nLab
Chevalley-Eilenberg chain complex

Given a kk-Lie algebra 𝔀\mathfrak{g} over a commutative unital ring kk which is free as a kk-module, the Chevalley–Eilenberg chain complex is a particular projective resolution V *(𝔀)β†’kV_*(\mathfrak{g})\to k of the trivial 𝔀\mathfrak{g}-module kk in the abelian category of 𝔀\mathfrak{g}-modules (what is the same as U𝔀U\mathfrak{g}-modules, where U𝔀U\mathfrak{g} is the universal enveloping algebra of 𝔀\mathfrak{g}). Graded components of the underlying kk-module of this resolution is given by

V p(𝔀)=U(𝔀)βŠ— kΞ› p𝔀 V_p(\mathfrak{g}) = U(\mathfrak{g})\otimes_k \Lambda^p{\mathfrak{g}}

and it has the obvious U𝔀U\mathfrak{g}-module structure by multiplication in the first tensor factor, because Ξ› p𝔀\Lambda^p{\mathfrak{g}} is free as a kk-module.

If u∈U𝔀u \in U\mathfrak{g} and x iβˆˆπ”€x_i\in \mathfrak{g} then the differential is given by

d(uβŠ—x 1βˆ§β‹―βˆ§x p)=βˆ‘ i=1 p(βˆ’1) i+1ux iβŠ—x 1βˆ§β‹―βˆ§x^ iβˆ§β‹―βˆ§x p+βˆ‘ i<j(βˆ’1) i+juβŠ—[x i,x j]βˆ§β‹―βˆ§x^ iβ‹―βˆ§x^ jβ‹―βˆ§x p d(u\otimes x_1 \wedge \cdots \wedge x_p) = \sum_{i = 1}^p (-1)^{i+1} u x_i \otimes x_1 \wedge \cdots \wedge \hat{x}_i\wedge \cdots \wedge x_p + \sum_{i\lt j} (-1)^{i+j} u\otimes [x_i, x_j] \wedge \cdots \wedge \hat{x}_i\cdots \wedge \hat{x}_j\cdots \wedge x_p

and

Last revised on August 22, 2018 at 10:15:08. See the history of this page for a list of all contributions to it.