Algebro-geometric methods in the theory of Yang-Baxter equations
Final Report Abstract
This project was focused on developing algebro-geometric tools in the theory of Yang- Baxter equations (YBEs). The outcome can be divided into two parts. In the first part, significant contributions to the structure theory of non-associative bialgebra structures were achieved, while the second part dealt with the development of novel quantum group theoretic methods in the geometric Langlands correspondence. The first part was based on the algebro-geometric theory of the classical YBE (CYBE). In a joint work with S. Maximov, A. Stolin, and E. Zelmanov, we utilized this theory in order to classify all topological Lie bialgebra structures on formal power series with coefficients in a simple Lie algebra. Building on this work, I derived a similar classification result for other non-associative analogs of such bialgebra structures. Furthermore, the results of the aforementioned joint work motivated the study of two generalizations of Lie bialgebras in the setting of formal power series, namely Lie quasi-bialgebras and Lie dialgebras. Together with S. Maximov and A. Stolin we also achieved classification results for these generalizations in two follow-up papers. The development of quantum group theoretic methods in the geometric Langlands correspondence arose from the attempt of extending the algebro-geometric methods in the theory of YBEs to higher-genus. I demonstrated that a solution to a dynamical CYBE, originally constructed by Felder in the context of conformal field theory, is the r-matrix governing the integrability of a punctured Hitchin system. I further explained how this r-matrix can be naturally interpreted in terms of a higher-genus version of the algebrogeometric approach to the CYBE and related it to the Beilinson-Drinfeld quantization of the Hitchin system, which is important to the geometric Langlands correspondence. Beyond that, in a collaboration with W. Niu that was inspired by a 4d-gauge field theory, we linked the geometry of equivariant affine Grassmannians to the Yangian of cotangent Lie algebras and constructed solutions to the quantum YBE from this. Later these two directions of investigation converged in another joint work with W. Niu. It is well-known that the equivariant Grassmannian acts on the moduli space of principal bundles via Hecke modifications, a central concept in the geometric Langlands correspondence. We were able to realize the Hecke modification locally in terms of the aforementioned Yangian, by constructing a dynamical twist of the Yangian into a quantum groupoid over a neighborhood of the moduli space of G-bundles. Additionally, we derived a corresponding solution to the quantum dynamical YBE that essentially quantizes the previously mentioned r-matrix of the Hitchin system.
Publications
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Classification of D-bialgebra structures on power series algebras. Journal of Algebra, 635, 137-202.
Abedin, Raschid
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Topological Manin pairs and (n,s)-type series. Letters in Mathematical Physics, 113(3).
Abedin, Raschid; Maximov, Stepan & Stolin, Alexander
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Quantum groupoids from the moduli space of G-bundles.
VII. R. Abedin & W. Niu
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Topological Lie Bialgebras, Manin Triples and Their Classification Over g[[x]]. Communications in Mathematical Physics, 405(1).
Abedin, Raschid; Maximov, Stepan; Stolin, Alexander & Zelmanov, Efim
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Yangians for cotangent Lie algebras and the affine Grassmannian. Journal of Physics: Conference Series, 2912(1), 012024.
Abedin, Raschid & Niu, Wenjun
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Generalized classical Yang-Baxter equation and regular decompositions. Letters in Mathematical Physics, 115(3).
Abedin, R.; Maximov, S. & Stolin, A.
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The r-Matrix Structure of Hitchin Systems via Loop Group Uniformization. Annales Henri Poincaré.
Abedin, Raschid
