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Vector bundles on degenerations of elliptic curves and Yang–Baxter equations
About this Title
Igor Burban, Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany and Bernd Kreussler, Mary Immaculate College, South Circular Road, Limerick, Ireland
Publication: Memoirs of the American Mathematical Society
Publication Year:
2012; Volume 220, Number 1035
ISBNs: 978-0-8218-7292-5 (print); 978-0-8218-9207-7 (online)
DOI: https://doi.org/10.1090/S0065-9266-2012-00654-X
Published electronically: March 9, 2012
Keywords: Yang–Baxter equations,
vector bundles on genus one curves and elliptic fibrations,
derived categories of coherent sheaves,
Massey products
MSC: Primary 14F05, 14H60, 14H70, 16T25
Table of Contents
Chapters
- Introduction
- 1. Yang–Baxter Equations
- 2. Massey Products and AYBE – a Single Curve
- 3. Massey Products and AYBE – Families of Curves
- 4. Explicit Calculations – Smooth Curves
- 5. Explicit Calculations – Singular Curves
- 6. Summary
Abstract
In this paper we introduce the notion of a geometric associative $r$-matrix 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.- Marcelo Aguiar, On the associative analog of Lie bialgebras, J. Algebra 244 (2001), no. 2, 492–532. MR 1859038, DOI 10.1006/jabr.2001.8877
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