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- Title
On the Koszul map of Lie algebras.
- Authors
Cornulier, Yves
- Abstract
We motivate and study the reduced Koszul map, relating the invariant bilinear maps on a Lie algebra and the third homology. We show that it is concentrated in degree 0 for any grading in a torsion-free abelian group, and in particular it vanishes whenever the Lie algebra admits a positive grading. We also provide an example of a 12-dimensional nilpotent Lie algebra whose reduced Koszul map does not vanish. In an appendix, we reinterpret the results of Neeb and Wagemann about the second homology of current Lie algebras, which are closely related to the reduced Koszul map.
- Subjects
LIE algebras; KOSZUL algebras; HOMOLOGY theory; BILINEAR forms; NILPOTENT groups
- Publication
Forum Mathematicum, 2016, Vol 28, Issue 1, p101
- ISSN
0933-7741
- Publication type
Article
- DOI
10.1515/forum-2014-0048