Commutative Lie algebras and commutative cohomology in characteristic 2. (English) Zbl 1475.17030
The article under review deals with commutative Lie algebras in characteristic \(2\). That is, a vector space \(L\) over a base field \(K\) of characteristic \(2\) which carries a bracket such that for all \(x,y,z\in L\), we have \[
[x,y]=[y,x]
\]
and the Jacobi identity
\[
[[x,y],z]+[[y,z],x]+[[z,x],y]=0.
\]
This is a notion which lies in between Lie algebras in characteristic \(2\) (where \([x,x]=0\) for all \(x\in L\)) and (is a special case of) Leibniz algebras. The authors develop cohomology theory for these commutative Lie algebras, called commutative cohomology and based on symmetric tensor products. They have some sample computations, for example for the Heisenberg Lie algebra and the Zassenhaus Lie algebra. They argue that commutative cohomology constitutes another invariant which should be helpful in the classification of Lie algebras in characteristic \(2\).
In the follow-up work [the reviewer, Commun. Math. 28, No. 2, 123–137 (2020; Zbl 1480.17021)], some standard spectral sequences for commutative cohomology are developed, and the link to Chevalley-Eilenberg cohomology is explored further.
In the follow-up work [the reviewer, Commun. Math. 28, No. 2, 123–137 (2020; Zbl 1480.17021)], some standard spectral sequences for commutative cohomology are developed, and the link to Chevalley-Eilenberg cohomology is explored further.
Reviewer: Friedrich Wagemann (Nantes)
MSC:
| 17B50 | Modular Lie (super)algebras |
| 17B56 | Cohomology of Lie (super)algebras |
| 17B99 | Lie algebras and Lie superalgebras |
Citations:
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