Searcharxiv⌕ Search

arXiv subjects

D. V. Artamonov

Publications and source records attributed to D. V. Artamonov.

15 recordsLinked to original sources
arXiv2025

Classical $6j$-symbols for finite dimensional representation of the algebra $\mathfrak{gl}_3$

In the paper an explicit formula for an arbitrary $6j$-symbol for finite-dimensional irreducible representations of the algebra $\mathfrak{gl}_3$ is derived. A $6j$-symbol is written as a result of substitution of $\pm 1$ into a series of hypergeometric type, which is similar to a $Γ$-series, which is a simplest example of a multivariate series of hypergeometric type. The selection rulers for a $6j$-symbol are derived.

math.RT↗
arXiv2024

Models of representations for classical series of Lie algebras

A model of representations of a Lie algebra is a representation which a direct sum of all irreducible finite dimensional representations taken with multiplicity $1$. In the paper an explicit construction of a model of representation for all series of classical Lie algebras is given. The construction does not differ much for different series. The space of the model is constructed as a space of polynomial solutions of a system of partial differential equations. The equations in this system are constructed form relations between minors of matrices from the corresponding Lie group. This system has a simplification which is very close to the GKZ system, that is satisfied by $A$-hypergeometric functions.

math.RT↗
arXiv2022

A functional realization of the Gelfand-Tsetlin base

In the paper we consider a realization of a finite dimensional irreducible representation of the Lie algebra $\mathfrak{gl}_n$ in the space of functions on the group $GL_n$. It is proved that functions corresponding to Gelfand-Tsetlin diagrams are linear combinations of some new functions of hypergeometric type which are closely related to $A$-hypergeometric functions. These new functions are solution of a system of partial differential equations which one obtains from the Gelfand-Kapranov-Zelevinsky by an "antisymmetrization". The coefficients in the constructed linear combination are hypergeometric constants i.e. they are values of some hypergeometric functions when instead of all arguments ones are substituted.

math.RT↗
arXiv2017

A unified approach to construction of Gelfand-Tsetlin-Zhelobenko base vectors for series $A$, $B$, $C$, $D$

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 type $A$, where in this case we put $g_{n}=\mathfrak{gl}_{n+1}$, $g_{n-1}=\mathfrak{gl}_{n-1}$. We give a new explicit description of the space of the $g_{n-1}$-highest vectors, then we construct a base in this space. The case $n=2$ is considered separately for different algebras, but a passage from $n=2$ to an arbitrary $n$ is the same for all series $A$, $B$, $C$, $D$. This new procedure has the following advantage: it establishes a relation between spaces of $g_{n-1}$-highest vectors for different series of algebras. This procedure describes an extension of Gelfand-Tsetlin tableaux to the left.

math.RT↗
arXiv2017

The Gelfand-Tsetlin-Zhelobenko base vectors for the series $B$

In the paper using the method of $Z$-invariants of Zhelobenko we construct base vectors of Gelfand-Tsetlin type in a representation of $\mathfrak{o}_{2n+1}$, based on restrictions $\mathfrak{o}_{2n+1}\downarrow\mathfrak{o}_{2n-1}$. The construction is based on a discovered relation between restriction problems $\mathfrak{o}_{2n+1}\downarrow\mathfrak{o}_{2n-1}$ and $\mathfrak{gl}_{n+1}\downarrow\mathfrak{gl}_{n-1}$.

math.RT↗
arXiv2016

The comparison Gelfand-Tsetlin-Molev and Gelfand-Tsetlin-Zhelobenko bases for $\mathfrak{sp}_{2n}$

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 a construction of Gelfand-Tsetlin type base vectors and derived formulas for the action of generators of the algebra in this base. These two approaches use different ideas. In the present paper we compare these two approaches. Also we show that the Molev's base vectors can be obtained using a construction based on a relation between restriction problems $\mathfrak{sp}_{2n}\downarrow\mathfrak{sp}_{2n-2}$ and $\mathfrak{gl}_{n+1}\downarrow\mathfrak{gl}_{n-1}$, analogous to the construction giving the Zhelobenko's base.

math.RT↗
arXiv2016

Introduction to finite $W$-algebras

These are notes of lectures given at UN Encuentro 2016 at the Colombia National University. We begin with the definition of infinite $W$-algebras. Then we explain the motivation for the definition if finite $W$-algebras. Then we present basic facts about the structure and representations of finite $W$-algebras. In these lectures we follow the historical development of the subject.

math.RT↗
arXiv2014

$W$-algebras and higher analogs of the Kniznik-Zamolodchikov equations

The key role in the derivation of the Knizhnik-Zamolodchikov equations in the $WZW$-theory is played by the energy-momentum tensor, that is constructed from a central Casimir element of the second order in a universal enveloping algebra of a corresponding Lie algebra. In the paper a possibility of construction of analogs of Knizhnik-Zamolodchikov equations using higher order central elements is investigated. The Gelfand elements of the third order for a simple Lie algebra of series $A$ and Capelli elements of the fourth order for the a simple Lie algebra of series $B$, $D$ are considered. In the first case the construction is not possible a the second case the desired equation is derived.

math.RT↗
arXiv2013

Wigner coefficient for Lie algebras of series $B,C,D$ and a base of Gelfand-Tsetlin type

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 and the technique of Wigner coefficients, which was applied by Biedenharn and Baird to the construction of a Gelfand-Tsetlin base in the case $\mathfrak{gl}_n$. A relation between matrix elements and Wigner coefficients for $g_n$ and analogous objects for $\mathfrak{gl}_{n+1}$ is established.

math.RT↗
arXiv2012

Knizhnik-Zamolodchikov type equations for the root system $B$ and Capelli central elements

The construction of the well-known Knizhnik-Zamolodchikov equations uses the central element of the second order in the universal enveloping algebra for some Lie algebra. But in the universal enveloping algebra there are central elements of higher orders. It seems desirable to use these elements for the construction of Knizhnik-Zamolodchikov type equations. In the present paper we give a construction of such Knizhnik-Zamolodchikov type equations for the root system $B$ associated with Capelli central elements in the universal enveloping algebra for the orthogonal algebra.

math.RT↗
arXiv2012

Noncommutative pfaffians and representations

Noncommutative pfaffians associated with an orthogonal algebra are some special elements of the universal enveloping algebra. In the paper it is suggested to use some pfaffians as raising operators. The images of these pfaffians in the Mickelson-Zhelobenko algebra are calculated. It allows to find a place of pfaffians among other raising operators. As a byproduct the action of the pfaffians on the Gelfand-Tsetlin-Molev bases is found. The action of pfaffians in the tensor realization of representation is considered in the appendix.

math.RT↗
arXiv2012

The Schlesinger system and isomonodromic deformations of bundles with connections on Riemann surfaces

We introduce a way of presentation of pairs $(E,\nabla)$, where $E$ is a bundle on a Riemann surface and $\nabla$ is a logarithmic connection in $E$, which is based on a presentation of the surface as a factor of the exterior of the unit disc. In this presentation we write the local equation of isomonodormic deformation of pairs $(E,\nabla)$. These conditions are written as a modified Schlesinger system on a Riemann sphere (and in the typical case just as an ordinary Schlesinger system) plus some linear system.

math.CA↗