nLab
Lie algebra homology

The abelian homology of a kk-Lie algebra 𝔤\mathfrak{g} with coefficients in the left 𝔤\mathfrak{g}-module MM is defined as H * Lie(𝔤,M)=Tor * U𝔤(k,M)H_*^{Lie}(\mathfrak{g},M) = Tor^{U\mathfrak{g}}_*(k,M) where kk is the ground field understood as a trivial module over the universal enveloping algebra U𝔤U\mathfrak{g}. In particular it is a derived functor. It can be computed using Chevalley-Eilenberg chain complex V(𝔤)V(\mathfrak{g}) as the homology of the chain complex

M⊗ U𝔤V(𝔤)=M⊗ U𝔤U𝔤⊗ kΛ *𝔤=M⊗ kΛ *𝔤. M \otimes_{U\mathfrak{g}} V(\mathfrak{g}) = M\otimes_{U\mathfrak{g}} U\mathfrak{g}\otimes_k \Lambda^* \mathfrak{g} = M\otimes_k \Lambda^* \mathfrak{g}.
  • C. Chevalley, S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63, (1948). 85–124.
Revised on July 21, 2010 20:57:05 by Zoran Å koda (161.53.130.104)