A quantum cluster algebra of Kronecker type and the dual canonical basis

Details A quantum cluster algebra of Kronecker type and the dual canonical basis A quantum cluster algebra of Kronecker type and the dual canonical basis

2010-02-14

arxiv.org/abs/1002.2762v2

Int. Math. Res. Notices 2011, no. 13, 2970-3005

Mathematics

Representation Theory

32 pages

Scientific paper

10.1093/imrn/rnq162

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.

Affiliated with

Also associated with

No associations

Rating

A quantum cluster algebra of Kronecker type and the dual canonical basis does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with A quantum cluster algebra of Kronecker type and the dual canonical basis, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and A quantum cluster algebra of Kronecker type and the dual canonical basis will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-555051