Vector bundles on degenerations of elliptic curves and Yang-Baxter equations

Abstract

In this thesis, we study connections between vector bundles on degenerations of elliptic curves and the classical, quantum and associative Yang-Baxter equation. Let g denote n by n traceless matrices and let U denote the universal enveloping algebra of g. The classical Yang-Baxter equation (CYBE) over g plays an important role in mathematical physics, representation theory and integrable systems.<br /> In 1982, Belavin and Drinfeld gave a classification of solutions of the CYBE. In particular, they proved that any solution of the CYBE is either elliptic, trigonometric or rational. Moreover, they described all elliptic and trigonometric solutions. Their work has been extended by Stolin, who gave a certain classification of rational solutions.<br /> Result A. Let E be a Weierstrass cubic curve, 0<d<n a pair of coprime integers and A=Ad(P), where P is a simple vector bundle of rank n and degree d on E. Let H denote the zero-th cohomology of A twisted by x1. Then we have an isomorphism res(x1) from H to the fibre of A over x1 and ev(x2) from H to the fibre of A over x2. Also, there are certain trivialization of A yielding ismorphisms of each fibre of A with g. Hence the concatenation of the inverse of res(x1) with ev(x2) yields a map s from g to g. Then the tensor r in g tensor g, obtained from s above using the Killing form, is a solution of the CYBE.<br /> This result extends an earlier construction given in works of Polishchuk and Burban-Kreußler. The core of our method is the computation of certain triple Massey products in the bounded derived category of coherent sheaves on E.<br /> Result B. Let E be a cuspidal cubic curve. Then the solution r from above is rational. We explicitly describe the Stolin triple (L,B,k) (where L is a Lie subalgebra of g, B is a 2-cocycle of L and k is a natural number) such that r corresponds to (L,B,k).<br /> Result C. We have found new elliptic solutions of the associative Yang-Baxter equation with higher order poles. This leads to new identities for the higher derivatives of the Kronecker function.<br /> Result D. We elaborate a relation between solutions of the associative, classical and quantum Yang-Baxter equations, generalizing results of Polishchuk.