Constructing homomorphisms between Verma modules

From Jantzen to Andersen Filtration via Tilting Equivalence

The space of homomorphisms from a projective object to a Verma module in category O inherits an induced filtration from the Jantzen filtration on the Verma module. On the other hand there is the Andersen filtration on the space of homomorphisms from a Verma module to a tilting module as described in [Soe07]. The tilting equivalence from [Soe98] induces an isomorphism of these kinds of Hom-spaces. We will show that this equivalence even identifies both filtrations.

Homomorphisms between scalar Vema modules of ${\mathfrak gl}(n, {\mathbb C})$

In this article, we classify the homomorphisms between scalar generalized Verma modules of ${\mathfrak gl}(n, {\mathbb C})$. In fact such homomorphisms are compositions of elementary homomorphisms.

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 …

Systems of Differential Operators and Generalized Verma Modules

In this paper we close the cases that were left open in our earlier works on the study of conformally invariant systems of second-order differential operators for degenerate principal series. More precisely, for these cases, we find the special values of the systems of differential operators, and determine the standardness of the homomorphisms between the generalized Verma modules, that come from the conformally invariant systems.

On Reducible Verma Modules over Jacobi Algebra

With this paper we start the study of reducible representations of the Jacobi algebra with the ultimate goal of constructing differential operators invariant w.r.t. the Jacobi algebra. In this first paper we show examples of the low level singular vectors of Verma modules over the Jacobi algebra. According to our methodology these will produce the invariant differential operators.

Generalized Imaginary Verma and Wakimoto modules

We develop a general technique of constructing new irreducible weight modules for any affine Kac-Moody algebra using the parabolic induction, in the case when the Levi factor of a parabolic subalgebra is infinite-dimensional and the central charge is nonzero. Our approach uniforms and generalizes all previously known results with imposed restrictions on inducing modules. We also define generalized Imaginary Wakimoto modules which provide an explicit realization for generic generalized Imaginary Verma modules.

奇数Verma定理

Odd Verma's Theorem

本文研究了正则对称化Kac-Moody李超代数上Verma超模的性质，通过改变Borel子代数揭示$\mathfrak{gl}(1|1)$结构，并用半砖精确描述主块，改进相伴簇和射影维数结果。

Andersen Filtration and Hard Lefschetz

On the space of homomorphisms from a Verma module to an indecomposable tilting module of the BGG-category O we define a natural filtration following Andersen and establish a formula expressing the dimensions of the filtration steps in terms of coefficients of Kazhdan-Lusztig polynomials.

Suzuki functor at the critical level

In this paper we define and study a critical-level generalization of the Suzuki functor, relating the affine general linear Lie algebra to the rational Cherednik algebra of type A. Our main result states that this functor induces a surjective algebra homomorphism from the centre of the completed universal enveloping algebra at the critical level to the centre of the rational Cherednik algebra at t=0. We use this homomorphism to obtain several results about the functor. We compute it on Verma modules, Weyl modules, and their restricted versions. We describe the maps between endomorphism rings i…

Homomorphisms Between Specht Modules of the Ariki Koike Algebra

In this paper we generalize a theorem due to Lyle, extending its application to the setting of the Ariki-Koike algebra, and in doing so establish an analogue of the kernel intersection theorem. This in turn provides us with a means towards constructing homomorphisms between Specht modules for the Ariki-Koike algebra.

Homomorphisms between different quantum toroidal and affine Yangian algebras

This paper concerns the relation between the quantum toroidal algebras and the affine Yangians of $\mathfrak{sl}_n$, denoted by $\mathcal{U}^{(n)}_{q_1,q_2,q_3}$ and $\mathcal{Y}^{(n)}_{h_1,h_2,h_3}$, respectively. Our motivation arises from the milestone work of Gautam and Toledano Laredo, where a similar relation between the quantum loop algebra $U_q(L \mathfrak{g})$ and the Yangian $Y_h(\mathfrak{g})$ has been established by constructing an isomorphism of $\mathbb{C}[[\hbar]]$-algebras $Φ:\widehat{U}_{\exp(\hbar)}(L\mathfrak{g})\to \widehat{Y}_\hbar(\mathfrak{g})$ (with $\ \widehat{}\ $ sta…

量子对称对子代数中的权重模块

Weight modules for quantum symmetric pair subalgebras

本文研究量子对称对子代数中的权重模块理论，通过构造权重基和分类不可约有限维模块，拓展了经典表示论的应用范围。

Semi-infinite cohomology and the linkage principle for $W$-algebras

Let $\mathfrak{g}$ be a simple Lie algebra, and let $W_κ$ be the affine ${W}$-algebra associated to a principal nilpotent element of $\mathfrak{g}$ and level $κ$. We explain a duality between the categories of smooth ${W}$ modules at levels $κ+ κ_c$ and $-κ+ κ_c$, where $κ_c$ is the critical level. Their pairing amounts to a construction of semi-infinite cohomology for the ${W}$-algebra. As an application, we determine all homomorphisms between the Verma modules for ${W}$, verifying a conjecture from the conformal field theory literature of de Vos--van Driel. Along the way, we determine the li…

赋范模与积分、级数展开及微分的范畴化

Normed modules and the categorification of integrations, series expansions, and differentiations

本文通过为有限维代数上的模赋予范数，构造了两个新范畴来建立积分、级数展开及导数范畴化的框架，并证明了 $\mathscr{A}^p$ 意义下的 Stone--Weierstrass 逼近定理。

仿射胞腔代数与渐近代数

Affine cellular algebras and asymptotic algebras

扩展仿射胞腔代数理论以纳入其渐近代数，明晰经典与仿射情形差异并与Lusztig渐近Hecke代数比较，有关于二者双中心化子性质等新成果。