78. 15/07/2024 Andrey Lazarev (Lancaster University, UK) 

Title: Cohomology of Lie coalgebras
Abstract: 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.