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Lie algebra cohomology

Definition

The abelian cohomology of a Lie algebra $𝔤$ is the cochain cohomology of its Chevalley-Eilenberg algebra $\mathrm{CE}(𝔤)$. See also nonabelian Lie algebra cohomology.

The degree $n$ Lie algebra cohomology computes the homotopy classes of $L_{\mathrm{\infty }}$-algebra morphisms

$$𝔤\to b^{n-1}𝔲(1).$$\mathfrak{g} \to b^{n-1} \mathfrak{u}(1) \,.

See also nonabelian Lie algebra cohomology.

Extensions

Every Lie algebra degree $n$ cocycle $\mu$ gives rise to an extension

$$b^{n-2}𝔲(1)\to {𝔤}_{\mu }\to 𝔤$$b^{n-2} \mathfrak{u}(1) \to \mathfrak{g}_{\mu} \to \mathfrak{g}

of the Lie algebra by an $L_{\mathrm{\infty }}$-algebra. This is Theorem 55 here:

• John Baez and Alissa Crans, Higher-Dimensional Algebra VI: Lie 2-Algebras, Theory and Applications of Categories 12 (2004), 492-528. arXiv

Ordinary Lie algebras

• String Lie 2-algebra

Super Lie algebras

• supergravity Lie 3-algebra

• C. Weibel, An introduction to homological algebra (Cambridge studies in advanced mathematics 38, 1994), chapter 7: Lie algebra homology and cohomology

References

Ordinary Lie algebras

• Springer online enc:Lie algebra cohomology

Super Lie algebras

• J. A. de Azcárraga and P. K. Townsend, Superspace geometry and classification of supersymmetric extended objects, Phys. Rev. Lett. 62, 2579–2582 (1989)