Results 1  10
of
15
Quantum cluster variables via Serre polynomials
, 2010
"... Abstract. For skewsymmetric acyclic quantum cluster algebras, we express the quantum Fpolynomials and the quantum cluster monomials in terms of Serre polynomials of quiver Grassmannians of rigid modules. As byproducts, we obtain the existence of counting polynomials for these varieties and the po ..."
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Abstract. For skewsymmetric acyclic quantum cluster algebras, we express the quantum Fpolynomials and the quantum cluster monomials in terms of Serre polynomials of quiver Grassmannians of rigid modules. As byproducts, we obtain the existence of counting polynomials for these varieties and the positivity conjecture with respect to acyclic seeds. These results complete previous work by Caldero and Reineke and confirm a recent conjecture by Rupel.
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.
A quantum analogue of generic bases for affine cluster algebras
 Science China Mathematics.55 (2012
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The multiplication theorem and bases in finite and affine quantum cluster algebras
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THE INTEGRAL CLUSTER CATEGORY
"... Integral cluster categories of acyclic quivers have recently been used in the representationtheoretic approach to quantum cluster algebras. We show that over a principal ideal domain, such categories behave much better than one would expect: They can be described as orbit categories, their indeco ..."
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Integral cluster categories of acyclic quivers have recently been used in the representationtheoretic approach to quantum cluster algebras. We show that over a principal ideal domain, such categories behave much better than one would expect: They can be described as orbit categories, their indecomposable rigid objects do not depend on the ground ring and the mutation operation is transitive.