Title: Quillen-Barr-Beck cohomology for restricted Lie algebras
Abstract: The Hochschild cohomology for restricted Lie algebras classifies strongly abelian extensions of restricted Lie algebras. In this talk we define Quillen-Barr-Beck cohomology for the category of restricted Lie algebras and we prove that Quillen-Barr-Beck’s cohomology classifies general abelian extensions. Moreover, using Duskin-Glenn’s torsors cohomology theory, we prove a classification theorem for the second Quillen-Barr-Beck cohomology group in terms of 2-fold extensions of restricted Lie algebras. Finally, we give an interpretation of Cegarra-Aznar’s exact sequence for torsor cohomology. Thus,we obtain for a short exact sequence of restricted Lie algebras, an eight-term exact sequence for Quillen-Barr-Beck cohomology. This sequence replaces the five-term exact sequence proved by Eckmann-Stammbach in the context of Hochschild cohomology.