有限维基本Hopf代数的分类及相关主题
摘要
本论文的主要目的是分类有限维的Hopf代数，特别地去分类有限维的基本Hopf代数。我们的思想是通过其表示型来分类他们，我们的方法主要依赖于有限维代数的表示理论。
为了分类有限维的Hopf代数，我们给出了以下四个步骤：
(1)给出一个有效的方法来决定基本Hopf代数的表示型；
(2)通过表示型来分类有限维基本Hopf代数；
(3)确定一个Hopf代数什么时候会Morita等价于一个基本Hopf代数；
(4)寻找新的途径来将(2)的结果推广到一般的有限维Hopf代数。
为了解决步骤(1)，我们为每一个基本Hopf代数H配备了一个被称为表示型数的数n_H并且证明(ⅰ)H是有限型当且仅当n_H=0或n_H=1；(ⅱ)如果H是Tame型，则n_H=2；(ⅲ)如果n_H≥3，则H是Wild型。
对于(2)，目前，我们已经给出了有限型的完整分类。具体地讲，它们共分三类：①如果H是半单的，则H同构与一个群代数的对偶；②如果H是非半单的并且基础域的特征是0的话，则H同构一个所谓Andruskiewitsch-Schneider代数与一个群代数交差积的对偶；③如果H是非半单的并且基础域的特征不是0的话，则H同构于某个特定代数与一个群代数交差积的对偶。对于Tame型的Basic Hopf代数，我们可以给出的是根分次情形的结构定理。我们将看到根分次的情形至多只有五类。我们还给出了一些关于TameHopf代数的例子。
广义路(余)代数为我们解决步骤(3)(4)提供了一种可能。我们首先研究了所谓的广义路余代数的同构问题，证明了两个广义路余代数k(△，C)≌k(△′，D)当且仅当存在quiver的同构ψ：△→△′使得S_i≌T_ψ(i)对任意i∈△_0。我们还给出了广义路(余)代数的Gabriel's定理。关于一个广义路余代数上什么时候具有Hopf代数结构的问题也被解决。
The main aim of this paper is to classify finite dimensional Hopf algebras, especially basic Hopf algebras. Our idea is to classify them through their representation type and our methods relay heavily on the representation theory of finite dimensional algebras.
In order to do so, we give four programs to classify finite dimensional Hopf algebras as follows.
(1) Give an effective way to determine the representation type of a finite dimensional basic Hopf algebra;
(2) Classify finite dimensional basic Hopf algebras through their representation type;
(3) Determine that when a finite dimensional Hopf algebra is Morita equivalent to a finite dimensional basic Hopf algebra;
(4) Find some new ways to generalize the conclusions in (2) to general finite dimensional Hopf algebras.
In order to resolve program (1), we attache every finite dimensional basic Hopf algebra H a number nh which is called representation type number of H and proved that (i) H is of finite representation type if and only if nH = 0 or nH = 1; (ii) H is tame then nh = 2 and (iii) H is wild if nH≥ 3.
For program (2), we can classify finite dimensional basic Hopf algebras of finite representation type completely now. Explicitly, they are consist of three classes: (i) If H is semisimple, then H (?) k(G)* for some finite group; (ii) If H is not semisimple and the characteristic of k is zero, then H is isomorphic the dual of the cross product between one so called Andruskiewitsch-Schneider algebra and a group algebra; (iii) If H is not semisimple and the characteristic of k is not zero, then H is isomorphic the dual of the cross product between one special algebra and a group algebra. We also can give the structure theorem for finite dimensional basic Hopf algebra of tame type in the radical graded case. We can see that in this case they are consist of five classes at most. More examples about tame Hopf algebras are also given in this paper.
Generalized path (co) algebras give us one possibility to solve programs (3) (4). We study so called isomorphism problem for generalized path coalgebras at first and prove that two normal generalized path coalgebras k(△,C) (?)(△',D) as coalgebras if and only if there is an isomorphism of quivers (?) : △ →△' such that Si (?) T(?)(i) as coalgebras for i ∈ △0. The Gabriel's Theorem for generalized path (co)algebras are also given in this paper. The problem of when there is a Hopf structure on generalized path coalgebra is settled.
为了分类有限维的Hopf代数，我们给出了以下四个步骤：
(1)给出一个有效的方法来决定基本Hopf代数的表示型；
(2)通过表示型来分类有限维基本Hopf代数；
(3)确定一个Hopf代数什么时候会Morita等价于一个基本Hopf代数；
(4)寻找新的途径来将(2)的结果推广到一般的有限维Hopf代数。
为了解决步骤(1)，我们为每一个基本Hopf代数H配备了一个被称为表示型数的数n_H并且证明(ⅰ)H是有限型当且仅当n_H=0或n_H=1；(ⅱ)如果H是Tame型，则n_H=2；(ⅲ)如果n_H≥3，则H是Wild型。
对于(2)，目前，我们已经给出了有限型的完整分类。具体地讲，它们共分三类：①如果H是半单的，则H同构与一个群代数的对偶；②如果H是非半单的并且基础域的特征是0的话，则H同构一个所谓Andruskiewitsch-Schneider代数与一个群代数交差积的对偶；③如果H是非半单的并且基础域的特征不是0的话，则H同构于某个特定代数与一个群代数交差积的对偶。对于Tame型的Basic Hopf代数，我们可以给出的是根分次情形的结构定理。我们将看到根分次的情形至多只有五类。我们还给出了一些关于TameHopf代数的例子。
广义路(余)代数为我们解决步骤(3)(4)提供了一种可能。我们首先研究了所谓的广义路余代数的同构问题，证明了两个广义路余代数k(△，C)≌k(△′，D)当且仅当存在quiver的同构ψ：△→△′使得S_i≌T_ψ(i)对任意i∈△_0。我们还给出了广义路(余)代数的Gabriel's定理。关于一个广义路余代数上什么时候具有Hopf代数结构的问题也被解决。
The main aim of this paper is to classify finite dimensional Hopf algebras, especially basic Hopf algebras. Our idea is to classify them through their representation type and our methods relay heavily on the representation theory of finite dimensional algebras.
In order to do so, we give four programs to classify finite dimensional Hopf algebras as follows.
(1) Give an effective way to determine the representation type of a finite dimensional basic Hopf algebra;
(2) Classify finite dimensional basic Hopf algebras through their representation type;
(3) Determine that when a finite dimensional Hopf algebra is Morita equivalent to a finite dimensional basic Hopf algebra;
(4) Find some new ways to generalize the conclusions in (2) to general finite dimensional Hopf algebras.
In order to resolve program (1), we attache every finite dimensional basic Hopf algebra H a number nh which is called representation type number of H and proved that (i) H is of finite representation type if and only if nH = 0 or nH = 1; (ii) H is tame then nh = 2 and (iii) H is wild if nH≥ 3.
For program (2), we can classify finite dimensional basic Hopf algebras of finite representation type completely now. Explicitly, they are consist of three classes: (i) If H is semisimple, then H (?) k(G)* for some finite group; (ii) If H is not semisimple and the characteristic of k is zero, then H is isomorphic the dual of the cross product between one so called Andruskiewitsch-Schneider algebra and a group algebra; (iii) If H is not semisimple and the characteristic of k is not zero, then H is isomorphic the dual of the cross product between one special algebra and a group algebra. We also can give the structure theorem for finite dimensional basic Hopf algebra of tame type in the radical graded case. We can see that in this case they are consist of five classes at most. More examples about tame Hopf algebras are also given in this paper.
Generalized path (co) algebras give us one possibility to solve programs (3) (4). We study so called isomorphism problem for generalized path coalgebras at first and prove that two normal generalized path coalgebras k(△,C) (?)(△',D) as coalgebras if and only if there is an isomorphism of quivers (?) : △ →△' such that Si (?) T(?)(i) as coalgebras for i ∈ △0. The Gabriel's Theorem for generalized path (co)algebras are also given in this paper. The problem of when there is a Hopf structure on generalized path coalgebra is settled.
引文
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[31] E.Green, 0.Solberg, Basic Hopf Algebras and Quantum Groups, Math.Z, 229(1998), 45-76
[32] I.Heckenberger, Finite Dimensional Rank 2 Nichols Algebras of Diagonal Type I: Examples, arXiv:math.QA/0402350 v2
[33] I.Heckenberger, Finite Dimensional Rank 2 Nichols Algebras of Diagonal Type II: Classification, arXiv:math.QA/0404008 vl
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[41] F.Li, Characterization of Left Artinian Algebras Through Pseudo Path Algebras, submmitted
[42] F.Li, Weak Hopf Algebras and Some New Solutions of Quantum Yang-Baxter Equation, J. Algebra, 208 (1998), 72-100
[43] F.Li, Solutions of Yang-Baxter Equation in an Endomorphism Semigroup and Quasi- (Co)braided Almost Bialgebras, Comm in Alg, 28 (5), 2253-2270 (2000)
[44] F.Li, The Right Braids, Quasi-Braided Pre-Tensor Ctegories and General Yang- Baxter Operators, Comm in Alg, Volume 32, Number 2, (2004)
[45] F.Li, S.Duplij, Weak Hopf Algebras and Singular Solutions of Quantum Yang-Baxter Equation, Comm.Math.Phys, 225(1): 192-217 (2002)
[46] G.X.Liu, F.Li, Basic Hopf Algebras of Finite Representation Type and Their Classification, submitted to Proc of AMS. (available at http://www.cms.zju.edu.cn/index.asp?ColumnName=pdfbook)
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[2] N.Andruskiewitsch, P.Etingof, S.Gelaki, Triangular Hopf Algebras with the Chevalley Property, math.QA/0008232
[3] N.Andruskiewitsch, M.Grana, Braided Hopf Algebras over Non-abelian Groups, Bol. Acad. Ciencias (Cordoba) 63 (1999), 45-78
[4] N.Andruskiewitsch, H.- J.Schneider, Pointed Hopf Algebras, in "New direction in Hopf algebras", 1-68, Math. Sci. Res. Inst. Publ. 43, Cambridge Univ. Press, Cambridge, 2002
[5] N.Andruskiewitsch, H.-J.Schneider, Finite Quantum Groups and Cartan Matrices, Adv.Math, 154(2000), 1-45
[6] N.Andruskiewitsch, H.-J.Schneider, Hopf Algebras of Order p~2 and Braided Hopf Algebras of Order p, J.Algerba, 199(1998), 430-454
[7] N.Andruskiewitsch, H.-J.Schneider, Lifting of Quantum Linear Spaces and Pointed Hopf Algebras of Order p~3, J.Algebra, 209(1998), 659-691
[8] M.Auslander, I.Reiten, Smal?, Representation Theory of Artin Algebras, Cambridge University Press, 1995
[9] G.Bohm, F.Nill, K.Szlachanyi, Weak Hopf Algebras I. Integral Theory and C*- structure, J.Algebra, 221(1999), 385-438
[10] G.Bohm, K.Szlachanyi, A Coassociative C*-quantum Group with Nonintegral Dimensions, Lett, in Math.Phys, 35(1996), 437-456
[11] X.W.Chen, H.L.Huang, P.Zhang, Dual Gabriel Theorem and Quivers, preprint
[12] Xiao-Wu Chen, Hua-Lin Huang, Yu Ye, Pu Zhang, Monomial Hopf Algerbas, J.Algerba, 275(2004), 212-232
[13] W.Chin, S.Montgomery, Basic Coalgebra, AMS/IP Studies in Adv. Math, 4, 1997, 41-47
[14] C.Cibils, A Quiver Quantum Group, Comm.Math.Phys, 157(1993), no.3, pp 459-477
[15] C.Cibils, M.Rosso, Hopf Quiver, J.Algebra, 254(2002), 241-251
[16] F.U.Coelho, S.X.Liu, Generalized Path Coalgebras, Interactions between ring theory and repersentations of algebras (Muricia), Leture Notes in Pure and Appl. Math., Dekker, New York, 210, 2000, 53-66
[17] M.Cohen, D.Fishman, Hopf Algebra Actions, J.Algebra, 100(1986), 363-379
[18] W.W. Crawley-Boevey, Tame Algebras and Generic Godules, Proc. L.M.S, 63(1991), 241-264
[19] W.W. Crawley-Boevey, Lectures on Representations of Quivers, Lectures in Oxford, 1992
[20] Y.Doi, Homological Coalgebra, J. Math. Soc. Japan, 33(1), 1981, 31-50
[21] Yu.A.Drozd, Tame and Wild Matrix Problems, Representations and Quadratic Forms. Inst.Math., Acad.Sciences.Ukrainian SSR, Kiev 1979, 39-74. Amer.Math.Soc.
Transl. 128(1986), 31-55
[22] Yu.A.Drozd, V.V.Kirichenko, Finite Dimensional Algebras, Springer-Verlag, Berlin- Heidelberg-New York-Tokyo, 1993
[23] K.Erdmann, Blocks of Tame Representation Type and Related Algebras, LNM 1428, Springer-Verlag, 1990
[24] K.Erdmann, E.L.Green, N.Snashall, R.Taillefer, Representation Theory of the Drin- feld Double of a Family of Hopf Algebras, arXiv:math.RT/0410017 v1
[25] P.Etingof, S.Gelaki, On Finite-dimensional Semisimple and Cosemisimple Hopf Algebras in Positive Characteristic, International Mathematics Research Notices, no 16, 1998, 851-864
[26] P.Etingof, S.Gelaki, The Classification of Triangular Semisimple and Cosemisimple Hopf Algebras over an Algebraically Closed Field, International Mathematics Re- search Notices, no 5 (1997), 223-234
[27] P.Etingof, D.Nikshych, Dynamical Quantum Groups at Roots of 1, Duke Math.J, 108 (2001), no.1, 135-168
[28] P.Etingof, O.Schiffmann, Lecture on the Dynamical Yang-Baxter Equations, Quantum groups and Lie theory ( Durham, 1999), 89-129. London Math Soc. Lecture Note Ser., 290, Cambridge Univ. Press, Cambridge, 2001
[29] D.Fischman, S.Montgomery, H.-J.Schneider, Frobenius Extensions of Subalgebras of Hopf Algebras, Trans. Amer. Math. Soc, 349(1997), 4857-4895
[30] M.Grana, On Nichols Algebras of Low Dimension, in "New Trends in Hopf Algebra Theorey", Contemp.Math, 267(2000), 111-134
[31] E.Green, 0.Solberg, Basic Hopf Algebras and Quantum Groups, Math.Z, 229(1998), 45-76
[32] I.Heckenberger, Finite Dimensional Rank 2 Nichols Algebras of Diagonal Type I: Examples, arXiv:math.QA/0402350 v2
[33] I.Heckenberger, Finite Dimensional Rank 2 Nichols Algebras of Diagonal Type II: Classification, arXiv:math.QA/0404008 vl
[34] R.G.Heyneman, D.E.Radford, Reflexivity and Coalgebras of Finite Type, J. Algebra, 28(2), 1974, 215-246
[35] A.Joyal, R.Street, Braided Tensor Categories, Adv. Math 102 (1993), 20-78
[36] L.Kadison, D.Nikshych, Frobenius Extensions and Weak Hopf Algebra, J.Algebra, 244 (2001), 312-342
[37] C.Kassel, Quantum groups, GTM 155, Springer-Verlag, New York, 1995
[38] H.Krause, Stable Equivalnece Preserves Representation Type, Comment. Math. Helv, 72(1997), 266-284
[39] R.G.Larson, D.E.Radford, Semisimple Cosemisimple Hopf Algebras. Amer. J.Math, 1988, 110, 187-195
[40] R.G.Larson, D.E.Radford, Finite Dimensional Cosemisimple Hopf Algebras in Char-acteristic 0 Are Semisimple. J. Algebra, 117(1988), 267-289
[41] F.Li, Characterization of Left Artinian Algebras Through Pseudo Path Algebras, submmitted
[42] F.Li, Weak Hopf Algebras and Some New Solutions of Quantum Yang-Baxter Equation, J. Algebra, 208 (1998), 72-100
[43] F.Li, Solutions of Yang-Baxter Equation in an Endomorphism Semigroup and Quasi- (Co)braided Almost Bialgebras, Comm in Alg, 28 (5), 2253-2270 (2000)
[44] F.Li, The Right Braids, Quasi-Braided Pre-Tensor Ctegories and General Yang- Baxter Operators, Comm in Alg, Volume 32, Number 2, (2004)
[45] F.Li, S.Duplij, Weak Hopf Algebras and Singular Solutions of Quantum Yang-Baxter Equation, Comm.Math.Phys, 225(1): 192-217 (2002)
[46] G.X.Liu, F.Li, Basic Hopf Algebras of Finite Representation Type and Their Classification, submitted to Proc of AMS. (available at http://www.cms.zju.edu.cn/index.asp?ColumnName=pdfbook)
[47] M.Lorenz, Representations of Finite Dimensional Hopf Algebras, J.Algebra, 188(1997), 476-505
[48] M.E.Lorenz, M. Lorenz, On Crossed Products of Hopf Algebras. Proc of AMS, 123, 1995, 33-38
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