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Coherent sheaves on quiver varieties and categorification. (English) Zbl 1284.14016

The paper under review studies the construction of Kac-Moody algebra representations on the \(K\)-theory of Nakajima’s quiver varieties leading to a categorification of the notion. Nakajima constructed representations of the quantum affine algebra on the equivariant \(K\)-theory of quiver varieties and this article lifts this action to the derived category of coherent sheaves. The action on the Lie algebra \(\mathfrak{g}\) on cohomology is defined in terms of correspondences which can be seen as Fourier-Mukai kernels, hence they induce functors in the derived categories, providing an example of the idea that geometrization lifts to categorification.
To a finite graph \(\Gamma\), it is associated its simply-laced Kac-Moody algebra \(\mathfrak{g}\) and its quantized universal enveloping algebra \(U_{g}(\mathfrak{g})\). A representation of this algebra is given by a collection of weight spaces for the vertex of the Dynkin diagram and generators and relations which resemble on the commutator relations of the representations of \(\mathfrak{sl}_{2}(\mathbb{C})\). The notion of geometrical categorical \(\mathfrak{g}\)-action is given by associating to each weight space a variety and to each generator a Fourier-Mukai kernel, such that the kernels define functors between the derived categories of coherent sheaves over the varieties. These objects satisfy several properties (listed on pages 812-813) which are the categorical version of the usual presentation of the Kac-Moody Lie algebra.
After the introduction in section 1, sections 2 and 3 are devoted to present the notion of geometrical categorical \(\mathfrak{g}\)-action and state the main theorem, that quiver varieties, their deformations and the Hecke correspondences yield a categorical \(\mathfrak{g}\)-action. Sections 4, 5 and 6 are devoted to prove the technical properties of such an action. In section 7, it is obtained an action of the affine braid group on the derived category of quiver varieties. Section 8 relates how to categorify the irreducible representations of the universal enveloping algebra on the K-theory of the quiver variety (since the geometrical categorical action is, in general, reducible) and section 9 provides examples.

MSC:

14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
16G20 Representations of quivers and partially ordered sets
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
16E20 Grothendieck groups, \(K\)-theory, etc.
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