The article concerns the dual of Lusztig's canonical basis of a subalgebra of the positive part Uq(n) of the universal enveloping algebra of a Kac-Moody Lie algebra of type A(1) 1 . The examined subalgebra is associated with a terminal module M over the path algebra of the Kronecker quiver via an Weyl group element w of length four.Geiß-Leclerc-Schröer attached to M a category CM of nilpotent modules over the preprojective algebra of the Kronecker quiver together with an acyclic cluster algebra A(CM ). The dual semicanonical basis contains all cluster monomials. By construction, the cluster algebra A(CM ) is a subalgebra of the graded dual of the (non-quantized) universal enveloping algebra U (n).We transfer to the quantized setup. Following Lusztig we attach to w a subalgebra U + q (w) of Uq(n). The subalgebra is generated by four elements that satisfy straightening relations; it degenerates to a commutative algebra in the classical limit q = 1. The algebra U + q (w) possesses four bases, a PBW basis, a canonical basis, and their duals. We prove recursions for dual canonical basis elements. The recursions imply that every cluster variable in A(CM ) is the specialization of the dual of an appropriate canonical basis element. Therefore, U + q (w) is a quantum cluster algebra in the sense of Berenstein-Zelevinsky. Furthermore, we give explicit formulae for the quantized cluster variables and for expansions of products of dual canonical basis elements.

No abstract

No abstract

The article concerns the subalgebra U + v (w) of the quantized universal enveloping algebra of the complex Lie algebra sln+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 An. The quantum cluster algebra is a deformation of the ordinary cluster algebra Geiß-Leclerc-Schröer 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.

Cluster algebras are a class of commutative algebras whose generators are defined by a recursive process called mutation. We give a brief introduction to cluster algebras, and explain how discrete integrable systems can appear in the context of cluster mutation. In particular, we give examples of birational maps that are integrable in the Liouville sense and arise from cluster algebras with periodicity, as well as examples of discrete Painlevé equations that are derived from Y-systems.