nLab
Chevalley-Eilenberg cochain complex

Idea

A finite dimensional Lie algebra gg or degreewise finite-dimensional L-∞ algebra gg is encoded in a a differential D: g gD : \vee^\bullet g \to \vee^\bullet g on the cofree co-commutative coalgebra generated by gg.

The dual of this is a differential graded algebra ( g *,d)(\wedge^\bullet g^*, d). The underlying cochain complex (forgetting the monoidal structure) is the Chevalley-Eilenberg cochain complex.

There is in fact a bijection between quasi-free cochain differential graded algebras in non-negative degree and L-∞ algebras.

The Chevalley-Eilenberg complex is usually defined a bit more generally for Lie algebras equipped with a Lie module? gEndVg \to End V. In the above language this more general cochain complex is the one underlying the Lie ∞-algebroid that encodes this action in the sense of Lie ∞-algebroid representations.

Lie algebra cohomology

The cohomology of the Chevalley-Eilenberg cochain complex agrees with the Lie algebra cohomology with trivial coefficients. The Lie algebra is however defined also for infinite-dimensional Lie algebras and arbitrary module MM coefficients. Namely the Lie algebra cohomology is H Lie *(𝔤,M)=Ext U(𝔤)(𝔤,M)=Hom U(𝔤)(V(𝔤),M)Hom k(Λ *𝔤,M)H^*_{Lie}(\mathfrak{g},M) = Ext_{U(\mathfrak{g})}(\mathfrak{g},M) = Hom_{U(\mathfrak{g})}(V(\mathfrak{g}),M)\cong Hom_k(\Lambda^*\mathfrak{g},M) where U(𝔤)U(\mathfrak{g}) is the universal enveloping of 𝔤\mathfrak{g} and V(𝔤)=U(𝔤)Λ *𝔤V(\mathfrak{g}) = U(\mathfrak{g})\otimes \Lambda^* \mathfrak{g} (with the appropriate differential) is the Chevalley-Eilenberg chain complex. Now if 𝔤\mathfrak{g} is finite-dimensional then Hom U(𝔤)(V(𝔤),M)CE(𝔤,M)Hom_{U(\mathfrak{g})}(V(\mathfrak{g}),M)\cong CE(\mathfrak{g},M) and CE(𝔤)=CE(𝔤,k)=Λ *𝔤 *CE(\mathfrak{g}) = CE(\mathfrak{g},k) = \Lambda^* \mathfrak{g}^*.

References

Revised on July 21, 2010 20:55:48 by Zoran Škoda (161.53.130.104)