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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Cohomology operations for Lie algebras


Authors: Grant Cairns and Barry Jessup
Journal: Trans. Amer. Math. Soc. 356 (2004), 1569-1583
MSC (2000): Primary 17B56, 55P62
Published electronically: November 4, 2003
MathSciNet review: 2034319
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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 $0\to Z\to L\to L/Z\to 0$.


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Additional Information

Grant Cairns
Affiliation: Department of Mathematics, La Trobe University, Melbourne, Australia 3086
Email: G.Cairns@latrobe.edu.au

Barry Jessup
Affiliation: Department of Mathematics and Statistics, University of Ottawa, Ottawa, Canada K1N 6N5
Email: bjessup@uottawa.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03392-0
PII: S 0002-9947(03)03392-0
Keywords: Lie algebra, cohomology, representation, Hirsch-Brown model, Borel construction, toral rank, harmonic form
Received by editor(s): December 4, 2002
Received by editor(s) in revised form: February 26, 2003
Published electronically: November 4, 2003
Additional Notes: This research was supported in part by NSERC and the ARC
Article copyright: © Copyright 2003 American Mathematical Society