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Spectral sequences for commutative Lie algebras. (English) Zbl 1480.17021

A commutative Lie algebra over a field \(\mathbb{F}\) of characteristic 2 is a vector space over \(\mathbb{F}\) with an \(\mathbb{F}\)-bilinear product that is commutative and satisfies the Jacobi identity. Every Lie algebra over a field of characteristic 2 is a commutative Lie algebra, but not conversely. Moreover, every commutative Lie algebra is a left and a right Leibniz algebra. Commutative Lie algebras and their cohomology were introduced recently by V. Lopatkin and P. Zusmanovich [Commun. Contemp. Math. 23, No. 5, Article ID 2050046, 20 p. (2021; Zbl 1475.17030)]. A left module for a commutative Lie algebra \(\mathfrak{g}\) is a left module for \(\mathfrak{g}\) considered as a left Leibniz algebra. Then the cohomology of \(\mathfrak{g}\) with coefficients in a left \(\mathfrak{g}\)-module \(M\) is the cohomology of the cochain complex \((\mathrm{Hom}_\mathbb{F}(S^n\mathfrak{g},M),d^n)_{n\ge 0}\), where \(d^n\) satifies the same formula as the usual coboundary operator \(\delta^n\) on the Chevalley-Eilenberg cochain complex \((\mathrm{Hom}_\mathbb{F} (\Lambda^n L,V),\delta^n)_{n\ge 0}\) for an ordinary Lie algebra \(L\) and an \(L\)-module \(V\).
The author of the paper under review constructs several spectral sequences for the cohomology of a commutative Lie algebra and discusses some of their applications. First, for a commutative Lie algebra and any of its subalgebras a Hochschild-Serre type spectral sequence is investigated. For every subalgebra the \(E_0\)-term of this spectral sequence and its differential are computed. In the case that the subalgebra is an ideal, the \(E_1\)-term and its differential are identified which also leads to a description of the \(E_2\)-term. As an application the cohomologies of certain two-dimensional commutative Lie algebras with trivial coefficients are computed, and more generally, some vanishing results for the cohomology of a commutative Lie algebra with an ideal of (co)dimension one are derived. (Remark of the reviewer: In the final version of the paper [J. Feldvoss and F. Wagemann, J. Algebra 569, 276–317 (2021; Zbl 1465.17006)] Example D of the paper under review is Example C and Example E of the paper under review is Example A.)
T. Pirashvili [Ann. Inst. Fourier 44, No. 2, 401–411 (1994; Zbl 0821.17023)] constructed a spectral sequence comparing Chevalley-Eilenberg homology and Leibniz homology of a Lie algebra. (The cohomological analogue of this spectral sequence was obtained by the author of the paper under review and the reviewer in [J. Algebra 569, 276–317 (2021; Zbl. 1465.17006)].) Similarly, one can construct a spectral sequence comparing Chevalley-Eilenberg cohomology and commutative cohomology and a spectral sequence comparing commutative cohomology and Leibniz cohomology. Of course, the former only makes sense for a Lie algebra, but the latter applies to any commutative Lie algebra. All these spectral sequences are constructed by the author. Note that for commutative cohomology and Leibniz cohomology the left module has to be considered as a symmetric Leibniz bimodule. As a consequence, it is shown that the vanishing of Chevalley-Eilenberg cohomology in degrees \(0\le k\le n\) implies the vanishing of commutative cohomology in degrees \(0\le k\le n\) and isomorphisms of commutative cohomology and Chevalley-Eilenberg cohomology in degrees \(n+1\) and \(n+2\), respectively. Similarly, the vanishing of commutative cohomology in degrees \(0\le k\le n\) implies the vanishing of Leibniz cohomology in degrees \(0\le k\le n\) and isomorphisms of Leibniz cohomology and commutative cohomology in degrees \(n+1\) and \(n+2\), respectively.

MSC:

17B50 Modular Lie (super)algebras
17A30 Nonassociative algebras satisfying other identities
17A32 Leibniz algebras
17B55 Homological methods in Lie (super)algebras
17B56 Cohomology of Lie (super)algebras
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References:

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