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Dimitry Gurevich

Publications and source records attributed to Dimitry Gurevich.

9 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

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.

math.QA↗
arXiv2024

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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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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