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

Abstract

In this paper we introduce the notion of a geometric associative rmatrix attached to a genus one fibration with a section and irreducible fibres. It allows us to study degenerations of solutions of the classical Yang-Baxter equation using the approach of Polishchuk. We also calculate certain solutions of the classical, quantum and associative Yang-Baxter equations obtained from moduli spaces of (semi-)stable vector bundles on Weierstraß cubic curves.Since the complex manifold M (n,d) E is a homogeneous space over the algebraic group J = Pic 0 (E), it turns out that r(v 1 , v 2 ; y 1 , y 2 ) ∼ r(v 1 − v 2 ; y 1 , y 2 ) = r(v; y 1 , y 2 ),

Tilting on non-commutative rational projective curves

“…We hope that the results of this article will find applications to the homological mirror symmetry for degenerations of elliptic curves [25] and to the theory of integrable systems, in particular to the study of solutions of Yang-Baxter equations [14,32].…”

Parameter-dependent associative Yang–Baxter equations and Poisson brackets

“…An interesting relations of CYBE and AYBE with vector bundle geometry on singular and degenerated cubic curves were described in papers of Burban with collaborators [11]. They extend the ideas and results of Polishschuk to much wider class of plane cubic curves and their degenerations.…”

“…Definition (Cf. the matrix algebra case in [9] and [11]). A classical parameter-dependent r-matrix on an associative algebra A is any solution of the following functional equations…”

“…This definition is the straightforward generalization of the four parameterdependent unitary AYBE which was defined in [9] and extensively studied in [11]. Let us remind the context of the Polischuk's definition.…”

Simple vector bundles on plane degenerations of an elliptic curve

“…By 1.2(i), Λ is a moduli space of simple vector bundles of given rank r and multidegree provided gcd(r, d) = 1. By an observation of Burban and Kreußler [BK4], for a given tuple of integers (r, ) ∈ N × Z N such that gcd(r, d) = 1, our method yields an explicit construction of a vector bundle P = P(r, ) ∈ VB E×Λ satisfying in the general case only the following universality properties:…”