V. K. Dobrev
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.
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In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebra $G_{2(2)}$ which is split real form of $G_2$. We give the classification of reducible Verma modules $G_2$. We give also the singular vectors between these modules, thus setting the stage for construction of the invariant differential operators over $G_{2(2)}$.
In the present paper we study the representations of the Jacobi algebra. More concretely, we define, analogously to the case of semi-simple Lie algebras, the Verma modules over the Jacobi algebra ${\cal G}_2$. We study their reducibility and give explicit construction of the reducible Verma modules exhibiting the corresponding singular vectors. Using this information we give a complete classification of the reducible Verma modules. More than this we exhibit their interrelation of embeddings between these modules. These embeddings are illustrated by diagrams of the embedding patterns so that ea…
Reducibility of Scalar Generalized Verma Modules of Minimal Parabolic Type II
本文研究了例外李代数中最小抛物子代数相关标量广义韦尔玛模的可约性，通过计算盖尔范-基里洛夫维数确定其第一个对角可约点。
Reducibility of scalar generalized Verma modules of minimal parabolic type
本文针对经典复单李代数的最小抛物子代数，通过计算最高权模的Gelfand-Kirillov维数，确定了标量广义Verma模的第一个对角可约点。
Let $\preceq$ be a compatible total order on the additive group $\mathbb{Z}^2$, and $L$ be the rank two Heisenberg-Virasoro algebra. For any $\mathbf{c}=(c_1,c_2,c_3,c_4) \in \mathbb{C}^4$, we define $\mathbb{Z}^2$-graded Verma module $M(\mathbf{c}, \preceq)$ for the Lie algebra $L$. A necessary and sufficient condition for the Verma module $M(\mathbf{c}, \preceq)$ to be irreducible is provided. Moreover, the maximal $\mathbb{Z}^2$-graded submodules of the Verma module $M(\mathbf{c}, \preceq)$ are characterized when $M(\mathbf{c}, \preceq)$ is reducible.
A well known theorem of Duflo claims that the annihilator of a Verma module in the enveloping algebra of a complex semisimple Lie algebra is generated by its intersection with the centre. For a Lie superalgebra this result fails to be true. For instance, in the case of the orthosymplectic Lie superalgebra osp(1,2), Pinczon gave in [Pi] an example of a Verma module whose annihilator is not generated by its intersection with the centre of universal enveloping algebra. More generally, Musson produced in [Mu1] a family of such "singular" Verma modules for osp(1,2l) cases. In this article we give a…
On the reducibility of scalar generalized Verma modules associated to two-step nilpotent parabolic subalgebras
本文利用Gelfand-Kirillov维数研究非极大型二步幂零抛物子代数对应的标量广义Verma模可约性，限定三类李代数，并将可约点表示在二维复平面中。
A parabolic subalgebra $\mathfrak{p}$ of a complex semisimple Lie algebra $\mathfrak{g}$ is called a parabolic subalgebra of abelian type if its nilpotent radical is abelian. In this paper, we provide a complete characterization of the parameters for scalar generalized Verma modules attached to parabolic subalgebras of abelian type such that the modules are reducible. The proofs use Jantzen's simplicity criterion, as well as the Enright-Howe-Wallach classification of unitary highest weight modules.
Little $q$-Jacobi polynomials and symmetry breaking operators for $U_q(sl_2)$
本文研究了$U_q(sl_2)$量子群作用于张量积Verma模的 intertwining 算子，利用小$q$-雅可比多项式和$q$-变形的Rankin-Cohen算子，揭示了非交换变量与Clebsch-Gordan系数之间的联系。
Realisations of Racah algebras using Jacobi operators and convolution identities
本文利用 sl(2) 的表示理论与最低权 Verma 模张量积模型，通过 Jacobi 微分算子实现了 Hahn 代数、Racah 代数及高秩 Racah 代数，并恢复了相关的 Jacobi 多项式卷积公式。
In the present paper we construct explicitly the intertwining differential operators for the Jacobi algebra ${\cal G}_2.$ For the construction we use the singular vectors of the Verma modules over ${\cal G}_2$ which we have constructed earlier. We construct the function spaces on which the operators act. We display two versions of the left (representation) action and the right action. The latter is inserted in the singular vectors to provide the intertwining differential operators.
Simple restricted modules over a new Lie superalgebra extended by the Ovsienko--Roger algebra
本文介绍了基于新李超代数的简单受限模块构造，并研究了其广义Verma模的分类与可约性。
We give a complete classification of the irreducible quasifinite modules for algebras of the form Vir \otimes A, where Vir is the Virasoro algebra and A is a Noetherian commutative associative unital algebra over the complex numbers. It is shown that all such modules are tensor products of generalized evaluation modules. We also give an explicit sufficient condition for a Verma module of Vir \otimes A to be reducible. In the case that A is an infinite-dimensional integral domain, this condition is also necessary.
Invariant Differential Operators for the Real Exceptional Lie Algebra $F"_4$
本文以非紧例外李代数$F"_4$（$F_4$的分裂秩一形式）为对象，通过分类$F_4$上与该归纳相容的可约Verma模，完成了对应不变微分算子的分类，推进了不变微分算子的系统构建研究。
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…
We derive non-linear commutator HS symmetry algebra, which encode unitary irreducible representations of AdS group subject to Young tableaux $Y(s_1,...,s_k)$ with $k\geq 2$ rows on $d$-dimensional anti-de-Sitter space. Auxiliary representations for specially deformed non-linear HS symmetry algebra in terms of generalized Verma module in order to additively convert a subsystem of second-class constraints in the HS symmetry algebra into one with first-class constraints are found explicitly for the case of HS fields for $k=2$ Young tableaux. The oscillator realization over Heisenberg algebra for …
Invariant Differential Operators for the Real Exceptional Lie Algebra $F'_4$
本文以实例外李代数 $F'_4$ 为例，通过极大抛物诱导分类可约 Verma 模，从而得到不变微分算子的完整分类。
Gauge Invariant Lagrangian Formulations for Mixed Symmetry Higher Spin Bosonic Fields in AdS Spaces
推导了AdS空间中混合对称高旋场的非线性对称代数与BRST算子，构建了规范不变拉格朗日表述，并发展了用于暗物质候选大质量高旋场的离壳三次相互作用顶点的BRST构建方法。
The aim of this paper is to apply systematically to AdS_4 some modern tools in the representation theory of Lie algebras which are easily generalised to the supersymmetric and quantum group settings and necessary for applications to string theory and integrable models. Here we introduce the necessary representations of the AdS_4 algebra and group. We give explicitly all singular (null) vectors of the reducible AdS_4 Verma modules. These are used to obtain the AdS_4 invariant differential operators. Using this we display a new structure - a diagram involving four partially equivalent reducible …
Let $\mathcal{O}_{25}$ be the vertex algebraic braided tensor category of finite-length modules for the Virasoro Lie algebra at central charge $25$ whose composition factors are the irreducible quotients of reducible Verma modules. We show that $\mathcal{O}_{25}$ is rigid and that its simple objects generate a semisimple tensor subcategory that is braided tensor equivalent to an abelian $3$-cocycle twist of the category of finite-dimensional $\mathfrak{sl}_2$-modules. We also show that this $\mathfrak{sl}_2$-type subcategory is braid-reversed tensor equivalent to a similar category for the Vir…
Structure of Virasoro tensor categories at central charge $13-6p-6p^{-1}$ for integers $p > 1$
本文研究了中心荷为13−6p−6p⁻¹的Virasoro张量范畴结构，证明其刚性并构造了射影包，同时推导了三重顶点算子代数的相关结果。
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