The b-functions for prehomogeneous vector spaces of commutative parabolic type and generalized universal Verma modules

Prehomogeneous modules of commutative linear algebraic groups

Let $A$ be a finite dimensional commutative associative algebra with unit over an algebraically closed field of characteristic zero. The group $G(A)$ of invertible elements is open in $A$ and thus $A$ has a structure of a prehomogeneous $G(A)$-module. We show that every prehomogeneous module of a commutative linear algebraic group appears this way. In particular, the number of equivalence classes of prehomogeneous $G$-modules is finite if and only if the corank of $G$ is at most $5$.

Differential Calculi on Some Quantum Prehomogeneous Vector Spaces

This paper is devoted to study of differential calculi over quadratic algebras, which arise in the theory of quantum bounded symmetric domains. We prove that in the quantum case dimensions of the homogeneous components of the graded vector spaces of k-forms are the same as in the classical case. This result is well-known for quantum matrices. The quadratic algebras, which we consider in the present paper, are q-analogues of the polynomial algebras on prehomogeneous vector spaces of commutative parabolic type. This enables us to prove that the de Rham complex is isomorphic to the dual of a quan…

On Homomorphisms Between Global Weyl Modules

Global Weyl modules for generalized loop algebras $\lie g\tensor A$, where $\lie g$ is a simple finite dimensional Lie algebra and A is a commutative associative algebra were defined, for any dominant integral weight $λ$, by generators and relations. They are expected to play the role similar to that of Verma modules in the study of categories of representations of these algebras. One of the fundamental properties of Verma modules is that the space of morphisms between two Verma modules is either zero or one--dimensional and also that any non--zero morphism is injective. The aim of this paper …

杂交量子群的Category $\mathcal{O}$和非交换Springer解析

Category $\mathcal{O}$ for hybrid quantum groups and non-commutative Springer resolutions

本文研究了杂交量子群的Category $\mathcal{O}$，通过非交换Springer解析构建其主块模型，并证明主块与仿射Hecke范畴导出等价，同时揭示了简单模在泛化Kazhdan-Lusztig多项式中的分级乘数。

Differential equations in vertex algebras and simple modules for the Lie algebra of vector fields on a torus

We study irreducible representations for the Lie algebra of vector fields on a 2-dimensional torus constructed using the generalized Verma modules. We show that for a certain choice of parameters these representations remain irreducible when restricted to a loop subalgebra in the Lie algebra of vector fields. We prove this result by studying vertex algebras associated with the Lie algebra of vector fields on a torus and solving non-commutative differential equations that we derive using the vertex algebra technique.

Vortices and Vermas

In three-dimensional gauge theories, monopole operators create and destroy vortices. We explore this idea in the context of 3d N=4 gauge theories in the presence of an Omega-background. In this case, monopole operators generate a non-commutative algebra that quantizes the Coulomb-branch chiral ring. The monopole operators act naturally on a Hilbert space, which is realized concretely as the equivariant cohomology of a moduli space of vortices. The action furnishes the space with the structure of a Verma module for the Coulomb-branch algebra. This leads to a new mathematical definition of the C…