Quantum integrable systems and differential Galois theory
Abstract
The purpose of this paper is to connect two subjects: the theory of quantum integrable systems (complete commutative rings of differential operators), and differential Galois theory. We define quantum completely integrable systems (QCIS), algebraically integrable QCIS, the differential Galois group of a QCIS. We show that the differential Galois group is always reductive and that a QCIS is algebraically integrable if and only if its differential Galois group is commutative. In particular, we show that a differential operator L in one variable is algebraic in the sense of Krichever (i.e. finite-zone) if and only if the differential Galois group of the differential equation Lf=af is commutative for a generic number a. As a by-product, we obtain a proof of the Veselov-Chalyh conjecture on the algebraic integrability of the elliptic Calogero-Moser system.
Keywords
Cite
@article{arxiv.alg-geom/9607012,
title = {Quantum integrable systems and differential Galois theory},
author = {Alexander Braverman and Pavel Etingof and Dennis Gaitsgory},
journal= {arXiv preprint arXiv:alg-geom/9607012},
year = {2008}
}
Comments
23 pages, amstex. The definition of the Hermite-Bethe variety was corrected, and it was explained that the centralizer commutatitvity theorem is due to Makar-Limanov