arXiv subjects

M. Matushko

Publications and source records attributed to M. Matushko.

5 recordsLinked to original sources

arXiv2025

R-matrix valued Lax pair for elliptic Calogero-Inozemtsev system and associative Yang-Baxter equations of ${\rm BC}_n$ type

We consider the elliptic Calogero-Inozemtsev system of ${\rm BC}_n$ type with five arbitrary constants and propose $R$-matrix valued generalization for $2n\times 2n$ Takasaki's Lax pair. For this purpose we extend the Kirillov's ${\rm B}$-type associative Yang-Baxter equations to the similar relations depending on the spectral parameters and the Planck constants. General construction uses the elliptic Shibukawa-Ueno $R$-operator and the Komori-Hikami $K$-operators satisfying reflection equation. Then, using the Felder-Pasquier construction the answer for the Lax pair is also written in terms of the Baxter's 8-vertex $R$-matrix. As a by-product of the constructed Lax pair we also propose ${\rm BC}_n$ type generalization for the elliptic XYZ long-range spin chain, and we present arguments pointing to its integrability.

math-ph↗

arXiv2023

Supersymmetric generalization of q-deformed long-range spin chains of Haldane-Shastry type and trigonometric GL(N|M) solution of associative Yang-Baxter equation

We propose commuting sets of matrix-valued difference operators in terms of trigonometric ${\rm GL}(N|M)$-valued $R$-matrices thus providing quantum supersymmetric (and possibly anisotropic) spin Ruijsenaars-Macdonald operators. Two types of trigonometric supersymmetric $R$-matrices are used for this purpose. The first is the one related to the affine quantized algebra ${\hat{\mathcal U}}_q({\rm gl}(N|M))$. The second is a graded version of the standard $\mathbb Z_n$-invariant $A_{n-1}$ type $R$-matrix. We show that being properly normalized the latter graded $R$-matrix satisfies the associative Yang-Baxter equation. Next, we discuss construction of long-range spin chains using the Polychronakos freezing trick. As a result we obtain a new family of spin chains, which extends the ${\rm gl}(N|M)$-invariant Haldane-Shastry spin chain to q-deformed case with possible presence of anisotropy.

math-ph↗

arXiv2022

On R-matrix identities related to elliptic anisotropic spin Ruijsenaars-Macdonald operators

We propose and prove a set of identities for ${\rm GL}_M$ elliptic $R$-matrix (in the fundamental representation). In the scalar case ($M=1$) these are elliptic function identities derived by S.N.M. Ruijsenaars as necessary and sufficient conditions for his kernel identity underlying construction of integral solutions to quantum spinless Ruijsenaars-Schneider model. In this respect the result of the present paper can be considered as the first step towards constructing solutions of quantum eigenvalue problem for the anisotropic spin Ruijsenaars model.

math-ph↗

arXiv2022

Elliptic generalization of integrable q-deformed anisotropic Haldane-Shastry long-range spin chain

We describe integrable elliptic q-deformed anisotropic long-range spin chain. The derivation is based on our recent construction for commuting anisotropic elliptic spin Ruijsenaars-Macdonald operators. We prove that the Polychronakos freezing trick can be applied to these operators, thus providing the commuting set of Hamiltonians for long-range spin chain constructed by means of the elliptic Baxter-Belavin ${\rm GL}_M$ $R$-matrix. Namely, we show that the freezing trick is reduced to a set of elliptic function identities, which are then proved. These identities can be treated as conditions for equilibrium position in the underlying classical spinless Ruijsenaars-Schneider model. Trigonometric degenerations are studied as well. For example, in $M=2$ case our construction provides q-deformation for anisotropic XXZ Haldane-Shastry model. The standard Haldane-Shastry model and its Uglov's q-deformation based on ${\rm U}_q({\widehat {\rm gl}_M})$ XXZ $R$-matrix are included into consideration by separate verification.

math-ph↗

arXiv2022

Anisotropic spin generalization of elliptic Macdonald-Ruijsenaars operators and R-matrix identities

We propose commuting set of matrix-valued difference operators in terms of the elliptic Baxter-Belavin $R$-matrix in the fundamental representation of ${\rm GL}_M$. In the scalar case $M=1$ these operators are the elliptic Macdonald-Ruijsenaars operators, while in the general case they can be viewed as anisotropic versions of the quantum spin Ruijsenaars Hamiltonians. We show that commutativity of the operators for any $M$ is equivalent to a set of $R$-matrix identities. The proof of identities is based on the properties of elliptic $R$-matrix including the quantum and the associative Yang-Baxter equations. As an application of our results, we introduce elliptic generalization of q-deformed Haldane-Shastry model.

math.QA↗