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15 April 2021 Integral quantum cluster structures
K. R. Goodearl, M. T. Yakimov
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Duke Math. J. 170(6): 1137-1200 (15 April 2021). DOI: 10.1215/00127094-2020-0061

Abstract

We prove a general theorem for constructing integral quantum cluster algebras over Z[q±1/2], namely, that under mild conditions the integral forms of quantum nilpotent algebras always possess integral quantum cluster algebra structures. These algebras are then shown to be isomorphic to the corresponding upper quantum cluster algebras, again defined over Z[q±1/2]. Previously, this was only known for acyclic quantum cluster algebras. The theorem is applied to prove that, for every symmetrizable Kac–Moody algebra g and Weyl group element w, the dual canonical form Aq(n+(w))Z[q±1] of the corresponding quantum unipotent cell has the property that Aq(n+(w))Z[q±1]Z[q±1]Z[q±1/2] is isomorphic to a quantum cluster algebra over Z[q±1/2] and to the corresponding upper quantum cluster algebra over Z[q±1/2].

Citation

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K. R. Goodearl. M. T. Yakimov. "Integral quantum cluster structures." Duke Math. J. 170 (6) 1137 - 1200, 15 April 2021. https://doi.org/10.1215/00127094-2020-0061

Information

Received: 6 May 2019; Revised: 20 August 2020; Published: 15 April 2021
First available in Project Euclid: 23 March 2021

Digital Object Identifier: 10.1215/00127094-2020-0061

Subjects:
Primary: 13F60
Secondary: 16S38 , 17B37 , 81R50

Keywords: dual canonical bases , integral quantum cluster algebras , integral upper quantum cluster algebras , quantum unipotent cells , symmetrizable Kac–Moody algebras

Rights: Copyright © 2021 Duke University Press

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Vol.170 • No. 6 • 15 April 2021
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