Quantum Unipotent Subgroup and dual canonical basis

Details Quantum Unipotent Subgroup and dual canonical basis Quantum Unipotent Subgroup and dual canonical basis

2010-10-20

arxiv.org/abs/1010.4242v1

Mathematics

Quantum Algebra

41 pages

Scientific paper

10.1215/21562261-1550976

Geiss-Leclerc-Schroer defined the cluster algebra structure on the coordinate ring $C[N(w)]$ of the unipotent subgroup, associated with a Weyl group element $w$ and they proved cluster monomials are contained in Lusztig's dual semicanonical basis $S^{*}$. We give a set up for the quantization of their results and propose a conjecture which relates the quantum cluster algebras to the dual canonical basis ${B}^{up}$. In particular, we prove that the quantum analogue $O_{q}[N(w)]$ of ${C}[N(w)]$ has the induced basis from ${B}^{up}$, which contains quantum flag minors and satisfies a factorization property with respect to the `$q$-center' of $O_{q}[N(w)]$. This generalizes Caldero's results from ADE cases to an arbitary symmetrizable Kac-Moody Lie algebra.

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