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Cohomology operations for Lie algebras

Authors:	Grant Cairns Barry Jessup

Institution:	Department of Mathematics, La Trobe University, Melbourne, Australia 3086 ; Department of Mathematics and Statistics, University of Ottawa, Ottawa, Canada K1N 6N5

Abstract:	If is a Lie algebra over $\mathbb{R}$ and 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 in the cohomology of . 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 $0\to Z\to L\to L/Z\to 0$ .

Keywords:	Lie algebra cohomology representation Hirsch-Brown model Borel construction toral rank harmonic form

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