Quiver varieties and cluster algebras

By H. Nakajima. 2009

169 citations recorded by the source corpus.

Motivated by a recent conjecture by Hernandez and Leclerc [arXiv:0903.1452], we embed a Fomin-Zelevinsky cluster algebra [arXiv:math/0104151] into the Grothendieck ring R of the category of representations of quantum loop algebras U_q(Lg) of a symmetric Kac-Moody Lie algebra g, studied earlier by the author via perverse sheaves on graded quiver varieties [arXiv:math/9912158]. Graded quiver varieties controlling the image can be identified with varieties which Lusztig used to define the canonical base. The cluster monomials form a subset of the base given by the classes of simple modules in R, or Lusztig's dual canonical base. The positivity and linearly independence (and probably many other) conjectures of cluster monomials [arXiv:math/0104151] follow as consequences, when there is a seed with a bipartite quiver. Simple modules corresponding to cluster monomials factorize into tensor products of `prime' simple ones according to the cluster expansion.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/568ac9d8b29495680b168104e658c1c55762bd9f

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Citations in the corpus

Representations of Algebras and Related Topics (2011, methodology): “chikov’s periodicity conjecture for Y-systems [13], also intimately related with the representation theory of Uq(Lg) [28]. Our conjecture, which involves a positive integer ℓ, is now proved when ℓ= 1 [24, 35], but the general case remains open. 1. Representations ofquantum loop algebras In this ﬁrst”
Total positivity and cluster algebras (2010, unknown): “∗ 0 0 · · · 0 0 ∗ 0 · · · 0 0 0 ∗ · · · 0 ... ... ... . . . ... 0 0 0 · · · ∗”
Positivity for quantum cluster algebras (2016, methodology): “vity conjecture have already been proved. In [KQ14, Cor.3.3.10] the conjecture is proved by Kimura and Qin in case the cluster algebra has a seed with an acyclic quiver, building on ideas of Nakajima [Nak11] in the classical case. By diﬀerent methods, which are much closer to the ones employed in th”
Positivity for cluster algebras (2013, background): “positivity has been shown in [17], building on [4], [16], [25], [26], using monoidal categorifications of quantum cluster algebras and perverse sheaves over graded quiver varieties.”
A cluster algebra approach to $q$-characters of Kirillov–Reshetikhin modules (2013, unknown): “e standard modules, which are isomorphic to tensor products of fundamental modules. Remark 4.11 Let gbe of type A, D, E. Let V and W be ﬁnite-dimensional vector spaces graded by V−. In [N1] (see also [N4]), Nakajima has introduced a graded quiver variety L•(V,W )and has endowed the sum of cohomologi”
Generic bases for cluster algebras from the cluster category (2011, background): “luster monomials. Good bases for cluster algebraswerealsoconstructedby P. Sherman andA. Zelevinsky[46], P. Calderoand B. Keller [6], G. Dupont [17] [16], D. Hernandez and B. Leclerc [29], H. Nakajima [41], M. Ding, J. Xiao and F. Xu [15], G. Cerulli Irelli [7], G. Cerulli Irelli and F. Esposito [8],”
Donaldson-Thomas theory and cluster algebras (2010, result): “and encouragement. In particular, the proof of Theorem 3.4 is due to him; PierreGuy Plamondon who kindly explained the results in his PhD thesis [Pla]; Hiraku Nakajima who explained me his results in [Naka] and encouraged me to promote the result of [KS]; Bernard Leclerc who recommended me to give a”
Cluster algebras and derived categories (2012, background): “He is very grateful to B. Leclerc for correcting an error in a previous version of example 4.3 and to H. Nakajima for a helpful conversation on the results of [30].”
Monoidal categorification of cluster algebras (2014, background): “Nakajima extended it to the cases of the cluster algebras of type A,D,E ([21]) (see also [10]).”
Simple tensor products (2009, background): “e representations. This factorization problem is related to the program of realization of cluster algebras in Rep(Uq(g)) initiated in [HL] when gis simply-laced (see more results in this direction in [N3]). Cluster algebras have a distinguished set of generators called cluster variables, and (in ﬁni”
Quivers with potentials and their representations II: Applications to cluster algebras (2009, background): “ discussion above. Now the ﬁrst projection Gr e(M) →Gr1(C3) = P2 identiﬁes Gr e(M) with the projective curve C given by the equation det(a1(m), a2(m), a3(m), a4(m)) = 0 (m∈C3). 2It was pointed out in [17] that the proof in [8] contains a gap. QUIVERS WITH POTENTIALS II 13 Since Cis a smooth curve of”