In this paper we prove that if G is a connected, simplyconnected, semi-simple algebraic group over an algebraically closed field of sufficiently large characteristic, then all the blocks of the restricted enveloping algebra (Ug)0 of the Lie algebra g of G can be endowed with a Koszul grading (extending results of Andersen, Jantzen, Soergel). We also give information about the Koszul dual rings....

We study the representation theory of the invariant subalgebra of the Weyl algebra under a torus action, which we call a “hypertoric enveloping algebra.” We define an analogue of BGG category O for this algebra, and identify it with a certain category of sheaves on a hypertoric variety. We prove that a regular block of this category is highest weight and Koszul, identify its Koszul dual, comput...

We want to present here the part of the work in common with Martin Markl [11] which concerns quadratic operads for n-ary algebras and their dual for n odd. We will focus on the ternary case (i.e n = 3). The aim is to underline the problem of computing the dual operad and the fact that this last is in general defined in the graded differential operad framework. We prove that the operad associate...

The main result of this paper is that over a noncommutative Koszul algebra, high truncations of finitely generated graded modules have linear free resolutions.

We provide a prorepresenting object for the noncommutative derived deformation problem of deforming module $X$ over differential graded algebra. Roughly, we show that corresponding functor is homotopy prorepresented by dual bar construction on endomorphism algebra $X$. specialise to case when one-dimensional base field, and introduce notion framed deformations, which rigidify slightly allow us ...

We prove a duality for factorization homology which generalizes both usual Poincaré duality for manifolds and Koszul duality for En-algebras. The duality has application to the Hochschild homology of associative algebras and enveloping algebras of Lie algebras. We interpret our result at the level of topological quantum field theory.

We extend the theory of chiral and factorization algebras, developed for curves by Beilinson and Drinfeld in [BD1], to higher-dimensional varieties. This extension entails the development of the homotopy theory of chiral and factorization structures, in a sense analogous to Quillen’s homotopy theory of differential graded Lie algebras. We prove the equivalence of higherdimensional chiral and fa...