Pointed irreducible bialgebras

Abstract

It is well-known [5, p. 274, Theorem 13.0.11 that a pointed irreducible cocommutative bialgebra B over a field of characteristic zero is isomorphic as a bialgebra to U(P(B)), the universal enveloping algebra of the Lie algebra P(B) of primitives of B. One method of proof is first to show that B and U@‘(B)) are each coalgebra-isomorphic to the cofree pointed irreducible cocommutative coalgebra on the vector space P(B), and then to use these coalgebra isomorphisms to induce a bialgebra isomorphism. In this paper we prove a dual version of the above. Let B be a pointed irreducible commutative bialgebra over a field of characteristic zero. Write I for its augmentation ideal, and Q(B) = l/r2. Then B is isomorphic as a bialgebra to the appropriate universal object UIC(Q(B)). We first show that each is isomorphic as an algebra to Sym(Q(B)), and then use these algebra isomorphisms to induce a bialgebra isomorphism. The algebra isomorphism B q Sym(Q(B)) that we obtain is well-known, at least in the atline case. When B is affine, B represents a unipotent algebraic group scheme G. The Lie algebra L of G defines an atie scheme L, , which is represented by Sym(L*). There is a natural scheme isomorphism exp: L, % G [2, IV, Sect. 2, no. 4.11, and thus a natural isomorphism B SI Sym(L*). The naturality allows the passage to the pro-affine case via direct limits, and the isomorphism so obtained is the same as the one we give here. The corresponding bialgebra isomorphism exhibited in [5] for the cocommutative case is not natural. We sketch at the end of Section 2 how a natural isomorphism could be given in that case as well. M. And& has given in [l] a proof of the main result of this paper for the case in which B is graded, with B, the field. Th e methods of that paper are quite different from ours; the gradation facilitates inductive arguments.