arXiv:math/9909014

Universal R-matrix formalism for the spin Calogero-Moser system and its difference counterpart

Dmitry Valer'evich Talalaev

Published Sep 2, 1999• 1 citation

Abstract

The expression of the quantum Ruijsenaars-Schneider Hamiltonian is obtained in the framework of the dynamical $R$-matrix formalism. This generalizes to the case of $U_q(sl_n)$ the result obtained by O. Babelon, D. Bernard and E. Billey for $U_q(sl_2)$ which is the higher difference Lame operator. The general method involved is the universal $R$-matrix construction.

DOI: 10.48550/arxiv.math/9909014

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arXiv:math/9909014

Universal R-matrix formalism for the spin Calogero-Moser system and its difference counterpart

Dmitry Valer'evich Talalaev

Published Sep 2, 1999• 1 citation

arXiv HTML PDF

Abstract

The expression of the quantum Ruijsenaars-Schneider Hamiltonian is obtained in the framework of the dynamical

R

-matrix formalism. This generalizes to the case of

U_q(sl_n)

the result obtained by O. Babelon, D. Bernard and E. Billey for

U_q(sl_2)

which is the higher difference Lame operator. The general method involved is the universal

R

-matrix construction.

DOI: 10.48550/arxiv.math/9909014

Related papers

Elliptic Ruijsenaars-Schneider and Calogero-Moser Models Represented by Sklyanin Algebra andsl(n) Gaudin Algebra

Kai Chen, Heng Fan, Bo-Yu Hou +3 more

arXiv:hep-th/0201211Progress of Theoretical Physics SupplementJan 1, 199911 citations

A Universal Construction of Calogero-Moser System with Two Types Spins Interaction

Jialiang Dai

Abstract We present universal construction for the Calogero-Moser system with two types spins interaction of trigonometric potential based on the root system of semi-simple Lie algebra. In this formalism, we successfully build up the correct Lax pair as well as the R-matrix for this generalized Calogero-Moser models. Moreover using the property of root system, we make a concise explanation that in the quantized model, the R-matrix takes the same form as the classical one, which is the main new result of this paper.Abstract We present universal construction for the Calogero-Moser system with two types spins interaction of trigonometric potential based on the root system of semi-simple Lie algebra. In this formalism, we successfully build up the correct Lax pair as well as the R-matrix for this generalized Calogero-Moser models. Moreover using the property of root system, we make a concise explanation that in the quantized model, the R-matrix takes the same form as the classical one, which is the main new result of this paper.

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V I Inozemtsev, Ryu Sasaki B

For any root system $Δ$ and an irreducible representation ${\cal R}$ of the reflection (Weyl) group $G_Δ$ generated by $Δ$ , a {\em spin Calogero-Moser model} can be defined for each of the potentials: rational, hyperbolic, trigonometric and elliptic. For each member $μ$ of ${\cal R}$ , to be called a "site", we associate a vector space ${\bf V}_μ$ whose element is called a "spin". Its dynamical variables are the canonical coordinates $\{q_j,p_j\}$ of a particle in ${\bf R}^r$ , ( $r=$ rank of $Δ$ ), and spin exchange operators $\{\hat{\cal P}_ρ\}$ ( $ρ\inΔ$ ) which exchange the spins at the sites $μ$ and $s_ρ(μ)$ . Here $s_ρ$ is the reflection generated by $ρ$ . For each $Δ$ and ${\cal R}$ a {\em spin exchange model} can be defined. The Hamiltonian of a spin exchange model is a linear combination of the spin exchange operators only. It is obtained by "freezing" the canonical variables at the equilibrium point of the corresponding classical Calogero-Moser model. For $Δ=A_r$ and ${\cal R}=$ vector representation it reduces to the well-known Haldane-Shastry model. Universal Lax pair operators for both spin Calogero-Moser models and spin exchange models are presented which enable us to construct as many conserved quantities as the number of sites for {\em degenerate} potentials.For any root system $Δ$ and an irreducible representation ${\cal R}$ of the reflection (Weyl) group $G_Δ$ generated by $Δ$, a {\em spin Calogero-Moser model} can be defined for each of the potentials: rational, hyperbolic, trigonometric and elliptic. For each member $μ$ of ${\cal R}$, to be called a "site", we associate a vector space ${\bf V}_μ$ whose element is called a "spin". Its dynamical variables are the canonical coordinates $\{q_j,p_j\}$ of a particle in ${\bf R}^r$, ($r=$ rank of $Δ$), and spin exchange operators $\{\hat{\cal P}_ρ\}$ ($ρ\inΔ$) which exchange the spins at the sites $μ$ and $s_ρ(μ)$. Here $s_ρ$ is the reflection generated by $ρ$. For each $Δ$ and ${\cal R}$ a {\em spin exchange model} can be defined. The Hamiltonian of a spin exchange model is a linear combination of the spin exchange operators only. It is obtained by "freezing" the canonical variables at the equilibrium point of the corresponding classical Calogero-Moser model. For $Δ=A_r$ and ${\cal R}=$ vector representation it reduces to the well-known Haldane-Shastry model. Universal Lax pair operators for both spin Calogero-Moser models and spin exchange models are presented which enable us to construct as many conserved quantities as the number of sites for {\em degenerate} potentials.

arXiv:hep-th/0105164Journal of Physics A Mathematical and GeneralSep 10, 200113 citations

Dynamicalr-matrix for the elliptic Ruijsenaars - Schneider system

Frank Willem Nijhoff, V B Kuznetsov, E K Sklyanin +1 more

arXiv:solv-int/9603006Journal of Physics A Mathematical and GeneralJul 7, 199636 citations

BC_n Ruijsenaars-Schneider models: R-Matrix structure and hamiltonians

Jean Avan, G. Rollet

This paper is replaced and superseded by nlin.SI/0106015 (Title: Structures in BC_N Ruijsenaars-Schneider models)

arXiv:hep-th/0008174Aug 22, 20002 citations

The expression of the quantum Ruijsenaars-Schneider Hamiltonian is obtained in the framework of the dynamical

R

-matrix formalism. This generalizes to the case of

U_q(sl_n)

the result obtained by O. Babelon, D. Bernard and E. Billey for

U_q(sl_2)

which is the higher difference Lame operator. The general method involved is the universal

R

-matrix construction.

For any root system

Δ

and an irreducible representation

{\cal R}

of the reflection (Weyl) group

G_Δ

generated by

Δ

, a {\em spin Calogero-Moser model} can be defined for each of the potentials: rational, hyperbolic, trigonometric and elliptic. For each member

μ

{\cal R}

, to be called a "site", we associate a vector space

{\bf V}_μ

whose element is called a "spin". Its dynamical variables are the canonical coordinates

\{q_j,p_j\}

of a particle in

{\bf R}^r

, (

r=

rank of

Δ

), and spin exchange operators

\{\hat{\cal P}_ρ\}

(

ρ\inΔ

) which exchange the spins at the sites

μ

and

s_ρ(μ)

. Here

s_ρ

is the reflection generated by

ρ

. For each

Δ

and

{\cal R}

a {\em spin exchange model} can be defined. The Hamiltonian of a spin exchange model is a linear combination of the spin exchange operators only. It is obtained by "freezing" the canonical variables at the equilibrium point of the corresponding classical Calogero-Moser model. For

Δ=A_r

and

{\cal R}=

vector representation it reduces to the well-known Haldane-Shastry model. Universal Lax pair operators for both spin Calogero-Moser models and spin exchange models are presented which enable us to construct as many conserved quantities as the number of sites for {\em degenerate} potentials.