Twist Automorphisms on Quantum Unipotent Cells and Dual Canonical Bases

Abstract

In this paper, we construct twist automorphisms on quantum unipotent cells, which are quantum analogues of the Berenstein-Fomin-Zelevinsky twist automorphisms on unipotent cells. We show that those quantum twist automorphisms preserve the dual canonical bases of quantum unipotent cells.Moreover we prove that quantum twist automorphisms are described by the syzygy functors for representations of preprojective algebras in the symmetric case. This is the quantum analogue of Geiß-Leclerc-Schröer's description, and Geiß-Leclerc-Schröer's results are essential in our proof. As a consequence, we show that quantum twist automorphisms are compatible with quantum cluster monomials. The 6-periodicity of specific quantum twist automorphisms is also verified.

Geometric crystals and cluster ensembles in Kac–Moody setting

“…A result similar to Proposition 4.6 for unipotent quantum minors of quantum unipotent cells is obtained in [12].…”

On Cacti and Crystals

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In the present work we study actions of various groups generated by involutions on the category O int q (g) of integrable highest weight U q (g)-modules and their crystal bases for any symmetrizable Kac-Moody algebra g. The most notable of them are the cactus group and (yet conjectural) Weyl group action on any highest weight integrable module and its lower and upper crystal bases. Surprisingly, some generators of cactus groups are anti-involutions of the Gelfand-Kirillov model for O int q (g) closely related to the remarkable quantum twists discovered by Kimura and Oya in [28].The following Lemmata are apparently well-known. We provide their proof for the reader's convenience.Proof.

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“…It follows from Lemmata 6.4 and 6.23 that κ lifts to an anti-involution κ on A q (g). By [28,Theorem 5.4], for any c = (c i ) i∈I ∈ (k × ) I the assignments…”

“…We do not expect an analogous result for J I; for example, for g = sl 3 , the σ i , i ∈ {1, 2} are not anti-automorphisms of C q (g). Our proof of Theorems 1.10 and 1.11 rely in a crucial way on properties of a remarkable quantum twist defined in [28].…”

“…The difference between our notation and that of[7,28] is in the linear automorphism of A q (g) defined on homogeneous elements x by x → q1 2 (|x|,|x|)−(|x|,ρ) x…”