Related papers: Shapovalov Elements For $U_q(\mathfrak{sl}(N+1))

Let G=GL_n be the general linear group over an algebraically closed field k and let g=gl_n be its Lie algebra. Let U be the subgroup of G which consists of the upper unitriangular matrices. Let k[g] be the algebra of regular functions on…

We study the crystal graphs of irreducible $U_{v}(\hat{sl}}_{e})$-modules of higher level l. Generalizing works of the first author, we obtain a simple description of the bijections between the classes of multipartitions which naturally…

It is demonstrated that decompositions of integrable highest weight modules of a simple Lie algebra with respect to its reductive subalgebra obey the set of algebraic relations leading to the recursive properties for the corresponding…

We construct reflection functors for Yetter-Drinfeld modules over Nichols systems and discuss their fundamental properties. We will obtain properties about the geometry of the support of Nichols systems and their Yetter-Drinfeld modules, by…

We describe a systematic method to construct arbitrary highest-weight modules, including arbitrary finite-dimensional representations, for any finite dimensional simple Lie algebra $\mathfrak{g}$. The Lie algebra generators are represented…

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…

Let X be the group of weights of a maximal torus of a simply connected semisimple group over C and let W be the Weyl group. The semidirect product W(Q\otimes X/X) is called the extended Weyl group. There is a natural C(v)-algebra H called…

We formulate and prove that there are "abundant" in nilpotent orbits in real semisimple Lie algebras, in the following sense. If S denotes the collection of hyperbolic elements corresponding the weighted Dynkin diagrams coming from…

This is a noncommutative version of the previous work entitled "Deformation Expression for Elements of Algebras (I)." In general in a noncommutative algebra, there is no canonical way to express elements in univalent way, which is often…

Let $\mathfrak{W}$ be the Lie algebra of vector fields on the line. Via computing extensions between all simple modules in the category $\mathcal{O}$, we give the block decomposition of $\mathcal{O}$, and show that the representation type…

Let $G$ be a reductive algebraic group over an algebraically closed field of characteristic $p>0$, and let ${\mathfrak g}$ be its Lie algebra. Given $\chi\in{\mathfrak g}^{*}$ in standard Levi form, we study a category ${\mathscr C}_\chi$…

The level-$1$ integrable highest weight modules of $U_q(\widehat{sl}_2)$ admit a level-$0$ action of the same algebra. This action is defined using the affine Hecke algebra and the basis of the level-$1$ module generated by components of…

A partition of the set of unary pp formulas into four regions is presented, which has a bearing on various structural properties of modules. The machinery developed allows for applications to IF, weakly coherent, nonsingular, and reduced…

In this paper, we study the representation theory for the affine Lie algebra $\H$ associated to the Nappi-Witten model $H_{4}$. We classify all the irreducible highest weight modules of $\H$. Furthermore, we give a necessary and sufficient…

A theory of cyclic elements in semisimple Lie algebras is developed. It is applied to an explicit construction of regular elements in Weyl groups.

We define the notion of a separable element in a finite Weyl group, generalizing the well-studied class of separable permutations. We prove that the upper and lower order ideals in weak Bruhat order generated by a separable element are…

For an admissible affine vertex algebra $V_k(\mathfrak{g})$ of type $A$, we describe a new family of relaxed highest weight representations of $V_k(\mathfrak{g})$. They are simple quotients of representations of the affine Kac-Moody algebra…

The weights are computed for the Bethe vectors of an RSOS type model with periodic boundary conditions obeying $U_q[sl(n)]$ ($q=\exp(i\pi/r)$) invariance. They are shown to be highest weight vectors. The q-dimensions of the corresponding…

We describe the structure of the irreducible highest weight modules for the twisted Heisenberg-Virasoro Lie algebra at level zero. We prove that such a module is either isomorphic to a Verma module or to a quotient of two Verma modules.

We study modules over a generalized Weyl algebra $R(\sigma,a)$ which are free when restricted to the base ring $R$. When $R$ is an integral domain, we construct all such finite-rank modules up to isomorphism, leading to new simple modules…