SIGMA 3 (2007), 033, 6 pages      nlin.SI/0701006      doi:10.3842/SIGMA.2007.033
Contribution to the Proceedings of the Coimbra Workshop on Geometric Aspects of Integrable Systems

Continuous and Discrete (Classical) Heisenberg Spin Chain Revised

Orlando Ragnisco and Federico Zullo
Dipartimento di Fisica, Università di Roma Tre and Istituto Nazionale di Fisica Nucleare Sezione di Roma Tre, Via Vasca Navale 84, I-00146 Roma, Italy

Received December 29, 2006; Published online February 26, 2007

Abstract
Most of the work done in the past on the integrability structure of the Classical Heisenberg Spin Chain (CHSC) has been devoted to studying the su(2) case, both at the continuous and at the discrete level. In this paper we address the problem of constructing integrable generalized ''Spin Chains'' models, where the relevant field variable is represented by a N × N matrix whose eigenvalues are the N^th roots of unity. To the best of our knowledge, such an extension has never been systematically pursued. In this paper, at first we obtain the continuous N × N generalization of the CHSC through the reduction technique for Poisson-Nijenhuis manifolds, and exhibit some explicit, and hopefully interesting, examples for 3 × 3 and 4 × 4 matrices; then, we discuss the much more difficult discrete case, where a few partial new results are derived and a conjecture is made for the general case.

Key words: integrable systems; Heisenberg chain; Poisson-Nijenhuis manifolds; geometric reduction; R-matrix; modified Yang-Baxter.

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