(h·m)=f((h·m)0)0?f((h·m)0)1S((h·m)1)=

        f(h2·m0)0?f(h2·m0)1S(β(h3)m1α(S-1(h1)))=

        h·f(m0)0?β(h3)β(S(h4))f(m0)1α(S-1(h2))·

        α(h1)S(m1)=h·f(m0)0?f(m0)1S(m1)=

        h·(π(f))(m))

        Thus,πis well defined. SinceMis a finitely generated projectiveH-module, we haveHhom(M,N?H)?Hhom(M,N)?H. So we obtain a map:

        ρ:Hhom(M,N)→Hhom(M,N)?H

        such thatρ(f)(m)=f(m0)0?f(m0)1S(m1), andHhom(M,N)∈MH.

        Now for anyf∈Hhom(M,N), iffisH-colinear, then

        ρ(f)(m)=f(m0)0?f(m0)1S(m1)=

        f(m0)?m1S(m2)=f(m)?1=(f?1)(m)

        Sofis coinvariant. Conversely, takef∈Hhom(M,N)coH, then we have

        ρN(f(m))=f(m0)0?f(m0)1ε(m1)=

        f(m0)0?f(m0)1S(m1)m2=f(m0)?m1

        for anym∈M, andfisH-linear. Thus,HhomH(M,N)=Hhom(M,N)coH.

        2) For anym∈M,h∈H, we have

        ((h·f)0?(h·f)1)(m)=(h·f(m0))0?

        (h·f(m0))1S(m1)=h2·f(m0)0?

        β(h3)f(m0)1α(S-1(h1))S(m1)=

        (h2·f0?β(h3)f1α(S-1(h1)))(m)

        Lemma3LetVbe ak-module andNbe anH-module, then

        1)Hhom(H?V,N) and hom(V,N) are isomorphic ask-modules, where the bijection is given byθ:Hhom(H?V,N)→hom(V,N),θ(f)(v)=f(1?v).

        2) IfVis a projectivek-module, thenH?Vis a projectiveH-module.

        Furthermore, ifV∈MH, thenH?Vis an object ofHYDH(α,β) via

        h·(h′?v)=hh′?v

        ρ(h?v)=h2?v0?β(h3)v1α(S-1(h1))

        Similar to Lemma 2, we can obtain the following lemmas.

        Lemma4LetV∈MHis a finitely generated projectivek-module. Then for anyH-comoduleN, we have hom(V,N)∈MH, where theH-coaction is given byρ(g)(v)=g(v0)0?g(v0)1S(v1). IfHis commutative, then for anyN∈HYDH(α,β), we can getHhom(H?V,N)∈HYDH(α,β).

        Lemma5Suppose thatHis commutative, andN∈HYDH(α,β).

        1) IfV∈MHis a finitely generated projectivek-module, thenHhom(H?V,N) and hom(V,N) are isomorphic asH-comodules.

        2) Ifkis a field,Vis a finite-dimensionalk-space and a projective rightH-comodule, thenH?Vis a projective object inHYDH(α,β).

        Proof1) It is straightforward.

        2) Obviously, we have

        HhomH(H?V,N)?Hhom(H?V,N)coH?

        hom(V,N)coH?homH(V,N)

        where the last isomorphism is due to the proof of Lemma 2. So the conclusion holds. From the above two lemmas, we have the following facts.

        Lemma6Letkbe a field, andM∈HYDH(α,β). ThenMis a finitely generatedH-module if and only if there exists a finite dimensionalH-comoduleVand anH-linearH-colinear epimorphismπ:H?V→M.

        LetH*be the linear dual ofH. IfM,N∈MH, then homk(M,N)∈H*Munder the followingH*-action

        (h*·f)(m)=h*(f(m0)1S(m1))·f(m0)0

        Lemma7Assume thatHis commutative, andM,N∈HYDH(α,β). ThenHhom(M,N) is a leftH*-submodule of homk(M,N).

        Furthermore,M∈H*Mis called rational if the leftH*-action onMis induced by a rightH-coaction onM.

        Proposition1Suppose thatHis commutative,kis a field,M,N∈HYDH(α,β), andMis a finitely generatedH-module. ThenHhom(M,N)∈HYDH(α,β).

        ProofBy Lemma 6, there exists a finite dimensionalH-comoduleVand anH-linearH-colinear epimorphismπ:H?V→M. So we obtain an injectivek-linear mapHhom(π,N):Hhom(M,N)→Hhom(H?V,N). For anyφ∈H*,v∈V,h∈H,f∈Hhom(M,N), we haveπ(h?v)=h·v,ρ(1?v)=1?v0?v1, and

        ((φ·f)°π)(1?v)=(φ·f)(v)=

        φ(f(v0)1S(v1))f(v0)0=

        φ(f(π(1?v0))1S(v1))f(π(1?v0))0=

        φ(f(π(1?v)0)1S(1?v)1)f(π(1?v)0)0=

        (φ·(f°π))(1?v)

        It follows thatHhom(π,N) isH*-linear. Then by Lemma 2,Hhom(H?V,N) is anH-comodule, and, therefore, a rationalH*-module. ThusHhom(M,N) is a rationalH*-submodule ofHhom(H?V,N). This means thatHhom(M,N) is anH-comodule. Then we obtainHhom(M,N)∈HYDH(α,β) by Lemma 2.

        We say thatHYDH(α,β) satisfies the exact condition if the following property holds: ifM∈HYDH(α,β) is a finitely generatedH-module, then the functorHhom(M,_):HYDH(α,β)→HYDH(α,β) is exact.

        By Proposition 1, ifHis commutative andMis a finitely generatedH-module, we haveHhom(M,N)∈HYDH(α,β) for anyN∈HYDH(α,β). ObviouslyHYDH(α,β) satisfies the exact condition ifHis semisimple.

        Proposition2Assume thatHis commutative, andHYDH(α,β) satisfies the exact condition and the functor (-)coH:HYDH(α,β)→kMis exact. Then any finitely generatedH-moduleM∈HYDH(α,β) is a projective object.

        Proofwe haveHhomH(M,_)?Hhom(M,_)coH=(-)coH°Hhom(M,_) which implies thatHhomH(M,_) is also an exact functor.

        Proposition3Under the same condition of Proposition 2, suppose thatHis noetherian. Then any finitely generatedH-moduleM∈HYDH(α,β) is a direct sum of a family of simple subobjects which are also finitely generated asH-modules inHYDH(α,β).

        ProofAssume thatNis a subobject ofM. ThenNandM/Nare finitely generatedH-modules sinceHis noetherian. Furthermore,NandM/Nare projective objects. So we have a split exact sequence inHYDH(α,β): 0→N→M→M/N→0.

        Thus the conclusion holds.

        TakeM∈HYDH(α,β) and anH-subcomoduleVofM. We set

        whereIis a finite set. ThenHVis a subobject ofMinHYDH(α,β) via:

        Theorem1LetHbe commutative and noetherian,HYDH(α,β) satisfies the exact condition and the functor (-)coH:HYDH(α,β)→kMis exact. Then everyM∈HYDH(α,β) is a direct sum of a family of simple subobjects ofMwhich are finitely generated asH-modules inHYDH(α,β). Therefore,HYDH(α,β) is a semisimple category.

        ProofFor anym∈M,mbelongs to a finite dimensionalH-subcomoduleVmofM. ThenVmis a finitely generatedH-module. By Proposition 3,Vmis a direct sum of a family of simple subobjects which are finitely generated. LetΩbe the set of all direct sumsN=?i∈INiwhere everyNiis both a finitely generatedH-module and a simple subobject ofMinHYDH(α,β). Then the sum of two elements inΩis also an object inΩ. ThusΩcontains a maximal elementM′ through Zorn’s Lemma. For anym∈M, we havem∈HVm∈Ω. This means thatHVm+M′=M′. SoM=M′. Thus, the conclusion holds.

        Corollary1LetHbe commutative and noetherian (particularly finite dimensional), semisimple and cosemisimple. Then eachM∈HYDH(α,β) is a direct sum of a family of simple subobjects ofMwhich are finitely generated asH-modules inHYDH(α,β). HenceHYDH(α,β) is a semisimple category.

        ProofSinceHis cosemisimple, the functor (-)coH:MH→kMis exact. Thus (-)coH:HYDH(α,β)→kMis exact. Furthermore, the semisimplicity implies thatHYDH(α,β) satisfies the exact condition. Then by Theorem 1, the conclusion holds.

        Theorem2As the disjoint union of allHYDH(α,β), the category of the generalized Yetter-Drinfeld modules YD(H) is also semisimple.

        [1]Panaite F, Staic M D. Generalized (anti) Yetter-Drinfeld modules as components of a braided T-category [J].IsraelJMath, 2007,158(1): 349-365.

        [2]Liu L, Wang S H. Constructing new braided T-categories over weak Hopf algebras [J].ApplCategorStruct, 2010,18(4): 431-459.

        [3]Turaev V.Homotopyquantumfieldtheory[M]. Bloomington: European Mathematical Society, 2010.

        [4]Etingof P, Nikshych D, Ostrik V. On fusion categories [J].AnnalsofMathematics, 2005,162(2): 581-642.

        [5]Drinfeld V, Gelaki S, Nikshych D, Ostrik V. On braided fusion categories Ⅰ [J].SelectaMathematica, 2010,16(1): 1-119.

        [6]Naidu D, Rowell E. A finiteness property for braided fusion categories [J].AlgebrRepresentTheor, 2011,14(5): 837-855.

        [7]Sweedler M.Hopfalgebras[M]. New York: Benjamin, 1969.

        [8]Montgomery S.Hopfalgebrasandtheiractionsonrings[M]. Rhode Island: American Mathematical Society, 1993.

        色老板精品视频在线观看| 蜜乳一区二区三区亚洲国产| 日本av在线一区二区| 久久99精品久久久久久秒播| 日本免费人成视频播放| 人妻少妇久久精品一区二区 | 国产精品美女自在线观看| 日本在线精品一区二区三区| 曰本大码熟中文字幕| 91网站在线看| 成年人视频在线播放麻豆| 精品国产午夜肉伦伦影院| 成人免费看www网址入口| 久久精品国产亚洲AV无码不| 日本精品少妇一区二区| 亚洲成av人片天堂网无码| 狠狠色丁香久久婷婷综合蜜芽五月 | 挺进邻居丰满少妇的身体| 久久精品国产精品亚洲| 野外性史欧美k8播放| 午夜在线观看有码无码| 国内精品国产三级国产| 亚洲熟妇少妇任你躁在线观看无码 | 国产一起色一起爱| 亚洲国产一区二区网站| 中文字幕一区日韩精品| 色综合自拍| 粗一硬一长一进一爽一a视频| 国产在线视频一区二区天美蜜桃| 人妻饥渴偷公乱中文字幕| 国产av无码专区亚洲av手机麻豆| 中文字幕色视频在线播放| 国内揄拍国内精品人妻久久| 又湿又紧又大又爽a视频国产| 亚洲av无码乱观看明星换脸va | 亚洲一区二区三区ay| 亚洲成a人片在线观看无码专区| 国产精品区一区第一页| 米奇亚洲国产精品思久久| 亚洲麻豆视频免费观看| 东京无码熟妇人妻av在线网址|