On the structure of commutative pointed Hopf algebras

Abstract

In this paper we study the general structure of a commutative pointed Hopf algebra over a field k (for example, the underlying Hopf algebra of a representationally solvable affine algebraic group). Sullivan’s work [6, 71 and the author’s [4] provide the basis for this investigation. The algebra structure of a commutative pointed Hopf algebra A seems to be best understood by a detailed analysis of the coalgebra structure. If B C A is a sub-Hopf algebra and A,, S B, then A as a B-algebra can be described in terms of the coradical filtration. In particular, if the ground field has characteristic 0, then A is a polynomial algebra with coefficients in B. In this case, if A is a finitely generated B-algebra, we examine properties of the function yB(C) (the cardinality of a set of indeterminants) defined on sub-Hopf algebras C containing B. Suppose iz is a commutative pointed Hopf algebra over any field k. If A is generated (as an algebra) by the nth term of its coradical filtration, then the same is true of any sub-Hopf algebra. This generalizes Theorem 3.4 of [4]. The result can be interpreted in terms of the representation of the (affine) group scheme G, determined by a sub-Hopf algebra B (see [4] or [8], for example). If -4 is any pointed Hopf algebra over a field k, there is a unique sub-Hopf algebra &? minimal with respect to the property that A,98 = A. Our proof is constructive and sheds light on the role G(A) (the grouplikes of A) plays in the structure of A. We examine the relationship between the function rB( ) and g when iz is a commutative pointed finitely generated &,-algebra in characteristic 0. Throughout we shall use rather freely the notation and results of [l] and [9]. All vector spaces will be over a field K. *This paper was written during the time the author was a summer visitor to the Institute for Advanced Study, Princeton, N. J. The author wishes to express his gratitude for the institution’s hospitality. The research was supported in part by National Science Foundation Grant MCS74-14580 A02.