Math 607: WINTER 2023, MWF 13:00-13:50, University 210

      CLUSTER ALGEBRAS
              Arkady Berenstein

Cluster algebras were introduced in 2001 by Fomin and Zelevinsky in order to study the total positivity in algebraic groups and canonical bases in their representations. 

A cluster algebra is a commutative algebra given by certain combinatorial data (an exchange matrix and mutations) and is naturally a subalgebra of the field of fractions in finitely many indeterminates. A spectacular "Laurent phenomenon" asserts that each cluster algebra, is, in fact, a subalgebra of the algebra of  Laurent polynomials in those indeterminates. 

Over the last two decades, the theory of cluster algebras became the most dynamic field in Algebra. It has links with several fields of study, including Teichmuller spaces and character varieties, integrable systems, field theories, mirror symmetry, higher category theory, toric varieties, Poisson geometry, quantum groups, and Lie/representation theory. 

In particular, all cluster algebras of finite type have been classified by the Dynkin diagrams and are closely related to the corresponding semisimple Lie algebras. 

REFERENCES.

[1] S. Fomin, L. Williams, A. Zelevinsky. Introduction to Cluster Algebras.
[2] R. Marsh. Lecture Notes on Cluster Algebras.
[3] M. Gekhtman, .M Shapiro, A. Vainshtein. Cluster algebras and Poisson geometry.
[4] I. Assem, S. Trepode. Homological Methods, Representation Theory, and Cluster Algebras.

Prerequisites: 600 Algebra.  Some knowledge of Category Theory and Representation Theory of Lie algebras & groups would also be helpful.