余半单Hopf代数的Frobenius性质.pdfVIP

273 Vol.27, No.3 20078 JOURNAL OF MATHEMATICAL RESEARCH AND EXPOSITION Aug., 2007 Article ID: 1000-341X(2007)03-0469-05 Document code: A Frobenius Property of a Cosemisimple Hopf Algebra 1 1 2 DONG Jing-cheng , DAI Li , LI Li-bin (1. Engineering College, Nanjing Agricultural University, Jiangsu 210031, China; 2. School of Math., Yangzhou University, Jiangsu 225002, China ) (E-mail: djcdl@) Abstract: Let H be a cosemisimple Hopf algebra over an algebraically closed field k of characteristic zero. We show that if H is of type l : 1 + m : p + 1 : q with p2 q , or of type 1 : 1 + 1 : m + 1 : n in the sense of Larson and Radford, then H has the Frobenius property, that is, Kaplansky conjecture is true for these Hopf algebras. Key words: Hopf algebra; type; Frobenius property. MSC(2000): 16W30; 57T05 CLC number: O153 1. Introduction In recent years, the theory of cosemisimple Hopf algebras over an algebraically closed field k of characteristic zero has been studied considerably. Many results have been found. For example, it was showed in [1, 2] that H is cosemisimple if H is semisimple and S2 = id. Recall that a cosemsimple Hopf algebra H is said to have the Frobenius property if the dimensions of simple H-comodules divide the dimension of H . It was conjectured by Kaplansky in [3] that every cosemisimple Hopf algebra had the Frobe- nius property. This is still an open problem, although it has been shown in [4] that any coqua- sitriangular cosemisimple Hopf algebra H has the Frobenius property. In several recent