COLLABORATIVE RESEARCH: CLUSTER STRUCTURES ON POISSON-LIE GROUPS AND COMPLETE INTEGRABILITY

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

This project lies in Algebra and focuses on cluster algebras and Poisson-Lie groups. Since the invention of cluster algebras by Fomin and Zelevinsky in 2001, the mathematical community witnessed an explosion of interest in the subject due to the deep connections that were revealed between cluster algebras and a variety of branches of mathematics and theoretical physics ranging from quiver representations and algebraic geometry to string theory and statistical physics. The PIs will build upon their previous collaborations to continue a systematic study of multiple cluster structures in coordinate rings of Poisson-Lie groups and a number of other varieties of importance in algebraic geometry, representation theory and mathematical physics and study an interaction between corresponding cluster algebras. The proposed research is linked to the development of undergraduate and graduate courses and research projects. Synergistic activities are planned with the goal to promote inter-institutional and inter-departmental cooperation, to attract graduate students from underrepresented groups and with diverse educational backgrounds, and, through community outreach, to expose high school students to mathematical research. The PIs will continue their work on applications of Poisson Geometry to the theory of cluster algebras. The main goals of the project include construction and study of (i) exotic cluster structures on simple Lie groups compatible with Poisson-Lie brackets described by the Belavin-Drinfeld classification; (ii) generalized cluster structures on the Drinfeld double and the Poisson-Lie dual of a simple Poisson-Lie group; (iii) combinatorics and inverse problems for higher genus nets dual to quivers arising in cluster structures above; (iv) discrete integrable systems arising as sequences of cluster transformations and elementary transformations of higher genus networks; (v) continuous limits for directed networks with applications to moduli spaces of flat connections.