Remarks on quantum unipotent subgroups and the dual canonical basis

Kimura, Yoshiyuki

doi:10.2140/pjm.2017.286.125

Cited by 13 publications

(3 citation statements)

References 12 publications

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“…They were defined by De Concini-Kac-Procesi [11] and Lusztig [30], who considered the antiisomorphic algebras U ± q [w] = * A q (n ± (w)) . It was proved in [2,28,36] that…”

Section: Quantum Groupsmentioning

confidence: 99%

Root of unity quantum cluster algebras and discriminants

Nguyen,

Trampel,

Yakimov

2020

Preprint

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We describe a connection between the subjects of cluster algebras, polynomial identity algebras and discriminants. For this, we define the notion of root of unity quantum cluster algebras and prove that they are polynomial identity algebras. Inside each such algebra we construct a (large) canonical central subalgebra, which can be viewed as a far reaching generalization of the central subalgebras of big quantum groups constructed by De Concini, Kac and Procesi and used in representation theory. Each such central subalgebra is proved to be isomorphic to the underlying classical cluster algebra of geometric type. When the root of unity quantum cluster algebra is free over its central subalgebra, we prove that the discriminant of the pair is a product of powers of the frozen variables times an integer. An extension of this result is also proved for the discriminants of all subalgebras generated by the cluster variables of nerves in the exchange graph. These results can be used for the effective computation of discriminants. As an application we prove an explicit formula for the discriminant of the integral form over Z[ε] of each quantum unipotent cells of De Concini, Kac and Procesi for arbitrary symmetrizable Kac-Moody algebras.

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“…They were defined by De Concini-Kac-Procesi [11] and Lusztig [30], who considered the antiisomorphic algebras U ± q [w] = * A q (n ± (w)) . It was proved in [2,28,36] that…”

Section: Quantum Groupsmentioning

confidence: 99%

Root of unity quantum cluster algebras and discriminants

Nguyen,

Trampel,

Yakimov

2020

Preprint

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show abstract

“…. , i ℓ ) ∈ I(w) (see also [16,Proposition 3.4]). Moreover, for a homogeneous element x ∈ U + q , we have S(x) = (−1) ht wt x q 1 2 (wt x,wt x)−(ρ,wt x) * (x)t wt x .…”

Section: Therefore We Havementioning

confidence: 99%

Quantum Twist Maps and Dual Canonical Bases

Kimura

Oya

2017

Algebr Represent Theor

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Abstract. In this paper, we show that quantum twist maps, introduced by Lenagan-Yakimov, induce bijections between dual canonical bases of quantum nilpotent subalgebras. As a corollary, we show the unitriangular property between dual canonical bases and Poincaré-Birkhoff-Witt type bases under the "reverse" lexicographic order. We also show that quantum twist maps induce bijections between certain unipotent quantum minors.

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“…In the present manuscript we have included a proof of Proposition 2.10 which works for the Kac-Moody case. We heard that Kimura also proved it by a different method (see Kimura [6]).…”

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confidence: 99%

Modules over quantized coordinate algebras and PBW-bases

Tanisaki¹

2017

J. Math. Soc. Japan

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Around 1990 Soibelman constructed certain irreducible modules over the quantized coordinate algebra. A. Kuniba, M. Okado, Y. Yamada [8] recently found that the relation among natural bases of Soibelman's irreducible module can be described using the relation among the PBW-type bases of the positive part of the quantized enveloping algebra, and proved this fact using case-by-case analysis in rank two cases. In this paper we will give a realization of Soibelman's module as an induced module, and give a unified proof of the above result of [8]. We also verify Conjecture 1 of [8] about certain operators on Soibelman's module.

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Remarks on quantum unipotent subgroups and the dual canonical basis

Cited by 13 publications

References 12 publications

Root of unity quantum cluster algebras and discriminants

Root of unity quantum cluster algebras and discriminants

Quantum Twist Maps and Dual Canonical Bases

Modules over quantized coordinate algebras and PBW-bases

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