Related papers: Face Functors for KLR Algebras

The parabolic category $\mathcal O$ for affine ${\mathfrak{gl}}_N$ at level $-N-e$ admits a structure of a categorical representation of $\widetilde{\mathfrak{sl}}_e$ with respect to some endofunctors $E$ and $F$. This category contains a…

We review the categorical representation of a Kac-Moody algebra on unipotent representations of finite unitary groups in non-defining characteristic given by the authors. Then, we extend this construction to finite reductive groups of types…

We consider the problem of representing the Kac-Moody algebra $\mathfrak{g}(N)$ specified by an $r\times r$ indecomposable generalised Cartan matrix $N$ as vector fields on the torus ${{\bb C}^*}^r$. It is shown that, if the representations…

The Lascoux-Leclerc-Thibon conjecture, reformulated and solved by S. Ariki, asserts that the K-group of the representations of the affine Hecke algebras of type A is isomorphic to the algebra of functions on the maximal unipotent subgroup…

Kashiwara and Saito have defined a crystal structure on the set of irreducible components of Lusztig's quiver varieties. This gives a geometric realization of the crystal graph of the lower half of the quantum group associated to a…

The paper is motivated by an analogy between cluster algebras and Kac-Moody algebras: both theories share the same classification of finite type objects by familiar Cartan-Killing types. However the underlying combinatorics beyond the two…

We describe quantum enveloping algebras of symmetric Kac-Moody Lie algebras via a finite field Hall algebra construction involving Z_2-graded complexes of quiver representations.

We construct a generalised notion of Kac-Moody algebras using smooth maps from the non-compact manifolds ${\cal M}=$SL$(2,\mathbb R)$ and ${\cal M}=$ SL$(2,\mathbb R)/U(1)$ to a finite-dimensional simple Lie group $G$. This construction is…

In this paper, we introduce the notion of abstract crystals for quantum generalized Kac-Moody algebras and study their fundamental properties. We then prove the crystal embedding theorem and give a characterization of the crystals…

We give a new interpretation and proof of the "quasi-particle" type character formulas for integrable representations of the simply-laced affine Kac-Moody algebras through a new "semi-infinite" construction of such representations. We…

In this note we give explicit isomorphisms of 2-categories between various versions of the categorified quantum group associated to a simply-laced Kac-Moody algebra. These isomorphisms are convenient when working with the categorified…

Let $C$ be a symmetrizable generalized Cartan Matrix, and $q$ an indeterminate. ${\fg}(C)$ is the Kac-Moody Lie algebra and $U=U_q({\fg}(C))$ the associated quantum enveloping algebra over $ k={\Bbb Q}(q)$. The quantum function algebra…

Let C be the category of finite-dimensional representations of a quantum affine algebra of simply-laced type. We introduce certain monoidal subcategories C_l (l integer) of C and we study their Grothendieck rings using cluster algebras.

The representation theory of involutory (or 'maximal compact') subalgebras of infinite-dimensional Kac-Moody algebras is largely terra incognita, especially with regard to fermionic (double-valued) representations. Nevertheless, certain…

We establish an embedding of the quantum enveloping algebra of a symmetric generalized Kac--Moody algebra into a localized Hall algebra of $\mathbb Z_2$-graded complexes of representations of a quiver with (possible) loops. To overcome…

Using face algebras (i.e. algebras of L-operators of IRF models), we construct modular tensor categories with positive definite inner product, whose fusion rules and S-matrices are the same as (or slightly different from) those obtained by…

We use induction from parabolic subalgebras with infinite-dimensional Levi factor to construct new families of irreducible representations for arbitrary Affine Kac-Moody algebra. Our first construction defines a functor from the category of…

Given the maximal compact subalgebra $\mathfrak{k}(A)$ of a split-real Kac-Moody algebra $\mathfrak{g}(A)$ of type $A$, we study certain finite-dimensional representations of $\mathfrak{k}(A)$, that do not lift to the maximal compact…

Attached to a weight space in an integrable highest weight representation of a simply-laced Kac-Moody algebra $\mathfrak{g}$, there are two natural commutative algebras: the cohomology ring of a quiver variety and the center of a cyclotomic…

This paper initiates a systematic study of the cyclotomic KLR algebras of affine types $A$ and $C$. We start by introducing a graded deformation of these algebras and the constructing all of the irreducible representations of the deformed…