Philipp Lampe
Published 2011-01-03, updated 2013-05-23Version 3
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.
Dual canonical basis for unipotent group and base affine space
Atomic basis of quantum cluster algebra of type $\widetilde{A}_{2n-1,1}$
Quantum cluster algebras associated to weighted projective lines
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