ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

2. Algebra

1991 MS Classification: 16, 13 Totok, Alexander, Moscow State University, Russia Actions of Hopf algebras on noncommutative algebras An action of a finite-dimensional Hopf algebra H on a noncommutative associative algebra A is considered. Properties of A^{H}, the subalgebra of invariants in A, are studied. A lot of information is available in the case A is commutative. We turn to noncommutative case. Some generalizations of results from [1] on integrality of extension A/A^{H} are obtained for some classes of noncommutative algebras, in particular for the case when A is integral over Z(A), the centre of an algebra A. Next the canonical action of H on the tensor algebra T (H) is considered. With the help of the technique developed some structure theorems for commutative Hopf algebras are proved: Let H be a commutative finitedimensional Hopf algebra and char k = 0 or > dim H. If the coradical H_{0} is a sub-Hopf algebra of H or cocommutative, then H is cosemisimple, i. e. H = H_{0}. In particular, a commutative pointed Hopf algebra with char k = 0 or > dim H is a group Hopf algebra. Also the example is constructed, showing that restrictions on char k are necessary. References: [1] A. Totok, Ob invariantakh konechnomernykh algebr Hopfa (On invariants of finite-dimensional Hopf algebras), Vestnik MGU: Matematika, Mekhanika 1997 N3, 31 – 34 (in russian). [2] A. Totok, Deistviya algebr Hopfa (Actions of Hopf algebras), Matematichesky Sbornik 189 (1998) N1, 149 – 160 (in russian).