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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.
From quantum Ore extensions to quantum tori via noncommutative UFDs, preprint
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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.