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QUANTUM UNIPOTENT SUBGROUP AND DUAL CANONICAL BASIS
, 2010
"... ... cluster algebra structure on the coordinate ring C[N(w)] of the unipotent subgroup, associated with a Weyl group element w. And they proved cluster monomials are contained in Lusztig’s dual semicanonical basis S ∗. We give a set up for the quantization of their results and propose a conjecture w ..."
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... cluster algebra structure on the coordinate ring C[N(w)] of the unipotent subgroup, associated with a Weyl group element w. And they proved cluster monomials are contained in Lusztig’s dual semicanonical basis S ∗. We give a set up for the quantization of their results and propose a conjecture which relates the quantum cluster algebras in [4] to the dual canonical basis B up. In particular, we prove that the quantum analogue Oq[N(w)] of C[N(w)] has the induced basis from B up, which contains quantum flag minors and satisfies a factorization property with respect to the ‘qcenter’ of Oq[N(w)]. This generalizes Caldero’s results [7, 8, 9] from ADE cases to an arbitary symmetrizable KacMoody Lie algebra.
Cluster structures on quantum coordinate rings,
 Selecta Math. (N.S.)
, 2013
"... Abstract We show that the quantum coordinate ring of the unipotent subgroup N(w) of a symmetric KacMoody group G associated with a Weyl group element w has the structure of a quantum cluster algebra. This quantum cluster structure arises naturally from a subcategory C w of the module category of t ..."
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Abstract We show that the quantum coordinate ring of the unipotent subgroup N(w) of a symmetric KacMoody group G associated with a Weyl group element w has the structure of a quantum cluster algebra. This quantum cluster structure arises naturally from a subcategory C w of the module category of the corresponding preprojective algebra. An important ingredient of the proof is a system of quantum determinantal identities which can be viewed as a qanalogue of a T system. In case G is a simple algebraic group of type A, D, E, we deduce from these results that the quantum coordinate ring of an open cell of a partial flag variety attached to G also has a cluster structure.
LINEAR RECURRENCE RELATIONS FOR CLUSTER VARIABLES OF AFFINE QUIVERS
"... Abstract. We prove that the frieze sequences of cluster variables associated with the vertices of an affine quiver satisfy linear recurrence relations. In particular, we obtain a proof of a recent conjecture by AssemReutenauerSmith. 1. ..."
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Abstract. We prove that the frieze sequences of cluster variables associated with the vertices of an affine quiver satisfy linear recurrence relations. In particular, we obtain a proof of a recent conjecture by AssemReutenauerSmith. 1.
Quantum cluster algebras of type A and the dual canonical basis
, 2012
"... The article concerns the subalgebraU+v (w) of the quantized universal enveloping algebra of the complex Lie algebra sln+1 associated with a particular Weyl group element of length 2n. We verify that U+v (w) can be endowed with the structure of a quantum cluster algebra of type An. The quantum cluste ..."
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The article concerns the subalgebraU+v (w) of the quantized universal enveloping algebra of the complex Lie algebra sln+1 associated with a particular Weyl group element of length 2n. We verify that U+v (w) can be endowed with the structure of a quantum cluster algebra of type An. The quantum cluster algebra is a deformation of the ordinary cluster algebra GeißLeclercSchröer attached to w using the representation theory of the preprojective algebra. Furthermore, we prove that the quantum cluster variables are, up to a power of v, elements in the dual of Lusztig’s canonical basis under Kashiwara’s bilinear form.