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Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

IMPACT FACTOR 2018: 0.867

CiteScore 2018: 0.71

SCImago Journal Rank (SJR) 2018: 0.898
Source Normalized Impact per Paper (SNIP) 2018: 0.964

Mathematical Citation Quotient (MCQ) 2018: 0.71

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Volume 28, Issue 1


On the Koszul map of Lie algebras

Yves Cornulier
Published Online: 2014-09-18 | DOI: https://doi.org/10.1515/forum-2014-0048


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.

Keywords: Lie algebra homology; nilpotent Lie algebra; current Lie algebra

MSC: 17B55; 17B30; 17B56; 17B70; 19C09; 15A63

About the article

Received: 2014-03-20

Revised: 2014-05-21

Published Online: 2014-09-18

Published in Print: 2016-01-01

Citation Information: Forum Mathematicum, Volume 28, Issue 1, Pages 101–128, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2014-0048.

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