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Association des Annales de l'Institut Fourier, Annales de l'Institut Fourier, 4(59), p. 1337-1358

DOI: 10.5802/aif.2466

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Transgression and Clifford algebras

Journal article published in 2009 by Rudolf Philippe Rohr ORCID
This paper is made freely available by the publisher.
This paper is made freely available by the publisher.

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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 $H(W/ )$ 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 $\qWg = \Ug ⊗ \Clg$ 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). ; Comment: 19 pages