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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.
A quantum analogue of generic bases for affine cluster algebras
 Science China Mathematics.55 (2012
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On a category of cluster algebras
, 2012
"... ... has countable coproducts, which can be constructed combinatorially, but no products. We characterise isomorphisms and monomorphisms in this category and provide combinatorial methods for constructing special classes of monomorphisms and epimorphisms. In the case of cluster algebras from surfaces ..."
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... has countable coproducts, which can be constructed combinatorially, but no products. We characterise isomorphisms and monomorphisms in this category and provide combinatorial methods for constructing special classes of monomorphisms and epimorphisms. In the case of cluster algebras from surfaces, we describe interactions between this category and the geometry of the
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"... Dedicated to Idun Reiten on the occasion of her seventieth birthday. Abstract. FominZelevinsky conjectured that in any cluster algebra, the cluster monomials are linearly independent and that the exchange graph is independent of the choice of coefficients. We confirm these conjectures for all skew ..."
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Dedicated to Idun Reiten on the occasion of her seventieth birthday. Abstract. FominZelevinsky conjectured that in any cluster algebra, the cluster monomials are linearly independent and that the exchange graph is independent of the choice of coefficients. We confirm these conjectures for all skewsymmetric cluster algebras.