On Chevalley-Eilenberg and Cyclic Homologies

where H;(_) denotes the Lie algebra homology, HC*(_) cyclic homology, and A is an algebra with unit over a field of characteristic zero. In this paper we give an alternative proof of this theorem which does not involve Weyl's invariant theory for GL(C). We use this approach to compute Chevalley-Eilenberg homology in some interesting new cases. In Section 1, we begin by proving (in characteristic 0) a Leday-QuillenTsygan theorem for complexes which occur as subcomplexes D* of the Chevalley-Eilenberg complex /\ * gl(A); the conditions we require of D * are minimal. In particular, D * need not be closed under the conjugation action of GL(Q), but only W(Q) = monomial matrices with entries in Q. Under the condition that this action is the identity map on homology, together with 3 other basic conditions (P1-P4, section 1) we show that there is an isomorphism of graded commutative Hopf algebras