Related papers: Crystals from categorified quantum groups
To each graph without loops and multiple edges we assign a family of rings. Categories of projective modules over these rings categorify $U^-_q(\mathfrak{g})$, where $\mathfrak{g}$ is the Kac-Moody Lie algebra associated with the graph.
For symmetric Kashiwara crystals of type $A$ and rank $e=2$, and for the canonical basis elements that we call external, corresponding to weights on the outer skin of the Kashiwara crystal, we construct the canonical basis elements in a…
We consider a category of $\gl_\infty$-crystals, whose objects are disjoint unions of extremal weight crystals of non-negative level with certain finite conditions on the multiplicity of connected components. We show that it is a monoidal…
We study monoidal categorifications of certain monoidal subcategories $\mathcal{C}_J$ of finite-dimensional modules over quantum affine algebras, whose cluster algebra structures coincide and arise from the category of finite-dimensional…
We show that the structure of blocks outside the critical hyperplanes of category O over any symmetrizable Kac-Moody algebra depends only on the corresponding integral Weyl group and its action on the parameters of the Verma modules by…
We describe the crystal bases of the modified quantum algebras and give the explicit form of the highest (or lowest) weight vector of its connected component $B_0(\lambda)$ containing the unit element for arbitrary rank 2 cases. We also…
Let $G$ be a reductive algebraic group with Lie algebra $\mathfrak{g}$ and $V$ a finite-dimensional representation of $G$. Costello-Gaiotto studied a graded Lie algebra $\mathfrak{d}_{\mathfrak{g}, V}$ and the associated affine Kac-Moody…
We derive the canonical structure and hamiltonian for arbitrary deformations of a higher-dimensional (quantum Hall) droplet of fermions with spin or color on a general phase space manifold. Gauge fields are introduced via a Kaluza-Klein…
The dual space of the Cartan subalgebra in a Kac-Moody algebra has a partial ordering defined by the rule that two elements are related if and only if their difference is a non-negative or non-positive integer linear combination of simple…
Let $\g$ be an arbitrary Kac-Moody algebra with a Cartan sualgebra $\h$. In this paper, we determine the category of $\g$-modules that are free $U(\h)$-modules of rank 1.
Hashimoto and Ueda determined the weights of generators of the graded ring of modular forms on the Cayley half-space of degree two. In this paper we describe explicit generators. We show that the graded ring can be generated by Eisenstein…
In this paper, we initiate the study of algebraic K-theory for non-commutative $\Gamma$-semirings, extending the classical constructions of Grothendieck and Bass to this setting. We first establish the categorical foundations by…
The aim of this note is to take benefit of the foam nature of the Khovanov-Kuperberg algebras to compute the Grothendieck groups of their categories of finitely generated projective modules. The computation relies on the Hattori-Stallings…
Given a vector bundle $A\to M$ we study the geometry of the graded manifolds $T^*[k]A[1]$, including their canonical symplectic structures, compatible Q-structures and Lagrangian Q-submanifolds. We relate these graded objects to classical…
We determine the graded composition multiplicity in the symmetric algebra S(V) of the natural GL_n(q)-module V, or equivalently in the coinvariant algebra of V, for a large class of irreducible modules around the Steinberg module. This was…
We give a new bijective interpretation of the Cauchy identity for Schur operators which is a commutation relation between two formal power series with operator coefficients. We introduce a plactic algebra associated with the Kashiwara's…
The problem is the classification of the ideals of ``free differential algebras", or the associated quotient algebras, the q-algebras; being finitely generated, unital C-algebras with homogeneous relations and a q-differential structure.…
In \cite{JKS} we gave an (additive) categorification of Grassmannian cluster algebras, using the category $\CM(A)$ of Cohen-Macaulay modules for a certain Gorenstein order $A$. In this paper, using a cluster tilting object in the same…
Suppose that we have a semisimple, connected, simply connected algebraic group $G$ with corresponding Lie algebra $\mathfrak{g}$. There is a Hopf pairing between the universal enveloping algebra $U(\mathfrak{g})$ and the coordinate ring…
We describe the upper seminormal crystal structure for the $\mu$-supported $\delta$-vectors for any quiver with potential with reachable frozen vertices, or equivalently for the tropical points of the corresponding cluster $\mc{X}$-variety.…