Representations of Hopf Algebras Arising fromLie Algebras of Cartan Type@c

Abstract

w x 1.1. This is a continuation of 13 in an effort to use algebraic group actions to study the representations of restricted Lie algebras of Cartan type. In this paper we study the representation theory of certain finite-diw x mensional Hopf algebras that arise from the construction in 13 . The representation theory of these Hopf algebras closely resembles the representation theory of Frobenius kernels of reductive algebraic groups in positive characteristics. Let G be a connected reductive algebraic group scheme over an algebraically closed field k of characteristic p ) 0. It is well known that if F : G a G is the r th iteration of the Frobenius map and G s ker F , r r there is a chain of normal subgroup schemes G eG e ??? eG. In earlier 1 2 terminology this is equivalent to the existence of a chain of finite-dimenŽ . Ž . sional cocommutative Hopf algebras Dist G ; Dist G ; ??? ; 1 2 Ž . Ž . Dist G , where Dist G is known as the r th hyperalgebra of the distribur Ž . Ž . tion algebra Dist G of G. Let g be the Lie algebra of G and u g be its restricted universal enveloping algebra. There exists an isomorphism of