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Dimitri Gurevich

Publications and source records attributed to Dimitri Gurevich.

At least 19 recordsLinked to original sources
arXiv2024

Wick theorem and matrix Capelli identity for quantum differential operators on Reflection Equation Algebras

Quantum differential operators on Reflection Equation Algebras, corresponding to Hecke symmetries R were introduced in previous publications. In the present paper we are mainly interested in quantum analogs of the Laplace and Casimir operators, which are invariant with respect to the action of the Quantum Groups U_q(sl(N)), provided R is the Drinfeld-Jimbo $R$-matrix. We prove that any such an operator maps the central characteristic subalgebra of a Reflection Equation algebra into itself. Also, we define the notion of normal ordering for the quantum differential operators and prove an analog of the Wick theorem for the product of partially ordered operators. As an important corollary we find a set of universal matrix Capelli identities generalizing the results of [Ok2] and [JLM]. Besides, we prove that the normal ordered form of any central differential operator from the characteristic subalgebra is also a central differential operator.

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arXiv2022

Matrix Capelli identities related to Reflection Equation algebra

By using the notion of a quantum double we introduce analogs of partial derivatives on a Reflection Equation algebra, associated with a Hecke symmetry of GL(N) type. We construct the matrix L=MD, where M is the generating matrix of the Reflection Equation algebra and D is the matrix composed of the quantum partial derivatives and prove that the matrices M, D and L satisfy a matrix identity, called the matrix Capelli one. Upon applying the quantum trace, it becomes a scalar relation, which is a far-reaching generalization of the classical Capelli identity. Also, we get a generalization of the some higher Capelli identities defined by A.Okounkov.

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arXiv2021

q-Casimir and q-cut-and-join operators related to Reflection Equation Algebras

In this paper we are dealing with the Reflection Equation algebra ${\cal M}(R)$, associated with a $GL_N$ type Hecke symmetry $R$. In this algebra we define the $q$-analogs of the partial derivatives $\partial_j^i$ in generators $m_i^j$ of ${\cal M}(R)$. The product $\hat L = MD$ of two matrices $M=\|m_i^j\|$ and $D=\|\partial_i^j\|$ turns out to be a generating matrix of a modified Reflection Equation algebra $\hat{\cal L}(R)$ which is similar to the universal enveloping algebra $U(gl_N)$ in many aspects. Central elements of the modified Reflection Equation algebra give rise to $q$-Casimir operators in a representation of $\hat{\cal L}(R)$ in the algebra ${\cal M}(R)$. We perform a spectral analysis of the first $q$-Casimir operator and formulate a conjecture about the spectrum of the higher ones. At last, we define the normal ordering for the $q$-differential operators and inroduce the $q$-cut-and-join operators. In several explicit examples we express some of $q$-cut-and-join operators via the $q$-Casimir ones by analogy with the classical case.

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arXiv2020

Noncommutative geometry on central extension of U(u(2))

In our previous publications we have introduced analogs of partial derivatives on the algebras U(gl(N)). In the present paper we compare two methods of introducing these analogs: via the so-called quantum doubles and by means of a coalgebraic structure. In the case N=2 we extend the quantum partial derivatives from U(u(2)) (the compact form of the algebra U(gl(2))) on a bigger algebra, constructed in two steps. First, we define the derivatives on a central extension of this algebra, then we prolongate them on some elements of the corresponding skew-field by using the Cayley-Hamilton identities for certain matrices with noncommutative entries. Eventual applications of this differential calculus are discussed.

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arXiv2020

Doubles of associative algebras and their applications

For a couple of associative algebras we define the notion of their double and give a set of examples. Also, we discuss applications of such doubles to representation theory of certain quantum algebras and to a new type of Noncommutative Geometry.

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arXiv2019

Determinants in Quantum Matrix Algebras and Integrable Systems

We define quantum determinants in Quantum Matrix Algebras, related to couples of compatible braidings following the scheme from [G]. We establish relations between these determinants and the so-called column-(row-)determinants, often used in the theory of integrable systems. Also, we generalize the quantum integrable spin systems from [CFRS] by using generalized Yangians, related to couples of compatible braidings. We demonstrate that such quantum integrable spin systems are not uniquely determined by the "quantum coordinate ring" of the basic space V. For instance, the "quantum plane" xy=qyx gives rise to two different integrable systems: rational and trigonometric ones.

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arXiv2018

KZ equations and Bethe subalgebras in generalized Yangians related to compatible R-matrices

The notion of compatible braidings was introduced by Isaev, Ogievetsky and Pyatov. On the base of this notion they defined certain quantum matrix algebras generalizing the RTT algebras and Reflection Equation ones. They also defined analogs of some symmetric polynomials in these algebras and showed that these polynomials generate commutative subalgebras, called Bethe. By using a similar approach we introduce certain new algebras called generalized Yangians and define analogs of some symmetric polynomials in these algebras. We claim that they commute with each other and thus generate a commutative Bethe subalgebra in each generalized Yangian. Besides, we define some analogs (also arising from couples of compatible braidings) of the Knizhnik-Zamolodchikov equation--classical and quantum.

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arXiv2018

Bethe subalgebras in Braided Yangians and Gaudin-type models

In \cite{GS1} the notion of braided Yangians of Reflection Equation type was introduced. Each of these algebras is associated with an involutive or Hecke symmetry $R$. Besides, the quantum analogs of certain symmetric polynomials (elementary symmetric ones, power sums) were suggested. In the present paper we show that these quantum symmetric polynomials commute with each other and consequently generate a commutative Bethe subalgebra. As an application, we get some Gaudin-type models and the corresponding Bethe subalgebras.

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arXiv2018

From Reflection Equation Algebra to Braided Yangians

In general, quantum matrix algebras are associated with a couple of compatible braidings. A particular example of such an algebra is the so-called Reflection Equation algebra. In this paper we analyse its specific properties, which distinguish it from other quantum matrix algebras (in first turn, from the RTT one). Thus, we exhibit a specific form of the Cayley-Hamilton identity for its generating matrix, which in a limit turns into the Cayley-Hamilton identity for the generating matrix of the enveloping algebra U(gl(m)). Also, we consider some specific properties of the braided Yangians, recently introduced by the authors. In particular, we establish an analog of the Cayley-Hamilton identity for the generating matrix of such a braided Yangian. Besides, by passing to a limit of the braided Yangian, we get a Lie algebra similar to that entering the construction of the rational Gaudin model. In its enveloping algebra we construct a Bethe subalgebra by the method due to D.Talalaev.

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arXiv2017

Centers in Generalized Reflection Equation algebras

In the Reflection Equation (RE) algebra associated with an involutive or Hecke symmetry $R$ the center is generated by elements ${\rm Tr}_R L^k$ (called the quantum power sums), where $L$ is the generating matrix of this algebra and ${\rm Tr}_R$ is the $R$-trace associated with $R$. We consider the problem: whether it is so in certain RE-like algebras depending on spectral parameters. Mainly, we deal with algebras similar to those considered in \cite{RS} (we call them algebras of RS type). These algebras are defined by means of some current (i.e. depending on parameters) $R$-matrices arising from involutive and Hecke symmetries via the so-called Baxterization procedure. We define quantum power sums in the algebras of RS type and show that the lowest quantum power sum in such an algebra is central iff the value of the "charge" $c$ entering its definition is critical. We exhibit the dependance of this critical value on the bi-rank of the initial symmetry $R$. Besides, we show that if the bi-rank of $R$ is $(m|m)$, and the value of $c$ is critical, then all quantum power sums are central.

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arXiv2017

Drinfeld-Sokolov reduction in quantum algebras

Applying the method of the paper [CT], we perform a quantum version of the Drinfeld-Sokolov reduction in Reflection Equation algebras and braided Yangians, associated with involutive and Hecke symmetries of general forms. This reduction is based on the Cayley-Hamilton identity valid for the generating matrices of these algebras.

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arXiv2017

Generalized Yangians and their Poisson counterparts

By a generalized Yangian we mean a Yangian-like algebra of one of two classes. One of these classes consists of the so-called braided Yangians, introduced in our previous paper. The braided Yangians are in a sense similar to the reflection equation algebra. The generalized Yangians of second class, called the Yangians of RTT type, are defined by the same formulae as the usual Yangians are but with other quantum $R$-matrices. If such an $R$-matrix is the simplest trigonometrical $R$-matrix, the corresponding Yangian of RTT type is the so-called q-Yangian. We claim that each generalized Yangian is a deformation of the commutative algebra ${\rm Sym}(gl(m)[t^{-1}])$ provided that the corresponding $R$-matrix is a deformation of the flip. Also, we exhibit the corresponding Poisson brackets.

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arXiv2016

Braided Yangians

Yangian-like algebras, associated with current R-matrices, different from the Yang ones, are introduced. These algebras are of two types. The so-called braided Yangians are close to the Reflection Equation algebras, arising from involutive or Hecke symmetries. The Yangians of RTT type are close to the corresponding RTT algebras. Some properties of these two classes of the Yangians are studied. Thus, evaluation morphisms for them are constructed, their bi-algebra structures are described, and quantum analogs of certain symmetric polynomials, in particular, quantum determinants, are introduced. It is shown that in any braided Yangian this determinant is always central, whereas in the Yangians of RTT type it is not in general so. Analogs of the Cayley-Hamilton-Newton identity in the braided Yangians are exhibited. A bozonization of the braided Yangians is performed.

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arXiv2015

Quantum geometry and quantization on U(u(2)) background. Noncommutative Dirac monopole

In our previous publications we introduced differential calculus on the enveloping algebras U(gl(m)) similar to the usual calculus on the commutative algebra Sym(gl(m)). The main ingredient of our calculus are quantum partial derivatives which turn into the usual partial derivatives in the classical limit. In the particular case m=2 we prolonged this calculus on a central extension A of the algebra U(gl(2)). In the present paper we consider the problem of a further extension of the quantum partial derivatives on the skew-field of the algebra A and define the corresponding de Rham complex. As an application of the differential calculus we suggest a method of transferring dynamical models defined on Sym(u(2)) to the algebra U(u(2)) (we call this procedure the quantization with noncommutative configuration space). In this sense we quantize the Dirac monopole and find a solution of this model.

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arXiv2014

Noncommutative Geometry and dynamical models on U(u(2)) background

In our previous publications we have introduced a differential calculus on the algebra $U(gl(m))$ based on a new form of the Leibniz rule which differs from that usually employed in Noncommutative Geometry. This differential calculus includes partial derivatives in generators of the algebra $U(gl(m))$ and their differentials. The orresponding differential algebra $Ω(U(gl(m)))$ is a deformation of the commutative algebra $Ω({\rm Sym}(gl(m)))$.A similar claim is valid for the Weyl algebra ${\cal W}(U(gl(m)))$ generated by the algebra $U(gl(m))$ and the mentioned partial derivatives. In the particular case $m=2$ we treat the compact form $U(u(2))$ of this algebra as a quantization of the Minkowski space algebra. Below we consider noncommutative versions of the Klein-Gordon equation and the Schrödinger equation for the hydrogen atom. We show that these quantum models become in a sense discrete.For the quantum Klein-Gordon model we get (under an assumption on momenta) an analog of the plane wave, for the quantum hydrogen atom model we find the first order corrections to the ground state energy and wave function.

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arXiv2013

Generalizations of Poisson structures related to rational Gaudin model

The Poisson structure arising in the Hamiltonian approach to the rational Gaudin model looks very similar to the so-called modified Reflection Equation Algebra. Motivated by this analogy, we realize a braiding of the mentioned Poisson structure, i.e. we introduce a "braided Poisson" algebra associated with an involutive solution to the quantum Yang-Baxter equation. Also, we exhibit another generalization of the Gaudin type Poisson structure by replacing the first derivative in the current parameter, entering the so-called local form of this structure, by a higher order derivative. Finally, we introduce a structure, which combines both generalizations. Some commutative families in the corresponding braided Poisson algebra are found.

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arXiv2009

Braided affine geometry and q-analogs of wave operators

The main goal of this review is to compare different approaches to constructing geometry associated with a Hecke type braiding (in particular, with that related to the quantum group U_q(sl(n))). We make an emphasis on affine braided geometry related to the so-called Reflection Equation Algebra (REA). All objects of such type geometry are defined in the spirit of affine algebraic geometry via polynomial relations on generators. We begin with comparing the Poisson counterparts of "quantum varieties" and describe different approaches to their quantization. Also, we exhibit two approaches to introducing q-analogs of vector bundles and defining the Chern-Connes index for them on quantum spheres. In accordance with the Serre-Swan approach, the q-vector bundles are treated as finitely generated projective modules over the corresponding quantum algebras. Besides, we describe the basic properties of the REA used in this construction and compare different ways of defining q-analogs of partial derivatives and differentials on the REA and algebras close to them. In particular, we present a way of introducing a q-differential calculus via Koszul type complexes. The lements of the q-calculus are applied to defining q-analogs of some relativistic wave operators.

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