Transgression and Clifford algebras

Abstract

Let W be a differential (not necessarily commutative) algebra which carries a free action of a polynomial algebra SP with homogeneous generators p 1,...,pr. We show that for W acyclic, the cohomology of the quotient H(W/) is isomorphic to a Clifford algebra C1(P, B), where the (possibly degenerate) bilinear form B depends on W. This observation is an analogue of an old result of Borel in a non-commutative context. As an application, we study the case of W given by the quantized Weil algebra W(g) = Ug ⊗ Clg for g a reductive Lie algebra. The resulting cohomology of the canonical Weil differential gives a Clifford algebra, but the bilinear form vanishes on the space of primitive invariants of the semi-simple part. As an application, we consider the deformed Weil differential (following Freed, Hopkins and Teleman).