Quillen–Barr–Beck (co-) homology for restricted Lie algebras
✍ Scribed by Ioannis Dokas
- Publisher
- Elsevier Science
- Year
- 2004
- Tongue
- English
- Weight
- 227 KB
- Volume
- 186
- Category
- Article
- ISSN
- 0022-4049
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✦ Synopsis
In this paper we use Quillen-Barr-Beck's theory of (co-) homology of algebras in order to deÿne (co-) homology for the category RLie of restricted Lie algebras over a ÿeld k of characteristic p = 0. In contrast with the cases of groups, associative algebras and Lie algebras we do not obtain Hochschild (co-) homology shifted by 1.
Precisely, we determine for L ∈ RLie the category of Beck L-modules and the group of Beck derivations of g ∈ RLie=L to a Beck L-module M . Moreover, we prove a classiÿcation theorem which gives a one-to-one correspondence between the one cohomology and the set of equivalent classes of p-extensions. Finally, a universal coe cient theorem is proved, relating the homology to the Hochschild homology via a short exact sequence. This shows that the new homology determines the Hochschild homology.