Quantum Unipotent Subgroup and dual canonical basis

Abstract

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 $S^{\ast }$. 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.

41 pages

References in corpus (5)

• The quantum dilogarithm and representations quantum cluster varieties
• Kac-Moody groups and cluster algebras
• Quiver varieties and cluster algebras
• Generic bases for cluster algebras and the Chamber Ansatz
• A quantum cluster algebra of Kronecker type and the dual canonical basis

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