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111
From triangulated categories to cluster algebras
"... Abstract. In the acyclic case, we establish a onetoone correspondence between the tilting objects of the cluster category and the clusters of the associated cluster algebra. This correspondence enables us to solve conjectures on cluster algebras. We prove a multiplicativity theorem, a denominator ..."
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Abstract. In the acyclic case, we establish a onetoone correspondence between the tilting objects of the cluster category and the clusters of the associated cluster algebra. This correspondence enables us to solve conjectures on cluster algebras. We prove a multiplicativity theorem, a denominator theorem, and some conjectures on properties of the mutation graph. As in the previous article, the proofs rely on the CalabiYau property of the cluster category. 1.
The HarderNarasimhan system in quantum groups and cohomology of Quiver Moduli
, 2002
"... Methods of Harder and Narasimhan from the theory of moduli of vector bundles are applied to moduli of quiver representations. Using the Hall algebra approach to quantum groups, an analog of the HarderNarasimhan recursion is constructed inside the quantized enveloping algebra of a KacMoody algebra. ..."
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Cited by 83 (12 self)
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Methods of Harder and Narasimhan from the theory of moduli of vector bundles are applied to moduli of quiver representations. Using the Hall algebra approach to quantum groups, an analog of the HarderNarasimhan recursion is constructed inside the quantized enveloping algebra of a KacMoody algebra. This leads to a canonical orthogonal system, the HN system, in this algebra. Using a resolution of the recursion, an explicit formula for the HN system is given. As an application, explicit formulas for Betti numbers of the cohomology of quiver moduli are derived, generalizing several results on the cohomology of quotients in ’linear algebra type’ situations.
On the quiver Grassmannian in the acyclic case
 J. Pure Appl. Algebra
"... Abstract. Let A be the path algebra of a quiver Q with no oriented cycle. We study geometric properties of the Grassmannians of submodules of a given Amodule M. In particular, we obtain some sufficient conditions for smoothness, polynomial cardinality and we give different approaches to Euler chara ..."
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Cited by 41 (3 self)
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Abstract. Let A be the path algebra of a quiver Q with no oriented cycle. We study geometric properties of the Grassmannians of submodules of a given Amodule M. In particular, we obtain some sufficient conditions for smoothness, polynomial cardinality and we give different approaches to Euler characteristics. Our main result is the positivity of Euler characteristics when M is an exceptional module. This solves a conjecture of Fomin and Zelevinsky for acyclic cluster algebras. Let M be a finite dimensional space on a field k. The Grassmannian Gre(M,k) of M is the set of subspaces of dimension e. It is well known that Gre(M,k) is an algebraic variety with nice properties. For instance, the linear group GLe(M,k) acts transitively on Gre(M,k) with parabolic stabilizer, hence the variety Gre(M,k) is smooth and projective.
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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Cited by 34 (3 self)
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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.
Finite Dimensional Algebras, Quantum Groups and Finite Groups of Lie Type
 225–244, Fields Inst. Commun
, 2004
"... We shall discuss generic extension monoids associated with finite dimensional (basic) hereditary algebras of finite or cyclic type and related applications to RingelHall algebras, (and hence, to quantum groups). We shall briefly review the geometric setting of quantum gl_n by Beilinson, Lusztig and ..."
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Cited by 27 (7 self)
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We shall discuss generic extension monoids associated with finite dimensional (basic) hereditary algebras of finite or cyclic type and related applications to RingelHall algebras, (and hence, to quantum groups). We shall briefly review the geometric setting of quantum gl_n by Beilinson, Lusztig and MacPherson and its connections to RingelHall algebras and qSchur algebras. In the second part of the paper, we shall survey the development of stratified algebras and their applications to the (generalized) qSchur algebra of finite groups of Lie type.