A Quantum Cluster Algebra of Kronecker Type and the Dual Canonical Basis

Lampe, P. (2011) A Quantum Cluster Algebra of Kronecker Type and the Dual Canonical Basis. International Mathematics Research Notices, 2011 (13). pp. 2970-3005. ISSN 1073-7928. (doi:https://doi.org/10.1093/imrn/rnq162) (Full text available)

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Abstract

The article concerns the dual of Lusztig's canonical basis of a subalgebra of the positive part U_q(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-Schroeer attached to M a category C_M of nilpotent modules over the preprojective algebra of the Kronecker quiver together with an acyclic cluster algebra A(C_M). The dual semicanonical basis contains all cluster monomials. By construction, the cluster algebra A(C_M) 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 U_q(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(C_M) 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.

Item Type: Article
Subjects: Q Science > QA Mathematics (inc Computing science) > QA150 Algebra
Q Science > QA Mathematics (inc Computing science) > QA171 Representation theory
Divisions: Faculties > Sciences > School of Mathematics Statistics and Actuarial Science
Depositing User: Philipp Lampe
Date Deposited: 16 Jul 2018 15:50 UTC
Last Modified: 19 Jul 2018 14:03 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/67633 (The current URI for this page, for reference purposes)
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