Quiver varieties and tensor products
By H. Nakajima. 2001
117 citations recorded by the source corpus.
Abstract.In this article, we give geometric constructions of tensor products in various categories using quiver varieties. More precisely, we introduce a lagrangian subvariety &?tilde; in a quiver variety, and show the following results: (1) The homology group of &?tilde; is a representation of a symmetric Kac-Moody Lie algebra ?, isomorphic to the tensor product V(λ1)⊗...⊗V(λN) of integrable highest weight modules. (2) The set of irreducible components of &?tilde; has a structure of a crystal, isomorphic to that of the q-analogue of V(λ1)⊗...⊗V(λN). (3) The equivariant K-homology group of &?tilde; is isomorphic to the tensor product of universal standard modules of the quantum loop algebra Uq(L?), when ? is of type ADE. We also give a purely combinatorial description of the crystal of (2). This result is new even when N=1.
Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/3d353376c258713cee64243cdd2843ee0abefd62
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Citations in the corpus
- Extremal weight modules of quantum affine algebras (2002, unknown): “Similarly we have N ′∨+ = 1−h.”
- Coproduct for Yangians of affine Kac–Moody algebras (2017, methodology): “This alternative also fits with the geometric construction in [MO12], once it is formulated as in [Nak13].”
- An approach to categorification of some small quantum groups II (2012, methodology): “product Op ×O2p of two integral domains. Categorification of tensor products of quantum group representations was obtained geometrically by Zheng[49], utilizing earlierideas andconstructions ofNakajima[36,37] and Malkin [31], and by Webster [47, 48] in an algebro-combinatorial fashion, who also exten”
- Cherednik algebras, W-algebras and the equivariant cohomology of the moduli space of instantons on A2 (2012, background): “In later works, a geometric construction of tensor products of representations in terms of both cohomology and K-theory of some quiver varieties was given in [33], [29].”
- Quantum Groups and Quantum Cohomology (2012, background): “C×-action. This embedding is, of course, well-known and played a central role in the work of M. Varagnolo and E. Vasserot [111, 112], H. Nakajima [83], and A. Malkin [65]. See in particular the paper [84] for further developments in this direction, closely related to our construction. 1.2.7 Suppose ”
- Geometry of canonical bases and mirror symmetry (2013, background): “L1], [L3]. Canonical bases in representations of GL3,Sp4 were defined by Gelfand-Zelevinsky-Retakh [GZ], [RZ]. Closely related, but in general different bases were considered by Kashiwara [K], Nakajima [N1], [N2], Malkin [?], Mirkovi´c-Vilonen [MV], and extensively studied afterwords. Abusing terminol”
- A categorification of finite-dimensional irreducible representations of quantum sl(2) and their tensor products (2005, background): “variety degenerates into the partial flag variety Fλ. It is important to note that the generalised Steinberg varieties Xλ,µ are precisely the tensor product varieties of Malkin ([Mal03]) and Nakajima ([Nak01]) in the special case of tensor products of finite dimensional irreducible Uq(sl2)-modules (se”