Transgression and Clifford algebras

TL;DR

This paper establishes a connection between the cohomology of certain differential algebras with group actions and Clifford algebras, extending classical results to non-commutative settings and exploring applications in Lie algebra cohomology.

Contribution

It demonstrates that for acyclic differential algebras with polynomial group actions, the cohomology of quotients is isomorphic to Clifford algebras, generalizing Borel's classical result to non-commutative cases.

Findings

Cohomology of quotients is isomorphic to Clifford algebras for acyclic W.

In the quantized Weil algebra, cohomology yields a Clifford algebra with a degenerate bilinear form.

Application to deformed Weil differentials shows new algebraic structures in Lie algebra cohomology.

Abstract

Let $W$ be a differential (not necessarily commutative) algebra which carries a free action of a polynomial algebra $SP$ with homogeneous generators $p_1, >..., p_r$. We show that for $W$ acyclic, the cohomology of the quotient $H(W/<p_1, ..., p_r>)$ is isomorphic to a Clifford algebra $\text{Cl}(P,B)$, 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 $\qWg = \Ug \otimes \Clg$ for $\Lieg$ 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).