$t$-Analog of $q$-Characters, Bases of Quantum Cluster Algebras, and a Correction Technique

Abstract

We first study a new family of graded quiver varieties together with a new $t$-deformation of the associated Grothendieck rings. This provides the geometric foundations for a joint paper by Yoshiyuki Kimura and the author. We further generalize the result of that paper to any acyclic quantum cluster algebra with arbitrary nondegenerate coefficients. In particular, we obtain the generic basis, the dual PBW basis, and the dual canonical basis. The method consists in a correction technique, which works for general quantum cluster algebras.

Journal version; 45 pages. Minor corrections; this article supports arXiv:1205.2066

References in corpus (1)

• On quantum cluster algebras of finite type

Cited by in corpus (2)

• Triangular bases in quantum cluster algebras and monoidal categorification conjectures
• Quantum groups via cyclic quiver varieties I