nLab
Chevalley-Eilenberg algebra

this entry needs to be merged with Chevalley-Eilenberg cochain complex

Ordinary concept

Chevalley–Eilenberg algebras are differential graded commutative algebras which are used to define cohomology of Lie algebras.

Synthetic version

see

• Chevalley-Eilenberg algebra in synthetic differential geometry

Generalization in $\mathrm{\infty }$-context

In the context of general Lie theory one notices that Chevalley–Eilenberg algebras generalized to arbitrary “quasi-free differential graded commutative algebras” are usefully thought of as providing a language that naturally allows to pass from Lie algebras to $L_{\mathrm{\infty }}$-algebroids: Chevalley–Eilenberg algebras are taken in this context to be the algebras of functions on NQ-supermanifolds.

Closely related are Weil algebras, which are the algebras of functions on the shifted tangent bundles of these NQ-supermanifolds.

Appearance in physics

In the physics literature Chevalley–Eilenberg algebras of $L_{\mathrm{\infty }}$-algebroids are addressed as BRST complexes.

In this context

• the generators in degree 0 are called fields;

• the generators in degree $1$ are called ghosts;

• the generators in degree $2$ are called ghosts of ghosts;

• etc.