nLab
Chevalley-Eilenberg chain complex

Given a kk-Lie algebra 𝔤\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 *(𝔤)kV_*(\mathfrak{g})\to k of the trivial 𝔤\mathfrak{g}-module kk in the abelian category of 𝔤\mathfrak{g}-modules (what is the same as U𝔤U\mathfrak{g}-modules, where U𝔤U\mathfrak{g} is the universal enveloping algera of 𝔤\mathfrak{g}). Graded components of the underlying kk-module this resolution is given by

V p(𝔤)=U(𝔤) kΛ p𝔤 V_p(\mathfrak{g}) = U(\mathfrak{g})\otimes_k \Lambda^p{\mathfrak{g}}

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

If uU𝔤u \in U\mathfrak{g} and x i𝔤x_i\in \mathfrak{g} then the differnetial is given by

d(ux 1x p)= d(u\otimes x_1 \wedge \cdots \wedge x_p) =
= i=1 p(1) i+1ux ix 1x^ ix p+ i<j(1) i+ju[x i,x j]x^ ix^ jx p = \sum_{i = 1}^p (-1)^{i+1} ux_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

See also Lie algebra homology, Lie algebra cohomology, Chevalley–Eilenberg cochain complex and Chevalley–Eilenberg algebra.

Revised on July 28, 2010 21:17:40 by Toby Bartels (173.190.150.114)