Generalized Imaginary Verma and Wakimoto modules

Imaginary Verma modules and Kashiwara algebras for $U_q(\widehat{\mathfrak{sl}(2)})$

We consider imaginary Verma modules for quantum affine algebra $U_q(\hat{\mathfrak{sl}(2)})$ and construct Kashiwara type operators and the Kashiwara algebra $\mathcal K_q$. We show that a certain quotient $\mathcal N_q^-$ of $U_q(\hat{\mathfrak{sl}(2)})$ is a simple $\mathcal K_q$-module.

Virasoro action on Imaginary Verma modules and the operator form of the KZ-equation

We define the Virasoro algebra action on imaginary Verma modules for affine sl(2) and construct the analogs of Knizhnik-Zamolodchikov equation in the operator form. Both these results are based on a free field realization of imaginary Verma modules.

Kashiwara Algebras and Imaginary Verma Modules for $U_q(\hat{\mathfrak{g}})$

We consider imaginary Verma modules for quantum affine algebra $U_q(\hat{\mathfrak{g}})$, where $\hat{\mathfrak{g}}$ is of type 1 i.e. of non-twisted type, and construct Kashiwara type operators and the Kashiwara algebra $\mathcal K_q$. We show that a certain quotient $\mathcal N_q^-$ of $U_q(\hat{\mathfrak{g}})$ is a simple $\mathcal K_q$-module.

未扭量子仿射代数的约化虚 Verma 模的晶体基

Crystal bases for reduced imaginary Verma modules of untwisted quantum affine algebras

本文研究未扭量子仿射代数的约化虚 Verma 模，借助 Kashiwara 代数定义虚晶体基，并证明适当范畴中所有对象均存在该基。

扭曲量子仿射代数的表示理论

Representations of twisted quantum affine algebras

本文发展扭曲量子仿射代数的虚Verma模表示理论，构造对应Kashiwara代数，证明其PBW型基、不可约性准则等核心结果，为扭曲量子与经典理论建立相容性。

On modules for double affine Lie algebras

Imaginary Verma modules, parabolic imaginary Verma modules, and Verma modules at level zero for double affine Lie algebras are constructed using three different triangular decompositions. Their relations are investigated, and several results are generalized from the affine Lie algebras. In particular, imaginary highest weight modules, integrable modules, and irreducibility criterion are also studied.

Imaginary crystal bases for $U_q(\widehat{\mathfrak{sl}(2)})$-modules in category $\mathcal O^q_{\text{red,im}}$

Recently we defined imaginary crystal bases for $U_q(\widehat{\mathfrak{sl}(2)})$- modules in category $\mathcal O^q_{\text{red,im}}$ and showed the existence of such bases for reduced quantized imaginary Verma modules for $U_q(\widehat{\mathfrak{sl}(2)})$. In this paper we show the existence of imaginary crystal basis for any object in the category $\mathcal O^q_{\text{red,im}}$.

New Irreducible Modules for Heisenberg and Affine Lie Algebras

We study $\mathbb Z$-graded modules of nonzero level with arbitrary weight multiplicities over Heisenberg Lie algebras and the associated generalized loop modules over affine Kac-Moody Lie algebras. We construct new families of such irreducible modules over Heisenberg Lie algebras. Our main result establishes the irreducibility of the corresponding generalized loop modules providing an explicit construction of many new examples of irreducible modules for affine Lie algebras. In particular, to any function $ϕ:\mathbb N\rightarrow \{\pm\}$ we associate a $ϕ$-highest weight module over the Heisen…

Relaxed singular vectors, Jack symmetric functions and fractional level $\widehat{\mathfrak{sl}}(2)$ models

The fractional level models are (logarithmic) conformal field theories associated with affine Kac-Moody (super)algebras at certain levels $k \in \mathbb{Q}$. They are particularly noteworthy because of several longstanding difficulties that have only recently been resolved. Here, Wakimoto's free field realisation is combined with the theory of Jack symmetric functions to analyse the fractional level $\widehat{\mathfrak{sl}}(2)$ models. The first main results are explicit formulae for the singular vectors of minimal grade in relaxed Wakimoto modules. These are closely related to the minimal gra…

Localization of free field realizations of affine Lie algebras

We use localization technique to construct new families of irreducible modules of affine Kac-Moody algebras. In particular, localization is applied to the first free field realization of the affine Lie algebra A_1^{(1)} or, equivalently, to imaginary Verma modules.

仿射老化代数 $\widehat{\mathfrak{age}}(1)$ 的表示

Representations of the affine ageing algebra $\widehat{\mathfrak{age}}(1)$

本文研究仿射老化代数 $\widehat{\mathfrak{age}}(1)$ 的 Verma 型模与限制模，刻画其单纯性并构造简单限制模。

Imaginary highest-weight representation theory and symmetric functions

Affine Lie algebras admit non-classical highest-weight theories through alternative partitions of the root system. Although significant inroads have been made, much of the classical machinery is inapplicable in this broader context, and some fundamental questions remain unanswered. In particular, the structure of the reducible objects in non-classical theories has not yet been fully understood. This question is addressed here for affine sl(2), which has a unique non-classical highest-weight theory, termed "imaginary". The reducible Verma modules in the imaginary theory possess an infinite desc…

负秩-2 Borcherds-Kac-Moody 李代数上模的特征标

Characters of modules over negative rank-2 Borcherds-Kac-Moody Lie algebras

本文研究负秩-2 Borcherds-Kac-Moody 李代数中非可积最高权模的结构和特征标，通过引入新的符号支配积分锥 $P^{\pm}$ 并分析 Verma 覆盖中的极大向量，推广了 Kac-Kazhdan 下界。

不可约 $\widehat{\mathfrak{gl}}_n$-模的薄化：具有主导最高权的薄化

Athinization of irreducible $\widehat{\mathfrak{gl}}_n$-modules with dominant highest weights

研究通过薄化方法将不可约 $\widehat{\mathfrak{gl}}_n$-模块转化为 affine Yangian 上的薄模块，提供新的组合实现。