arXiv subjects

Pavel Saponov

Publications and source records attributed to Pavel Saponov.

At least 19 recordsLinked to original sources

arXiv2026

Characteristic classes on a noncommutative background: a new approach

We introduce analogues of partial derivatives with respect to the generators of the enveloping algebras U(gl(N)) and their q-deformations, the so-called modified reflection equation algebras. Using these quantum partial derivatives, we introduce the corresponding differential algebras, define the quantum de Rham operator, and exhibit the Leibniz rule for its action. Our Leibniz rule differs from its classical analogue and stems from the construction of a quantum double. In the case of the algebra U(gl(N)), we present another form of the Leibniz rule that is more convenient for prolonging the differential calculus to certain extensions of this algebra. We then use analogues of the Cayley-Hamilton identity for the generating matrices of U(gl(N)) and of reflection equation algebras to construct projective modules over these algebras. For these projective modules, we introduce Grassmannian connections and define the corresponding Chern classes.

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arXiv2026

Representations of Spectrum of GL(m) type Quantum Matrices

In the present paper we are dealing with reflection equation algebras ${\cal L}(R)$ corresponding to even skew-invertible Hecke symmetries. Our main result consists in computing the characters of the spectral values of the generating matrix $L$ of ${\cal L}(R)$ in finite-dimensional representations labeled by partitions of integers. As is known, the spectral values belong to an algebraic extension of the center of the reflection equation algebra and elements of the center can be presented as symmetric functions in spectral values. As an application of our approach, we calculate the characters of the power sums $\mathrm{Tr}_R(L^n)$ in the mentioned finite dimensional representations. In a particular case of the Drinfeld-Jimbo $R$-matrix the enveloping algebra $U(gl(N))$ can be obtained as a specific limit of the reflection equation algebra. In this limit our results for power sums coincide with the those obtained in [PP].

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arXiv2025

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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arXiv2025

Braided Gelfand-Zetlin algebras and their semiclassical counterparts

We construct analogs of the Gelfand-Zetlin algebras in the Reflection Equation algebras, corresponding to Hecke symmetries, mainly to those coming from the quantum groups U_q(sl(N)). Corresponding semiclassical (i.e. Poisson) counterparts of the Gelfand-Zetlin algebras are described.

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arXiv2025

Generalized Harish-Chandra morphism on Reflection Equation algebras

We consider the so-called generalized Harish-Chandra morphism, taking the center of the enveloping algebra U(gl(N)) to the commutative algebra generated by eigenvalues of the generating matrix of this algebra, and generalize this construction to Reflection Equation algebras. To this end we introduce the eigenvalues of the generating matrix of the Reflection Equation algebra (modified or not), corresponding to a skew-invertible Hecke symmetry and define the generalized Harish-Chandra morphism in a similar way. We use this map in order to introduce quantum analogs of the so-called weight systems.

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arXiv2024

Yang-Mills models on Noncommutative background

In our previous publications we have developed some elements of Noncommutative calculus on the enveloping algebras of $A_m$ type, in particular, analogs of the partial derivatives and de Rham complex were defined. Also, we introduced the notion of quantization with Noncommutative configuration space and quantized a few dynamical models in this sense. In the current paper we propose a method of quantizing the Yang-Mills models in same sense.

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arXiv2023

Quantum Schur-Weyl duality and q-Frobenius formula related to Reflection Equation algebras

We establish a q-version of the Schur-Weyl duality, in which the role of the symmetric group is played by the Hecke algebra and the role of the enveloping algebra U(gl(N)) is played by the Reflection Equation algebra, associated with any skew-invertible Hecke symmetry. Also, in each Reflection Equation algebra we define analogues of the Schur polynomials and power sums in two forms: as polynomials in generators of a given Reflection Equation algebra and in terms of the so-called eigenvalues of the generating matrix $L$, defined by means of the Cayley-Hamilton identity. It is shown that on any Reflection Equation algebra there exists a formula, which brings into correlation the Schur polynomials and power sums by means of the characters of the Hecke algebras in the spirit of the classical Frobenius formula.

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arXiv2023

On Symmetrizers in Quantum Matrix Algebras

In this note we are dealing with a particular class of quadratic algebras -- the so-called quantum matrix algebras. The well-known examples are the algebras of quantized functions on classical Lie groups (the RTT algebras). We consider the problem of constructing some projectors on homogenous components of such algebras, which are analogs of the usual symmetrizers. The main objective of this note is to present a method, which hopefully enables one to construct symmetrizers on all homogenous components of the RTT algebras. We illustrate the construction by two low-dimensional examples. A way of extending this method onto other quantum matrix algebras is also discussed.

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arXiv2022

Quantum doubles of Fock type and bosonization

We introduce analogs of creation and annihilation operators, related to involutive and Hecke symmetries R, and perform bosonic and fermionic realization of the modified Reflection Equation algebras in terms of the so-called Quantum Doubles of Fock type. Also, we introduce Quantum Doubles of Fock type, associated with Birman-Murakami-Wenzl symmetries coming from orthogonal or simplectic Quantum Groups and exhibit the algebras obtained by means of the corresponding bosonization (fermionization). Besides, we apply this scheme to current braidings arising from Hecke symmetries R via the Baxterization procedure.

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arXiv2022

Quantum vector fields via quantum doubles and their applications

By treating generators of the reflection equation algebra corresponding to a Hecke symmetry as quantum analogs of vector fields, we exhibit the corresponding Leibniz rule via the so-called quantum doubles. The role of the function algebra in such a double is attributed to another copy of the reflection equation algebra. We consider two types of quantum doubles: these giving rise to the quantum analogs of left vector fields acting on the function algebra and those giving rise to quantum analogs of the adjoint vector fields acting on the same algebra. Also, we introduce quantum partial derivatives in the generators of the reflection equation algebra and then at the limit $q\rightarrow 1$ we get quantum partial derivatives on the enveloping algebra $U(gl_N)$ as well as on a certain its extension.

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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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arXiv2022

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

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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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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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

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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