Cohomology operations for Lie algebras

Abstract

If $L$ is a Lie algebra over $\mathbb {R}$ and $Z$ its centre, the natural inclusion $Z\hookrightarrow (L^{*})^{*}$ extends to a representation $i^{*} : \Lambda Z\to \operatorname {End} H^{*}(L,\mathbb {R})$ of the exterior algebra of $Z$ in the cohomology of $L$. We begin a study of this representation by examining its Poincaré duality properties, its associated higher cohomology operations and its relevance to the toral rank conjecture. In particular, by using harmonic forms we show that the higher operations presented by Goresky, Kottwitz and MacPherson (1998) form a subalgebra of $\operatorname {End} H^{*}(L,\mathbb {R})$, and that they can be assembled to yield an explicit Hirsch-Brown model for the Borel construction associated to $0\to Z\to L\to L/Z\to 0$.