Quantum cluster algebras of type A and the dual canonical basis

TL;DR

This paper demonstrates that a specific subalgebra of a quantum group related to sl_{n+1} can be structured as a quantum cluster algebra of type A_n, linking it to canonical bases and representation theory.

Contribution

It establishes the quantum cluster algebra structure on U_v^+(w) and connects quantum cluster variables to the dual canonical basis, advancing understanding of their interplay.

Findings

U_v^+(w) has a quantum cluster algebra structure of type A_n.

Quantum cluster variables relate to the dual of Lusztig's canonical basis.

The structure is a deformation of the classical cluster algebra via representation theory.

Abstract

The article concerns the subalgebra U_v^+(w) of the quantized universal enveloping algebra of the complex Lie algebra sl_{n+1} associated with a particular Weyl group element of length 2n. We verify that U_v^+(w) can be endowed with the structure of a quantum cluster algebra of type A_n. The quantum cluster algebra is a deformation of the ordinary cluster algebra Geiss-Leclerc-Schroeer attached to w using the representation theory of the preprojective algebra. Furthermore, we prove that the quantum cluster variables are, up to a power of v, elements in the dual of Lusztig's canonical basis under Kashiwara's bilinear form.