Transgression and Clifford algebras

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$\mathrm{.}Weshowthatfor$W$acyclic,thecohomologyofthequotient$H(W/<p_1, …, p_r>)$isisomorphictoaCliffordalgebra$$\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 $\text{qWg}=\text{Ug}\otimes \text{Clg}$ for $\text{Lieg}$ 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).