Classical Dynamical R-matrices Research Articles

Poisson Reductions of Master Integrable Systems on Doubles of Compact Lie Groups

We consider three ‘classical doubles’ of any semisimple, connected and simply connected compact Lie group G: the cotangent bundle, the Heisenberg double and the internally fused quasi-Poisson double. On each double we identify a pair of ‘master integrable systems’ and investigate their Poisson reductions. In the simplest cotangent bundle case, the reduction is defined by taking quotient by the cotangent lift of the conjugation action of G on itself, and this naturally generalizes to the other two doubles. In each case, we derive explicit formulas for the reduced Poisson structure and equations of motion and find that they are associated with well known classical dynamical r-matrices. Our principal result is that we provide a unified treatment of a large family of reduced systems, which contains new models as well as examples of spin Sutherland and Ruijsenaars–Schneider models that were studied previously. We argue that on generic symplectic leaves of the Poisson quotients the reduced systems are integrable in the degenerate sense, although further work is required to prove this rigorously.

Liouville Integrability of a Class of Integrable Spin Calogero-Moser Systems and Exponents of Simple Lie Algebras

In previous work, we introduced a class of integrable spin Calogero-Moser systems associated with the classical dynamical r-matrices with spectral parameter, as classified by Etingof and Varchenko for simple Lie algebras. Here the main purpose is to establish the Liouville integrability of these systems by a uniform method.

Quantum groupoids and dynamical categories

In this paper we realize the dynamical categories introduced in our previous paper as categories of modules over bialgebroids; we study the bialgebroids arising in this way. We define quasitriangular structure on bialgebroids and present examples of quasitriangular bialgebroids related to the dynamical categories. We show that dynamical twists over an arbitrary base give rise to bialgebroid twists. We prove that the classical dynamical r-matrices over an arbitrary base manifold are in one-to-one correspondence with a special class of coboundary Lie bialgebroids.

The local structure of Lie bialgebroids

We study the local structure of Lie bialgebroids at regular points. In particular, we classify all transitive Lie bialgebroids. In special cases, they are connected to classical dynamical r-matrices and matched pairs induced by Poisson group actions.

Twisted traces of intertwiners for Kac-Moody algebras and classical dynamical r-matrices corresponding to generalized Belavin-Drinfeld triples

In early eighties, Belavin and Drinfeld showed that nonskewsymmetric classical r-matrices for simple Lie algebras are classified by combinatorial objects which are now called Belavin-Drinfeld triples. Later the second author of the present paper generalized this result to the case of dynamical r-matrices, and showed that they correspond to generalized Belavin-Drinfeld triples. The dynamical r-matrix corresponding a triple is given by a certain explicit formula, which works for any symmetrizable Kac-Moody algebra. In special cases, this formula gives the Felder and the Belavin r-matrix. In 1994, Felder associated with every dynamical r-matrix a system of differential equations called the KZB equations. In the case of the Felder and Belavin r-matrix, solutions of these equations are known to have a representation theoretical presentation, as conformal blocks (i.e. traces of products of intertwining operators between representations of Lie algebras). In this paper, we propose such a presentation for ANY dynamical r-matrix given by the second author's formula. In particular, the paper sheds light on the Belavin-Drinfeld classical r-matrices of 1984. This paper begins a series of papers. In the next paper, joint with T.Schedler, we will give an explicit quantization of the dynamical r-matrices for simple Lie algebras (in particular, Belavin-Drinfeld r-matrices), which has been an open problem for a number of years. In another paper, we plan to generalize the results of the present paper to quantum groups. This should yield quantum KZB equations which involve quantum R-matrices obtained by quantizing the classical r-matrices that appear in this paper.