Tag - Cohomology of Lie algebras

Ioannis Dokas: On Quillen-Barr-Beck cohomology for restricted Lie algebras

22nd July, 2024

In this talk we define and study Quillen-Barr-Beck cohomology for the category of restricted Lie algebras. We prove that the first Quillen-Barr-Beck’s cohomology classifies general abelian extensions of restricted Lie algebras. 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.

Andrey Lazarev: Cohomology of Lie coalgebras

15th July, 2024

Associated to a Lie algebra 𝔤 and a 𝔤-module M is a standard complex C*(𝔤,M) computing the cohomology of 𝔤 with coefficients in M; this classical construction goes back to Chevalley and Eilenberg of the late 1940s. Shortly afterwards, it was realized that this cohomology is an example of a derived functor in the category of 𝔤-modules. The Lie algebra 𝔤 can be replaced by a differential graded Lie algebra and M – with a dg 𝔤-module with the same conclusion. Later, a deep connection with Koszul duality was uncovered in the works of Quillen (late 1960s) and then Hinich (late 1990s). In this talk I will discuss the cohomology of (dg) Lie coalgebras with coefficients in dg comodules. The treatment is a lot more delicate, underscoring how different Lie algebras and Lie coalgebras are (and similarly their modules and comodules). A definitive answer can be obtained for so-called conilpotent Lie coalgebras (though not necessarily conilpotent comodules). If time permits, I will also discuss some topological applications.