Cohomology operations for Lie algebras
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- by Grant Cairns and Barry Jessup PDF
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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 $0\to Z\to L\to L/Z\to 0$.References
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Additional Information
- Grant Cairns
- Affiliation: Department of Mathematics, La Trobe University, Melbourne, Australia 3086
- MR Author ID: 44265
- ORCID: 0000-0002-9011-4567
- Email: G.Cairns@latrobe.edu.au
- Barry Jessup
- Affiliation: Department of Mathematics and Statistics, University of Ottawa, Ottawa, Canada K1N 6N5
- MR Author ID: 265531
- Email: bjessup@uottawa.ca
- 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
- © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 356 (2004), 1569-1583
- MSC (2000): Primary 17B56, 55P62
- DOI: https://doi.org/10.1090/S0002-9947-03-03392-0
- MathSciNet review: 2034319