Related papers: Monoidal categorification and quantum affine algeb…
Kirillov-Reshetikhin crystals are colored directed graphs encoding the structure of certain finite-dimensional representations of affine Lie algebras. A tensor products of column shape Kirillov-Reshetikhin crystals has recently been…
We give a new construction of the algebraic $K$-theory of small permutative categories that preserves multiplicative structure, and therefore allows us to give a unified treatment of rings, modules, and algebras in both the input and…
We construct a new quantization $K_t(\mathcal{O}^{sh}_{\mathbb{Z}})$ of the Grothendieck ring of the category $\mathcal{O}^{sh}_{\mathbb{Z}}$ of representations of shifted quantum affine algebras (of simply-laced type). We establish that…
We study representations of the quantum affine superalgebra associated with a general linear Lie superalgebra. In the spirit of Hernandez-Jimbo, we construct inductive systems of Kirillov-Reshetikhin modules based on a cyclicity result that…
Let $\g$ be an untwisted affine Kac-Moody algebra of type $A^{(1)}_n$ $(n \ge 1)$ or $D^{(1)}_n$ $(n \ge 4)$ and let $\g_0$ be the underlying finite-dimensional simple Lie subalgebra of $\g$. For each Dynkin quiver $Q$ of type $\g_0$,…
We explicitly construct families of simple modules for Lie algebras of rank $2$, on which certain commutative subalgebra acts diagonally and has a simple spectrum. In type $A$ these modules are well known generic Gelfand-Tsetlin modules and…
In [arXiv:1509.02937], the notion of a module tensor category was introduced as a braided monoidal central functor $F\colon \mathcal{V}\longrightarrow \mathcal{T}$ from a braided monoidal category $\mathcal{V}$ to a monoidal category…
Let $J$ be a set of pairs consisting of good modules over an affine quantum algebra and invertible elements. The distribution of poles of the normalized R-matrices yields Khovanov-Lauda-Rouquier algebras $R^J$. We define a functor $F$ from…
We use Khovanov-Lauda-Rouquier (KLR) algebras to categorify a crystal isomorphism between a fundamental crystal and the tensor product of a Kirillov-Reshetikhin crystal and another fundamental crystal, all in affine type. The nodes of the…
In this paper, we define a number of closely related isomorphisms. On one side of these isomorphisms sit a number of of algebras generalizing the Hecke and affine Hecke algebras, which we call the "Hecke family"; on the other, we find…
This paper is a contribution to the construction of non-semisimple modular categories. We establish when M\"uger centralizers inside non-semisimple modular categories are also modular. As a consequence, we obtain conditions under which…
We define a family of graded restricted modules for the polynomial current algebra associated to a simple Lie algebra. We study the graded character of these modules and show that they are the same as the graded characters of certain…
For a quantum affine algebra of type A, we describe the composition series of the tensor product of a general minimal affinization with a Kirillov-Resehtikhin module associated to an extreme node of the Dynkin diagram of the underlying…
Cuspidal systems parameterize KLR algebra representations via root partitions $\pi$, where simple modules $L(\pi)$ arise as heads of proper standard modules. Working in affine type A with an arbitrary convex preorder, we construct explicit…
Snake modules introduced by Mukhin and Young form a family of modules of quantum affine algebras. The aim of this paper is to prove that the Hernandez-Leclerc conjecture about monoidal categorifications of cluster algebras is true for prime…
We develop abstract nonsense for module categories over monoidal categories (this is a straightforward categorification of modules over rings). As applications we show that any semisimple monoidal category with finitely many simple objects…
We construct Quillen equivalences between the model categories of monoids (rings), modules and algebras over two Quillen equivalent model categories under certain conditions. This is a continuation of our earlier work where we established…
To each category C of modules of finite length over a complex simple Lie algebra g, closed under tensoring with finite dimensional modules, we associate and study a category Aff(C)_\kappa of smooth modules (in the sense of Kazhdan and…
A cuspidal system for an affine Khovanov-Lauda-Rouquier algerba $R_\al$ yields a theory of standard modules. This allows us to classify the irreducible modules over $R_\al$ up to the so-called imaginary modules. We make a conjecture on…
We study the weight modules over affine Kac-Moody algebras from the view point of vertex algebras, and determine the abelian category of weight modules for the simple affine vertex algebra $L_k(\mathfrak{sl}_2)$ at any non-integral…