For a field k and two finite groups G and X, when G acts on X from the right by group automorphisms, there is a Hopf algebra structure on k-space （kX°P）＊ kG （see Theorem 2.1）, called a non-balanced quantum ...For a field k and two finite groups G and X, when G acts on X from the right by group automorphisms, there is a Hopf algebra structure on k-space （kX°P）＊ kG （see Theorem 2.1）, called a non-balanced quantum double and denoted by Dx（G）. In this paper, some Hopf algebra properties of Dx （G） are given, the representation types of Dx （G） viewed as a k-algebra are discussed, the algebra structure and module category over Dx（G） are studied. Since the Hopf algebra structure of non-balanced quantum double DX （G） generMizes the usual quantum double D（G） for a finite group G, all results about Dx（G） in this paper can also be used to describe D（G） as a special case and the universal R-matrix of Dx （G） provides more solutions of Yang-Baxter equation.展开更多
In this paper, we define the notion of self-dual graded weak Hopf algebra and self-dual semilattice graded weak Hopf algebra. We give characterization of finite-dimensional such algebras when they are in structually s...In this paper, we define the notion of self-dual graded weak Hopf algebra and self-dual semilattice graded weak Hopf algebra. We give characterization of finite-dimensional such algebras when they are in structually simple forms in the sense of E. L. Green and E. N. Morcos. We also give the definition of self-dual weak Hopf quiver and apply these types of quivers to classify the finite- dimensional self-dual semilattice graded weak Hopf algebras. Finally, we prove partially the conjecture given by N. Andruskiewitsch and H.-J. Schneider in the case of finite-dimensional pointed semilattice graded weak Hopf algebra H when grH is self-dual.展开更多
This is a contribution to the project of quiver approaches to quasi-quantum groups.We classify Majid bimodules over groups with 3-cocycles by virtue of projective representations.This leads to a theoretic classificati...This is a contribution to the project of quiver approaches to quasi-quantum groups.We classify Majid bimodules over groups with 3-cocycles by virtue of projective representations.This leads to a theoretic classification of graded pointed Majid algebras over path coalgebras,or equivalently cofree pointed coalgebras,and helps to provide a projective representation-theoretic description of the gauge equivalence of graded pointed Majid algebras.We apply this machinery to construct some concrete examples and obtain a classification of finite-dimensional graded pointed Majid algebras with the set of group-likes equal to the cyclic group of order 2.展开更多
基金Supported by the National Natural Science Foundation of China(11271318,11571173)the Zhejiang Provincial Natural Science Foundation of China(LZ13A010001)
基金Supported by Doctoral Foundation of Qingdao University of Science and Technology (20080022398)the National Natural Science Foundation of China (11271318, 11171296)the Specialized Research Fund for the Doctoral Program of Higher Education of China (20110101110010)
文摘For a field k and two finite groups G and X, when G acts on X from the right by group automorphisms, there is a Hopf algebra structure on k-space （kX°P）＊ kG （see Theorem 2.1）, called a non-balanced quantum double and denoted by Dx（G）. In this paper, some Hopf algebra properties of Dx （G） are given, the representation types of Dx （G） viewed as a k-algebra are discussed, the algebra structure and module category over Dx（G） are studied. Since the Hopf algebra structure of non-balanced quantum double DX （G） generMizes the usual quantum double D（G） for a finite group G, all results about Dx（G） in this paper can also be used to describe D（G） as a special case and the universal R-matrix of Dx （G） provides more solutions of Yang-Baxter equation.
基金the Program for New Century Excellent Talents in University (No 04-0522)the National Natural Science Foundation of China (No.10571153)
文摘In this paper, we define the notion of self-dual graded weak Hopf algebra and self-dual semilattice graded weak Hopf algebra. We give characterization of finite-dimensional such algebras when they are in structually simple forms in the sense of E. L. Green and E. N. Morcos. We also give the definition of self-dual weak Hopf quiver and apply these types of quivers to classify the finite- dimensional self-dual semilattice graded weak Hopf algebras. Finally, we prove partially the conjecture given by N. Andruskiewitsch and H.-J. Schneider in the case of finite-dimensional pointed semilattice graded weak Hopf algebra H when grH is self-dual.
基金supported by National Natural Science Foundation of China(Grant No. 10601052)Natural Science Foundation of Shandong Province(Grant No.2009ZRA01128)the Independent Innovation Foundation of Shandong University(Grant No.2010TS021)
文摘This is a contribution to the project of quiver approaches to quasi-quantum groups.We classify Majid bimodules over groups with 3-cocycles by virtue of projective representations.This leads to a theoretic classification of graded pointed Majid algebras over path coalgebras,or equivalently cofree pointed coalgebras,and helps to provide a projective representation-theoretic description of the gauge equivalence of graded pointed Majid algebras.We apply this machinery to construct some concrete examples and obtain a classification of finite-dimensional graded pointed Majid algebras with the set of group-likes equal to the cyclic group of order 2.
