SEPARATION OF VARIABLES AND THE XXZ GAUDIN MAGNET(2)

Abstract. In this work we generalise previous results connecting (rational) Gaudin magnet models and classical separation of variables. It is shown that the connection persists for the case of linear r-matrix algebra which corresponds to the trigonometric

2

E.G. KALNINS, V.B. KUZNETSOV AND WILLARD MILLER, JR.

recall the fundamental ideas of the r-matrix formalism (see 10,11] and references in there). For a classical mechanical system the basic (linear) r-matrix algebra is (1.1)

fL(u) I; I L(v)g= r(u? v); L(u) I+ I L(v)];L(u)11=?L(u)22= A(u); L(u)12= B (u); L(u)21= C (u);

where f; g is the Poisson bracket and;] the matrix commutator bracket. The operator L(u) is taken to be the 2 2 matrix and r(u) is a suitable 4 4 matrix of scalars solving the classical Yang-Baxter equation 10,11]; u being arbitrary constant is called the spectral parameter. In the case of the XXZ r-matrix algebra the non zero elements of r can be taken to be 1 (1.2) r(u)11= r(u)44= coth(u); r(u)23= r(u)32= sinh(u): In component form, the r-matrix algebra relations are (1.3)

1 fA(u); C (v)g= sinh(u? v) (? cosh(u? v)C (v)+ C (u));? fB (u); C (v)g= sinh(u2? v) (A(u)? A(v)): If we now ma

ke the ansatz A(u)= coth(u)S3; B (u)= (1= sinh(u))S+ and C (u)= (1= sinh(u))S? these relations imply (1.4)n X

fA(u); A(v)g= fB (u); B (v)g= fC (u); C (v)g= 0; 1 fA(u); B (v)g= sinh(u? v) (cosh(u? v)B (v)? B (u));

fS3; S g= S;

fS+; S?g= 2S3:n X

To relate this observation to the separation of variables methods, we form the L(u) operator with elements 1 B (u)= sinh(u? e ) S+;=1

C (u)=

(1.5) where (1.6) (1.7)

1 sinh(u? e ) S?;=1

A(u)=

n X=1

coth(u? e ) S3;

fS3; S g=

S;

fS+; S? g= 2 S3:

The r-matrix algebra relations, (1.1) or (1.3), imply

fdet L(u); det L(v)g= 0;