Reduction of the Elliptic SL(N,C) top

TL;DR

This paper establishes a connection between elliptic SL(N,C) tops and Toda systems, introducing a new integrable class via the Inozemtsev limit and symplectic maps, and proves their Liouville integrability.

Contribution

It introduces a novel relation between elliptic SL(N,C) tops and Toda systems using the Inozemtsev limit, extending the analysis to N>2 and establishing integrability of new systems.

Findings

For N=2, the limiting top is equivalent to Toda chains.

For N>2, the procedure generalizes using Lax matrices.

A new class of integrable systems is identified and proven to be Liouville integrable.

Abstract

We propose a relation between the elliptic SL(N,C) top and Toda systems and obtain a new class of integrable systems in a specific limit of the elliptic SL(N,C) top. The relation is based on the Inozemtsev limit (IL) and a symplectic map from the elliptic Calogero-Moser system to the elliptic SL(N,C) top. In the case when N=2 we use an explicit form of a symplectic map from the phase space of the elliptic Calogero-Moser system to the phase space of the elliptic SL(2,C) top and show that the limiting tops are equivalent to the Toda chains. In the case when N>2 we generalize the above procedure using only the limiting behavior of Lax matrices. In a specific limit we also obtain a more general class of systems and prove the integrability in the Liouville sense of a certain subclass of these systems. This class is described by a classical r-matrix obtained from an elliptic r-matrix.