Kyoto Journal of Mathematics

Quiver varieties and cluster algebras

Hiraku Nakajima

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Motivated by a recent conjecture by Hernandez and Leclerc, we embed a Fomin-Zelevinsky cluster algebra into the Grothendieck ring $\mathbf{R}$ of the category of representations of quantum loop algebras ${\mathbf{U}}_{q}({\mathbf{L}}\mathfrak{g})$ of a symmetric Kac-Moody Lie algebra, studied earlier by the author via perverse sheaves on graded quiver varieties. 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 $\mathbf{R}$, or Lusztig’s dual canonical base. The conjectures that cluster monomials are positive and linearly independent (and probably many other conjectures) of Fomin and Zelevinsky 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.

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Kyoto J. Math. Volume 51, Number 1 (2011), 71-126.

First available: 25 February 2011

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Mathematical Reviews number (MathSciNet)

Primary: 13F60: Cluster algebras
Secondary: 17B37: Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 81R50, 82B23] 14D21: Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) [See also 32L25, 81Txx] 16G20: Representations of quivers and partially ordered sets


Nakajima, Hiraku. Quiver varieties and cluster algebras. Kyoto Journal of Mathematics 51 (2011), no. 1, 71--126. doi:10.1215/0023608X-2010-021.

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