Some properties of Hopf algebras and H-module algebras.

摘要

In this dissertation we consider some properties of Hopf algebras and of H-module algebras. Some of these properties are analogues of the corresponding properties of group algebras, other properties are concerned with the actions of Hopf algebras on other algebras.;In the first chapter we give basic definitions.;In the second chapter we state and prove an analogue of Frobenius-Schur theorem for finite-dimensional split semisimple Hopf algebras over a field of characteristic zero. This also gives an independent proof of the classical Frobenius-Schur theorem for finite groups.;In the third chapter we give an algorithmic approach to the Kaplansky's sixth conjecture for finite-dimensional semisimple Hopf algebras.;In the fourth chapter we consider associative H-module algebras. We show that the Jacobson radical of a finite-dimensional H-module algebra is an H-ideal provided the Hopf algebra is finite-dimensional and the characteristic of the base field is zero. We prove some results on nilpotent H-module algebras.;In the fifth chapter we deal with generalized Lie algebras (which are also H-module algebras). Here we consider cotriangular Hopf algebras and use a theorem of P. Etingof and S. Gelaki on the structure of pseudoinvolutive cotriangular Hopf algebras over a field of characteristic zero to reduce studying of generalized Lie algebras over such Hopf algebras to studying Lie superalgebras with a locally finite regular action by automorphisms of a proalgebraic group. This allows us to prove analogues of the Poincare-Birkhoff-Witt and Ado theorems.;In the last fifth chapter we study the question about existence of a non-trivial Lie identity for a generalized Lie algebra in the case when its subalgebra of invariants satisfies a non-trivial Lie identity. We show that in the case when the characteristic of the ground field is zero the question has a positive answer if and only if the Hopf algebra is semisimple.