Extensions of Modules over Hopf Algebras Arising from Lie Algebras of Cartan Type

Abstract

In this paper we prove that there are no self-extensions of simple modules over restricted Lie algebras of Cartan type. The proof given by Andersen for classical Lie algebras not only uses the representation theory of the Lie algebra, but also representations of the corresponding reductive algebraic group. The proof presented in the paper follows in the same spirit by using the construction of a infinite-dimensional Hopf algebra D(G) u( $$\mathfrak{g}$$ ) containing u( $$\mathfrak{g}$$ ) as a normal Hopf subalgebra, and the representation theory of this algebra developed in our previous work. Finite-dimensional hyperalgebra analogs D(G r ) u( $$\mathfrak{g}$$ ) have also been constructed, and the results are stated in this setting.