Related papers: The Gelfand-Tsetlin-Zhelobenko base vectors for th…
A construction of Gelfand-Tsetlin type base vectors in a finite-dimensional representation of $\mathfrak{sp}_{2n}$ was firstly obtained in 60-th by Zhelobenko. But the final construction was obtained only in the year 1998 by Molev, who gave…
Using the Zhelobenko's approach we investigate a branching of an irreducible representation of $g_n$ under the restriction of algebras $g_n\downarrow g_{n-1}$, where $g_n$ is a Lie algebra of type $B_n$, $C_n$, $D_n$ or a Lie algebra of…
For the Lie algebras $g_n= \mathfrak{o}_{2n+1},\mathfrak{sp}_{2n},\mathfrak{o}_{2n}$ a simple construction of a base in an irreducible representation is given. The construction of this base uses the method of $Z$-invariants of Zhelobenko…
This paper completes a series devoted to explicit constructions of finite-dimensional irreducible representations of the classical Lie algebras. Here the case of odd orthogonal Lie algebras (of type B) is considered (two previous papers…
The paper presents a construction of finite-dimensional irreducible representations of the Lie algebra $\mathfrak{g}_2$. The representation space is constructed as the space of solutions to a certain system of partial differential equations…
A realization of representations of the Lie algebra $\mathfrak{o}_5$ in the space of functions on a group $Spin_5\simeq Sp_4$ is considered. In a representation we take a Gelfand-Tsetlin type base associated with a restriction…
In the paper a realization of representation of a Lie algebra $\mathfrak{sp}_4$ in the space of function on the Lie group $Sp_4$ is considered. We find a function corresponding to a Gelfand-Tsetlin type vector for $\mathfrak{sp}_4$…
A recursive method for construction of symmetric irreducible representations of O(2l+1) in the O(2l + 1) supset O(3) basis for identical boson systems is proposed. The formalism is realized based on the group chain U(2l + 1) supset U(2l- 1)…
A weight basis for each finite-dimensional irreducible representation of the orthogonal Lie algebra o(2n) is constructed. The basis vectors are parametrized by the D-type Gelfand--Tsetlin patterns. Explicit formulas for the matrix elements…
We construct functors categorifying the branching rules for $U_q(\mathfrak{g})$ for $\mathfrak{g}$ of type $B_n$, $C_n$, and $D_n$ for the embeddings $so_{2n+1}\supset so_{2n-1}$, $sp_{2n}\supset sp_{2n-2}$, and $so_{2n}\supset so_{2n-2}$.…
A new, so called odd Gel'fand-Zetlin basis is introduced for the irreducible covariant tensor representations of the Lie superalgebra gl(n|n). The related Gel'fand-Zetlin patterns are based upon the decomposition according to a particular…
We study Olshanski twisted Yangian based models, known as one-dimensional "soliton non-preserving" open spin chains, by means of algebraic Bethe ansatz. The even case, when the bulk symmetry is $\mathfrak{gl}_{2n}$ and the boundary symmetry…
Remarkable subalgebras of the Yangian for gl_n called the shifted Yangians were introduced in a recent work by Brundan and Kleshchev in relation to their study of finite W-algebras. In particular, in that work a classification of…
A Gel'fand-Zetlin basis is introduced for the irreducible covariant tensor representations of the Lie superalgebra gl(m|n). Explicit expressions for the generators of the Lie superalgebra acting on this basis are determined. Furthermore,…
Using a technique based on the Yangian Gelfand-Zetlin algebra and the associated Yangian Gelfand-Zetlin bases we construct an orthogonal basis of eigenvectors in the Calogero-Sutherland Model with spin, and derive product-type formulas for…
We study the effects of the branching $\mathfrak{osp}(1|2n)\supset \mathfrak{gl}(n)$ on a particular class of simple infinite-dimensional $\mathfrak{osp}(1|2n)$-modules $L(p)$ characterized by a positive integer $p$. In the first part we…
This is a review paper on the Gelfand-Tsetlin type bases for representations of the classical Lie algebras. Different approaches to construct the original Gelfand-Tsetlin bases for representations of the general linear Lie algebra are…
We generalize the famous weight basis constructions of the finite-dimensional irreducible representations of $\mathfrak{sl}(n,\mathbb{C})$ obtained by Gelfand and Tsetlin in 1950. Using combinatorial methods, we construct one such basis for…
This work is part of a project on weight bases for the irreducible representations of semisimple Lie algebras with respect to which the representation matrices of the Chevalley generators are given by explicit formulas. In the case of sl_n,…
We derive a bosonic formula for the character of the principal space in the level $k$ vacuum module for $\widehat{\mathfrak{sl}}_{n+1}$, starting from a known fermionic formula for it. In our previous work, the latter was written as a sum…