 Lie algebra cohomology

For a Lie algebra $\mathfrak{g}$ over a field $k$ and a Lie algebra representation $V$, set $C^n(\mathfrak{g},V)={\mathrm{Hom}}_k({\Lambda }^n\mathfrak{g},V)$. These alternating multilinear maps form a cochain complex with differentialThe hat means that the argument is omitted. The Jacobi identity and the Lie algebra representation identity give $d^2=0$. The Lie algebra cohomology is the quotient vector space $H^n(\mathfrak{g},V)=\ker (d:C^n\to C^{n+1})/\mathrm{im}(d:C^{n-1}\to C^n)$. In degree zero it consists of the invariant vectors of a Lie algebra representation; with a trivial Lie algebra representation as coefficients, $H^1(\mathfrak{g},k)={\mathrm{Hom}}_k(\mathfrak{g}/[\mathfrak{g},\mathfrak{g}],k)$.^[ref]

For trivial real coefficients, $H^2(\mathfrak{g},\mathbb{R})$ classifies central extensions of a Lie algebra by a fixed one-dimensional center, up to equivalences inducing the identity on the quotient and center. If $\mathfrak{g}$ is abelian and two-dimensional, every alternating bilinear form is a Lie algebra two-cocycle and every coboundary is zero; a nonzero such form produces the Heisenberg Lie algebra.

Lie algebra cohomology is a mathematical concept that arises in the study of Lie algebras, which are algebraic structures used extensively in mathematics and physics to describe symmetries and conservation laws. Cohomology, in this context, refers to a homological algebra framework that helps in analyzing the structure and properties of Lie algebras.