Results

**1 - 6**of**6**### A quantum analogue of generic bases for affine cluster algebras

- Science China Mathematics.55 (2012

"... ar ..."

(Show Context)
### The multiplication theorem and bases in finite and affine quantum cluster algebras

"... ar ..."

(Show Context)
### CLUSTER ALGEBRAS, REPRESENTATION THEORY, AND POISSON GEOMETRY

"... 1.1. Cluster algebras. Cluster algebras were introduced in 2000 by S. Fomin and A. Zelevinsky [26] as a tool for studying dual canonical bases and total positivity in semisimple Lie groups. They are constructively defined commutative algebras with a distinguished set of generators (cluster variables ..."

Abstract
- Add to MetaCart

(Show Context)
1.1. Cluster algebras. Cluster algebras were introduced in 2000 by S. Fomin and A. Zelevinsky [26] as a tool for studying dual canonical bases and total positivity in semisimple Lie groups. They are constructively defined commutative algebras with a distinguished set of generators (cluster variables) grouped into overlapping subsets (clusters) of fixed cardinality. Both the generators and the relations among them are not given from the outset, but are produced by an elementary iterative process called seed mutation. This procedure appears to be closely related to con-structions in various other fields, such as Poisson geometry, Teichmüller theory, representation theory of finite dimensional associative algebras and Lie theory and Coxeter groups. The theory of cluster algebras was further developed in the sub-sequent papers [27, 28, 3, 4, 29, 16, 17]. Remarkably, in the last two papers of this series superpotentials borrowed from mathematical physics play a prominent role. By now cluster algebras form a very active area of research which has obtained its own AMS classification number 13F60. A thematic semester in 2012 at the Mathe-matical Sciences Research Institute in Berkeley will be devoted to cluster algebras (more details on activities related to cluster algebras can be found on Fomin’s cluster algebra portal at