Integral quantum cluster structures
Ken R. Goodearl,Milen Yakimov +1 more
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TL;DR: In this paper, it was shown that the integral forms of quantum nilpotent algebras always possess integral quantum cluster algebra structures, and that the quantum quantum algebra over Z[q±1/2] is isomorphic to the corresponding upper quantum algebra.
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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].
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Citations
Prime Spectra of Abelian 2-Categories and Categorifications of Richardson Varieties
Kent B. Vashaw,Milen Yakimov +1 more
TL;DR: In this article, a general framework for prime, completely prime, semiprime, and primitive ideals of an abelian 2-category has been proposed, based on containment conditions in terms of Serre subcategories and Serre ideals of a tensor triangulated category.
8
An Analog of Leclerc's Conjecture for Bases of Quantum Cluster Algebras
TL;DR: In this article, an analogous conjecture for common triangular bases of quantum cluster algebras is proposed, and a weaker form of the analogous conjecture is shown to be true for quantum unipotent subgroups.
Poisson geometry and Azumaya loci of cluster algebras
Greg Muller,Bach Nguyen,Kurt Trampel,Milen Yakimov +3 more
- 23 Sep 2022
TL;DR: In this article , it was shown that the spectrum of a finitely generated upper cluster algebra with its GSV Poisson structure always has a Zariski open orbit of symplectic leaves and gave an explicit description of it.
2
A cluster structure on the coordinate ring of partial flag varieties
TL;DR: The main goal of as mentioned in this paper is to show that the coordinate ring of a partial flag variety C[G/PK−] contains a cluster algebra if G is any semisimple complex algebraic group.
Twist automorphisms and Poisson structures
Yoshiyuki Kimura,Fan Qin,Qiaoling Wei +2 more
- 25 Jan 2022
TL;DR: In this paper , the authors introduce twist automorphisms for upper cluster algebras and cluster Poisson alges with coefficients, which generalize the twist automomorphisms for quantum unipotent cells and study their existence and their compatibility with Poisson structures.
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On crystal bases of the $Q$-analogue of universal enveloping algebras
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