On Coverings and Hyperalgebras of Affine Algebraic Groups

Abstract

Over an algebraically closed field of characteristic zero, the universal group covering of a connected affine algebraic group, if such exists, can be constructed canonically from its Lie algebra only. In particular the isomorphism classes of simply connected affine algebraic groups are in 1-1 correspondence with the isomorphism classes of finite dimensional Lie algebras of some sort. We shall consider the counterpart of these results (due to Hochschild) in case of a positive characteristic, replacing the Lie algebra by the “hyperalgebra". We show that the universal group covering of a connected affine algebraic group scheme can be constructed canonically from its hyperalgebra only and hence, in particular, that the category of simply connected affine algebraic group schemes is equivalent to a subcategory of the category of hyperalgebras of finite type which contains all the semisimple hyperalgebras.