Semi-direct products of Hopf algebras

Abstract

The object of this paper is to investigate the notion of “semi-direct product” for Hopf algebras. We show that there are two such notions: the well-known concept of smash product, and the dual notion of smash coproduct introduced here. We investigate the basic properties of these notions and give several examples and applications. If G is an affine algebraic group (as defined in [4]) with coordinate ring A(G), then the coalgebra structure of A(G) “contains” the rational representation theory of G in the sense that the rational G-modules are precisely the A(G)comodules. Now if G is the semi-direct product of algebraic subgroups N and K (i.e., G = Nx,K as affine algebraic groups) then clearly A(G) = A(N) @ A(K) as algebras. But one would also like to know how the coalgebra structure of A(G) is related to the coalgebra structures of A(N) and A(K). In fact, the twisted multiplication on Nx,K induces a twisted comultiplication on A(N) @ A(K), called the smash coproduct of A(N) by A(K). This comultiplication is compatible with the tensor product algebra structure, and we have A(G) isomorphic to the smash coproduct of A(N) by A(K) (denoted A(N) x A(K)) as Hopf algebras. Similarly, a semi-direct product decomposition of a Lie algebra L induces a smash product decomposition of the universal enveloping algebra U(L), which in turn induces (by dualizing) a smash coproduct decomposition of the Hopf algebra U(L) of representative functions on U(L). This generalizes an earlier construction given by Hochschild [3, pp. 514151. In Section 2 we review the properties of the adjoint action and the notion of smash product for a Hopf algebra, and introduce the dual notions of coadjoint action and smash coproduct. (For an independent treatment in the special cast of commutative Hopf algebras see [l].) The smash product or smash co-