Dynamical R-Matrices for Calogero Models
Nuclear Physics B 621 [PM] (2002) 523–570 www.elsevier.com/locate/npe Dynamical R-matrices for Calogero models Michael Forger 1, Axel Winterhalder 2 Departamento de Matemática Aplicada, Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, BR-05315-970 São Paulo, SP, Brazil Received 17 July 2001; accepted 18 September 2001 Abstract We present a systematic study of the integrability of the Calogero models, degenerate as well as elliptic, associated with arbitrary (semi-)simple Lie algebras and with symmetric pairs of Lie algebras, where “integrability” is understood to encompass not only the existence of a Lax representation for the equations of motion but also the—more far-reaching—existence of a (dynamical) R-matrix. Using the standard group-theoretical machinery available in this context, we show that integrability of these models, in this sense, can be reduced to the existence of a certain function, denoted here by F , defined on the relevant root system and taking values in the respective Cartan subalgebra, subject to a rather simple set of algebraic constraints: these ensure, in one stroke, the existence of a Lax representation and of a dynamical R-matrix, all given by explicit formulas. We also show that among the simple Lie algebras, only those belonging to the A-series admit a solution of these constraints, whereas the AIII-series of symmetric pairs of Lie algebras, corresponding to the complex Grassmannians SU(p, q)/S(U(p) × U(q)), allows non-trivial solutions when |p − q| 1. Apart from reproducing all presently known dynamical R-matrices for Calogero models, our method provides new ones, namely for the degenerate models when |p − q|=1 and for the elliptic models when |p − q|=1orp = q.
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