相关论文: The Gelfand-Tsetlin type base for the algebra $\ma…

We explicitly construct families of simple modules for Lie algebras of rank $2$, on which certain commutative subalgebra acts diagonally and has a simple spectrum. In type $A$ these modules are well known generic Gelfand-Tsetlin modules and…

We give explicit actions of Drinfeld generators on Gelfand-Tsetlin bases of super Yangian modules associated with skew Young diagrams. In particular, we give another proof that these representations are irreducible. We study irreducible…

The universal enveloping algebra of any semisimple Lie algebra $\mathfrak{g}$ contains a family of maximal commutative subalgebras, called shift of argument subalgebras, parametrized by regular Cartan elements of $\mathfrak{g}$. For…

For the nonstandard $q$-deformed algebras $U_q(so_n)$, defined recently in terms of trilinear relations for generating elements, most general finite dimensional irreducible representations directly corresponding to those of nondeformed…

We study certain family of finite-dimensional modules over the Yangian $Y(gl_N)$. The algebra $Y(gl_N)$ comes equipped with a distinguished maximal commutative subalgebra $A(gl_n)$ generated by the centres of all algebras in the chain…

In 1960's I. Gelfand posed a problem: describe indecomposable representations of any simple infinite dimensional Lie algebra of polynomial vector fields. Here, by applying the elementary technique of Gelfand and Ponomarev, a toy model of…

Gaudin algebras form a family of maximal commutative subalgebras in the tensor product of $n$ copies of the universal enveloping algebra $U(\g)$ of a semisimple Lie algebra $\g$. This family is parameterized by collections of pairwise…

We address the problem of classifying of irreducible Gelfand-Tsetlin modules for gl(m|n) and show that it reduces to the classification of Gelfand-Tsetlin modules for the even part. We also give an explicit tableaux construction and the…

This paper focuses on the $GL_n$ tensor product algebra, which encapsulates the decomposition of tensor products of arbitrary finite dimensional irreducible representations of $GL_n$. We will describe an explicit basis for this algebra.…

Some introductory concepts and basic definitions of the Lie superalgebras and their quantum deformations are exposed. Especially the induced representation methods in both cases are described. Based on the Kac representation theory we have…

We study highest weight representations of shifted Yangians over an algebraically closed field of characteristic 0. In particular, we classify the finite dimensional irreducible representations and explain how to compute their…

We construct a nil algebra over a countable field which has finite but non-zero Gelfand-Kirillov dimension.

For an $n$-fold Kazhdan--Patterson cover or Savin's cover of a general linear group over a non-archimedean local field of residual characteristic $p$ with $\mathrm{gcd}(n,p)=1$, we realize the Gelfand--Graev representation as a Hecke…

First, I construct an isomorphism between the categories of (topological) groups of nilpotency class 2 with 2-divisible center and (topological) Lie rings of nilpotency class 2 with 2-divisible center. That isomorphism allows us to…

According to Kirillov's theory, the construction of a unitary irreducible representation of a nilpotent Lie group requires a precise computation of some polarizing subalgebra subordinated to a linear functional in the linear dual of the…

We establish a direct connection between the representation theories of Lie algebras and Lie superalgebras (of type A) via Fock space reformulations of their Kazhdan-Lusztig theories. As a consequence, the characters of finite-dimensional…

The Clebsh-Gordan coefficients for the Lie algebra $\mathfrak{gl}_3$ in the Gelfand-Tsetlin base are calculated. In contrast to previous papers the result is given as an explicit formula. To obtain the result a realization of a…

We modify the Hochschild $\phi$-map to construct central extensions of a restricted Lie algebra. Such central extension gives rise to a group scheme which leads to a geometric construction of unrestricted representations. For a classical…

The goal of this paper is to give an explicit construction of the Fock spaces of the parafermion and the paraboson algebra, for an infinite set of generators. This is equivalent to constructing certain unitary irreducible lowest weight…

Springer resolution of the set of nilpotent elements in a semisimple Lie algebra plays a central role in geometric representation theory. A new structure on this variety has arisen in several representation theoretic constructions, such as…