Cluster algebras in Lie and Knot theory

This is an expanded version of the notes of our lectures given at the conference "Current Developments in Mathematics 2003" held at Harvard University on November 21--22, 2003. We present an overview of the main definitions, results and applications of the theory of cluster algebras.

We apply the new theory of cluster algebras of Fomin and Zelevinsky to study some combinatorial problems arising in Lie theory. This is joint work with Geiss and Schr\"oer (3, 4, 5, 6), and with Hernandez (8, 9).

We survey some recent constructions of cluster algebra structures on coordinate rings of unipotent subgroups and unipotent cells of Kac-Moody groups. We also review a quantized version of these results.

We establish a connection between knot theory and cluster algebras via representation theory. To every knot diagram (or link diagram), we associate a cluster algebra by constructing a quiver with potential. The rank of the cluster algebra is $2n$, where $n$ is the number of crossing points in the knot diagram. We then construct $2n$ indecomposable modules $T(i)$ over the Jacobian algebra of the quiver with potential. For each $T(i)$, we show that the submodule lattice is isom...

To every knot (or link) diagram K, we associate a cluster algebra A that contains a cluster x with the property that every cluster variable in x specializes to the Alexander polynomial of K. We call x the knot cluster of A. Furthermore, there exists a cluster automorphism of A of order two that maps the initial cluster to the cluster x. We realize this connection between knot theory and cluster algebras in two ways. In our previous work, we constructed indecomposable repres...

This is an expanded version of the notes for the two lectures at the 2004 International Mathematics Conference (Chonbuk National University, August 4-6, 2004). The first lecture discusses the origins of cluster algebras, with the focus on total positivity and geometry of double Bruhat cells in semisimple groups. The second lecture introduces cluster algebras and discusses some basic results, open questions and conjectures.

We give a brief introduction to (upper) cluster algebras and their quantization using examples. Then we present several important families of bases for these algebras using topological models. We also discuss tropical properties of these bases and their relation to representation theory. This article is an extended version of the talk given at the 19th International Conference on Representations of Algebras (ICRA 2020).

Cluster algebras are commutative rings with a set of distinguished generators having a remarkable combinatorial structure. They were introduced by Fomin and Zelevinsky in 2000 in the context of Lie theory, but have since appeared in many other contexts, from Poisson geometry to triangulations of surfaces and Teichm\"uller theory. In this expository paper we give an introduction to cluster algebras, and illustrate how this framework naturally arises in Teichm\"uller theory. We...

This is a first step guide to the theory of cluster algebras. We especially focus on basic notions, techniques, and results concerning seeds, cluster patterns, and cluster algebras.

This is an extended abstract of my talk at the Oberwolfach-Workshop "Representation Theory of Finite-Dimensional Algebras" (February 6 - 12, 2005). It gives self-contained and simplified definitions of quantum cluster algebras introduced and studied in a joint work with A.Berenstein (math.QA/0404446).