Complex Analytic Groups and Hopf Algebras

Abstract

Let R be a commutative Hopf algebra with comultiplication 7 and antipode ti over the complex number field C. Then the C-algebra homomorphisms of R into C constitute a pro-affine algebraic group A = %R), the group composition being given in terms of comultiplications 7 of R by (x, y) -* (x y) ? 7 ([7]). When R is the algebra R(G) of representative functions of a complex analytic group G, then the proaffine algebraic group G* = 1 + 1 ? /?. If i? is an algebra of representative functions on a group G, then the primitive elements are precisely those functions which are homomorphisms of G into the additive group of the base field. An element u of the Hopf algebra R is called a group-like element if u # 0, and y(u) = u u. In the case where R is an algebra of representative functions on a group G, the group-like elements are precisely those functions which are homomorphisms of G into the multiplicative group of the nonzero elements of the base field. A subspace S of a Hopf algebra R is called left stable if 7(5) C S R right stable if 7(5) C R S, and bistable if it is both left and right