arXiv subjects

A. Silantyev

Publications and source records attributed to A. Silantyev.

12 recordsLinked to original sources

arXiv2022

Quantum Representation Theory and Manin matrices I: finite-dimensional case

We construct Quantum Representation Theory which describes quantum analogue of representations in frame of "non-commutative linear geometry" developed by Manin. To do it we generalise the internal hom-functor to the case of adjunction with a parameter and construct a general approach to representations of a monoid in a symmetric monoidal category with a parameter subcategory. Quantum Representation Theory is obtained by application of this approach to a monoidal category of some class of graded algebras with Manin product, where the parameter subcategory consists of connected finitely generated quadratic algebras. We formulate this theory in the language of Manin matrices and obtain quantum analogues of direct sum and tensor product of representations. Finally, we give some examples of quantum representations.

math.QA↗

arXiv2022

Manin matrices of type C: multi-parametric deformation

We constructed a multi-parametric deformation of the Brauer algebra representation related with the symplectic Lie algebras. The notion of Manin matrix of type C was generalised to the case of the multi-parametric deformation by using this representation and corresponding quadratic algebras. We derived pairing operators for these quadratic algebras and minors for the considered Manin matrices. The rank of pairing operators and dimensions of components of quadratic algebras were calculated.

math.QA↗

arXiv2021

General representation theory in relatively closed monoidal categories

We apply the notion of relative adjoint functor to generalise closed monoidal categories. We define representations in such categories and give their relation with left actions of monoids. The translation of these representations under lax monoidal functors is investigated. We introduce tensor product of representations of bimonoids as a functorial binary operation and show how symmetric lax monoidal functors act on this product. Finally we apply the general theory to classical and quantum representations.

math.CT↗

arXiv2019

Application of the reflection functor to integrable systems on quiver varieties

The theory of representations of quivers and of their preprojective algebras are reviewed. In particular, moduli spaces of representations of these algebras, quiver varieties and reflection functor are described. The proof that the bijection between moduli spaces induced by the reflection functor is an isomorphism of symplectic affine varieties is presented. Hamiltonian systems with complete flows on the quiver varieties are introduced. An application of the reflection functor to these systems is described. The results on the case of the cyclic quiver are reviewed and a role of the reflection functor in this case is discussed. In particular, "spin" integrable generalisations of the Calogero-Moser systems and their application to the generalised KP hierarchies are considered.

math-ph↗

arXiv2012

Algebraic properties of Manin matrices II: q-analogues and integrable systems

We study a natural q-analogue of a class of matrices with noncommutative entries, which were first considered by Yu. I. Manin in 1988 in relation with quantum group theory, (called Manin Matrices in [5]) . These matrices we shall call q-Manin matrices(qMMs). They are defined, in the 2x2 case, by the relations M_21 M_12 = q M_12 M_21; M_22 M_12 = q M_12 M_22; [M_11;M_22] = 1/q M_21 M_12 - q M_12 M_21: They were already considered in the literature, especially in connection with the q-Mac Mahon master theorem [16], and the q-Sylvester identities [25]. The main aim of the present paper is to give a full list and detailed proofs of algebraic properties of qMMs known up to the moment and, in particular, to show that most of the basic theorems of linear algebras (e.g., Jacobi ratio theorems, Schhur complement, the Cayley-Hamilton theorem and so on and so forth) have a straightforward counterpart for q-Manin matrices. We also show how this classs of matrices ?ts within the theory of quasi-determninants of Gel'fand-Retakh and collaborators (see, e.g., [17]). In the last sections of the paper, we frame our definitions within the tensorial approach to non-commutative matrices of the Leningrad school, and we show how the notion of q-Manin matrix is related to theory of Quantum Integrable Systems.

math.QA↗

arXiv2012

Singular polynomials from orbit spaces

We consider the polynomial representation S(V*) of the rational Cherednik algebra H_c(W) associated to a finite Coxeter group W at constant parameter c. We show that for any degree d of W and nonnegative integer m the space S(V*) contains a single copy of the reflection representation V of W spanned by the homogeneous singular polynomials of degree d-1+hm, where h is the Coxeter number of W; these polynomials generate an H_c(W) submodule with the parameter c=(d-1)/h+m. We express these singular polynomials through the Saito polynomials that are flat coordinates of the Saito metric on the orbit space V/W. We also show that this exhausts all the singular polynomials in the isotypic component of the reflection representation V for any constant parameter c.

math.RT↗

arXiv2011

Generalized Macdonald-Ruijsenaars systems

We consider the polynomial representation of Double Affine Hecke Algebras (DAHAs) and construct its submodules as ideals of functions vanishing on the special collections of affine planes. This generalizes certain results of Kasatani in types A_n, (C_n^\vee,C_n). We obtain commutative algebras of difference operators given by the action of invariant combinations of Cherednik-Dunkl operators in the corresponding quotient modules of the polynomial representation. This gives known and new generalized Macdonald-Ruijsenaars systems. Thus in the cases of DAHAs of types A_n and (C_n^\vee,C_n) we derive Chalykh-Sergeev-Veselov operators and a generalization of the Koornwinder operator respectively, together with complete sets of quantum integrals in the explicit form.

math.QA↗

arXiv2009

Groupes quantiques associes aux courbes rationnelles et elliptiques et leurs applications

The thesis was defended by the author in University of Angers (France). It consists of four parts. The fist part (in French) is introductory and is devoted to relation between quantum groups, integrable systems and statistical models. In the second part (in English) the transition function of the periodic Toda chain is interpreted in terms of the formalism of rational Lax operators. In the third part (in French) one compares two elliptic quantum groups and one conclude that they belong to two different bialgebra categories. The fourth part (in English) contains a construction of the partition function of the SOS model in terms of the projections of an elliptic quantum group.

math-ph↗

arXiv2008

SOS model partition function and the elliptic weight functions

We generalize a recent observation [arXiv:math/0610433] that the partition function of the 6-vertex model with domain-wall boundary conditions can be obtained by computing the projections of the product of the total currents in the quantum affine algebra $U_{q}(\hat{\mathfrak{sl}}_{2})$ in its current realization. A generalization is proved for the the elliptic current algebra [arXiv:q-alg/9703018,arXiv:q-alg/9601022]. The projections of the product of total currents are calculated explicitly and are represented as integral transforms of the product of the total currents. We prove that the kernel of this transform is proportional to the partition function of the SOS model with domain-wall boundary conditions.

math.QA↗

arXiv2007

On the universal weight function for the quantum affine algebra U_q(\hat{\mathfrak{gl}}_N)

We continue investigation of the universal weight function for the quantum affine algebra $U_q(\hat{\mathfrak{gl}}_N)$ started in arXiv:math/0610517 and arXiv:0711.2819. We obtain two recurrence relations for the universal weight function applying the method of projections developed in arXiv:math/0610398. On the level of the evaluation representation of $U_q(\hat{\mathfrak{gl}}_N)$ we reproduce both recurrence relations for the off-shell Bethe vectors calculated in arXiv:math/0702277 using combinatorial methods.

math.QA↗

arXiv2007

Classical elliptic current algebras

In this paper we discuss classical elliptic current algebras and show that there are two different choices of commutative test function algebras on a complex torus leading to two different elliptic current algebras. Quantization of these classical current algebras give rise to two classes of quantized dynamical quasi-Hopf current algebras studied by Enriquez-Felder-Rubtsov and Arnaudon-Buffenoir-Ragoucy-Roche-Jimbo-Konno-Odake-Shiraishi. Different degenerations of the classical elliptic algebras are considered. They yield different versions of rational and trigonometric current algebras. We also review the averaging method of Faddeev-Reshetikhin, which allows to restore elliptic algebras from the trigonometric ones.

math.QA↗

arXiv2006

Transition function for the Toda chain model

The method of Lambda-operators developed by S. Derkachov, G. Korchemsky, A. Manashov is applied to a derivation of eigenfunctions for the open Toda chain. The Sklyanin measure is reproduced using diagram technique developed for these Lambda-operators. The properties of the Lambda-operators are studied. This approach to the open Toda chain eigenfunctions reproduces Gauss-Givental representation for these eigenfunctions.

nlin.SI↗