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Pavel Saponov

Publications and source records attributed to Pavel Saponov.

29 records · Page 2Linked to original sources
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

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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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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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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arXiv2005

The GL(m|n) type quantum matrix algebras II: the structure of the characteristic subalgebra and its spectral parameterization

In our previous paper math.QA/0412192 the Cayley-Hamilton identity for the GL(m|n) type quantum matrix algebra was obtained. Here we continue investigation of that identity. We derive it in three alternative forms and, most importantly, we obtain it in a factorized form. The factorization leads to a separation of the spectra of the quantum supermatrix into the "even" and "odd" parts. The latter, in turn, allows us to parameterize the characteristic subalgebra (which can also be called the subalgebra of spectral invariants) in terms of the supersymmetric polynomials in the eigenvalues of the quantum supermatrix. For our derivation we use two auxiliary results which may be of independent interest. First, we calculate the multiplication rule for the linear basis of the Schur functions $s_λ(M)$ for the characteristic subalgebra of the Hecke type quantum matrix algebra. The structure constants in this basis are the Littlewood-Richardson coefficients. Second, we derive a series of bilinear relations in the graded ring $Λ$ of Schur symmetric functions in countably many variables.

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