A. I. Molev
The evaluation homomorphisms from the Yangian Y(gl_n) to the universal enveloping algebra U(gl_n) allow one to regard the irreducible finite-dimensional representations of gl_n as Yangian modules. We give necessary and sufficient conditions for irreducibility of tensor products of such evaluation modules.
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With any skew Young diagram one can associate a one parameter family of "elementary" modules over the Yangian $\Yg(\gł_N)$. Consider the twisted Yangian $\Yg(\g_N)\subset \Yg(\gł_N)$ associated with a classical matrix Lie algebra $\g_N\subset\gł_N$. Regard the tensor product of elementary Yangian modules as a module over $\Yg(\g_N)$ by restriction. We prove its irreducibility for generic values of the parameters.
On Irreducibility of Tensor Products of Yangian Modules
研究一般线性李代数 Yangian 初等不可约模张量积的循环性与不可约性，给出奇异向量张量积循环的充分条件，并针对矩形斜 Young 图情形建立不可约性判据。
We study the tensor product $W$ of any number of "elementary" irreducible modules $V_1,...,V_k$ over the Yangian of the general linear Lie algebra. Each of these modules is determined by a skew Young diagram and a complex parameter. For any indices $i,j=1,...,k$ there is a canonical non-zero intertwining operator $A_{ij}$ between the tensor products $V_i\otimes V_j$ and $V_j\otimes V_i$. This operator is defined up to a scalar multipler. We show that the tensor product $W$ is irreducible, if and only if all operators $A_{ij}$ with $i<j$ are invertible. This implies that the Yangian module $W$ …
Irreducibility of the tensor product of Yangian \( \mathrm{Y}(\mathfrak{gl}_{m|n}) \) evaluation modules
本文研究了超Yangian $ \mathrm{Y}(\mathfrak{gl}_{m|n}) $ 评估模块张量积的不可约性条件，基于Gelfand-Tsetlin基的存在性，给出了简单模块的判据。
We study the Verma modules M(mu(u)) over the Yangian Y(a) associated with a simple Lie algebra a. We give necessary and sufficient conditions for irreducibility of M(mu(u)). Moreover, regarding the simple quotient L(mu(u)) of M(mu(u)) as an a-module, we give necessary and sufficient conditions for finite-dimensionality of the weight subspaces of L(mu(u)).
In this thesis, we study the cyclicity condition for an ordered tensor product of fundamental representations and the local Weyl modules of Yangians. We provide a sufficient condition for the cyclicity of an ordered tensor product $L=V_{a_1}(ω_{b_1})\otimes V_{a_2}(ω_{b_2})\otimes...\otimes V_{a_k}(ω_{b_k})$ of fundamental representations of the Yangian $Y(\mathfrak{g})$. When $\mathfrak{g}$ is a classical simple Lie algebra or the simple Lie algebra of type $G$, we make the cyclicity condition concrete, which leads to an irreducibility criterion for the ordered tensor product $L$. In the case…
Representations of shifted super Yangians and finite $W$-superalgebras of type A
本文研究了移位超 Yangians 和类型 A 的有限 W-超代数的表示理论，给出了不可约模有限维性的判据，并为有限 W-超代数的 Verma 模提供了显式 Gelfand-Tsetlin 特征公式。
In this paper we discuss the structure of the tensor product V'_{α,β}\otimes L(c,h) of irreducible module from intermediate series and irreducible highest weight module over the Virasoro algebra. We generalize Zhang's irreducibility criterion, and show that irreducibility depends on the existence of integral roots of a certain polynomial, induced by a singular vector in the Verma module V(c,h). A new type of irreducible Vir-module with infinite-dimensional weight subspaces is found. We show how the existence of intertwining operator for modules over vertex operator algebra yields reducibility …
On the simplicity of the tensor product of two simple modules of quantum affine algebras
本文通过补充条件将 Lapid-Mínguez 关于抛物诱导不可约性的判据推广到任意 k≥1 的量子仿射代数情形，并在 Grassmannian 簇代数中建立了阶梯相容性的类似判据，推广了 Leclerc-Zelevinsky 的弱分离结果。
Whittaker modules over the loop Virasoro algebra
本文研究环面 Witt 代数与环面 Virasoro 代数上的 Whittaker 模，给出其单性/不可约性判据，并应用于仿射李代数 $\widehat{\mathfrak{sl}_2}$ 的泛 Whittaker 模。
We give an explicit criterion for the irreducibility of some induction products of evaluation modules of affine Hecke algebras of type A. This allows to describe the form of the singularities of the trigonometric R-matrix associated to any evaluation module of U_v(sl^_N).
Whittaker constructions for quantum affine algebras
本文通过抛物诱导从Heisenberg李代数的不可约Whittaker模构造了无扭仿射Kac-Moody李代数的虚Whittaker模，建立了不可约性判据，并推广到量子仿射代数情形，同时证明了$\mathcal{U}_q(A_1^{(1)})$上一类非量子形变模的不可约性。
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.
A criterion of irreducibility for induction products of evaluation modules of type A affine Hecke algebras is given. It is derived from multiplicative properties of the canonical basis of a quantum deformation of the Bernstein-Zelevinsky ring.
The modular representation theory of the queer Lie superalgebra q(n) over characteristic p>2 is developed. We obtain a criterion for the irreducibility of baby Verma modules with semisimple p-characters and a criterion for the semisimplicity of the corresponding reduced enveloping algebras. A (2p)-power divisibility of dimensions of q(n)-modules with nilpotent p-characters is established. The representation theory of q(2) is treated in detail. We formulate a Morita super-equivalence conjecture for q(n) with general p-characters which is verified for n=2.
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