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From quantum Ore extensions to quantum tori via noncommutative UFDs, preprint
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Graded quantum cluster algebras and an application to quantum Grassmannians
, 2013
"... We introduce a framework for Zgradings on cluster algebras (and their quantum analogues) that are compatible with mutation. To do this, one chooses the degrees of the (quantum) cluster variables in an initial seed subject to a compatibility with the initial exchange matrix, and then one extends t ..."
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We introduce a framework for Zgradings on cluster algebras (and their quantum analogues) that are compatible with mutation. To do this, one chooses the degrees of the (quantum) cluster variables in an initial seed subject to a compatibility with the initial exchange matrix, and then one extends this to all cluster variables by mutation. The resulting grading has the property that every (quantum) cluster variable is homogeneous. In the quantum setting, we use this grading framework to give a construction that behaves somewhat like twisting, in that it produces a new quantum cluster algebra with the same cluster combinatorics but with different quasicommutation relations between the cluster variables. We apply these results to show that the quantum Grassmannians Kq[Gr(k, n)] admit quantum cluster algebra structures, as quantizations of the cluster algebra structures on the classical Grassmannian coordinate ring found by Scott. This is done by lifting the quantum cluster algebra structure on quantum matrices due to Geiß–Leclerc–Schröer and completes earlier work of the authors on the finitetype cases.
PRIME FACTORS OF QUANTUM SCHUBERT CELL ALGEBRAS AND CLUSTERS FOR QUANTUM RICHARDSON VARIETIES
"... Abstract. The understanding of the topology of the spectra of quantum Schubert cell algebras hinges on the description of their prime factors by ideals invariant under the maximal torus of the ambient KacMoody group. We give an explicit description of these prime quotients by expressing their Cauc ..."
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Abstract. The understanding of the topology of the spectra of quantum Schubert cell algebras hinges on the description of their prime factors by ideals invariant under the maximal torus of the ambient KacMoody group. We give an explicit description of these prime quotients by expressing their Cauchon generators in terms of sequences of normal elements in chains of subalgebras. Based on this, we construct large families of quantum clusters for all of these algebras and the quantum Richardson varieties associated to arbitrary symmetrizable KacMoody algebras and all pairs of Weyl group elements. Along the way we develop a quantum version of the FominZelevinsky twist map for all quantum Richardson varieties. Furthermore, we establish an explicit relationship between the GoodearlLetzter and Cauchon approaches to the descriptions of the spectra of symmetric CGL extensions.