We describe categorifications of 512 and braid groups. In a first part, we give a survey of the
case of 512 (joint work with Joseph Chuang [ChRou]) and explain how it leads to the …

This is the first textbook treatment of work leading to the landmark 1979 Kazhdan–Lusztig
Conjecture on characters of simple highest weight modules for a semisimple Lie algebra gg …

We define and study category $\mathcal O $ for a symplectic resolution, generalizing the
classical BGG category $\mathcal O $, which is associated with the Springer resolution. This …

This book is directed primarily at undergraduate and postgraduate students interested to get
acquainted with the representation theory of Lie algebras. The book treats the case of the …

This article contains a review of categorifications of semisimple representations of various
rings via abelian categories and exact endofunctors on them. A simple definition of an …

The purpose of this paper is to study categorifications of tensor products of finite-
dimensional modules for the quantum group for sl _2. The main categorification is obtained …

This paper studies quadratic and Koszul duality for modules over positively graded
categories. Typical examples are modules over a path algebra, which is graded by the path …

For a fixed parabolic subalgebra 𝔭 of we prove that the centre of the principal block 𝒪0𝔭 of
the parabolic category 𝒪 is naturally isomorphic to the cohomology ring H*(ℬ𝔭) of the …

We develop a graded version of the theory of cyclotomic q-Schur algebras, in the spirit of the
work of Brundan-Kleshchev on Hecke algebras and of Ariki on q-Schur algebras. As an …

This paper presents categorifications of (right) cell modules and induced cell modules for
Hecke algebras of finite Weyl groups. In type A we show that these categorifications depend …