Quiver Varieties and Yangians

By M. Varagnolo. 2000

112 citations recorded by the source corpus.

We prove a conjecture of Nakajima (for type A it was announced by Ginzburg and Vasserot) giving a geometric realization, via quiver varieties, of the Yangian of type ADE (and more in general of the Yangian associated to every symmetric Kac–Moody Lie algebra). As a corollary, we get that the finite-dimensional representation theory of the quantized affine algebra and that of the Yangian coincide.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/853c62f749352cb33365b45dfdafb1a4b95f6040

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Citations in the corpus

The cohomological Hall algebra of a preprojective algebra (2014, methodology): “Following a similar method, Varagnolo constructed an action of the Yangian Y (gQ) on the equivariant Borel–Moore homology of quiver varieties [51].”
Yangians and cohomology rings of Laumon spaces (2008, background): “Varagnolo [24] to construct the action of Yangians in the equivariant cohomology of quiver varieties.”
Yangians and quantum loop algebras (2010, background): “Following [17] and [22], consider the rings S(m) = C[q±1, A±1 1 , .”
Induced Representations of Affine Hecke Algebras and Canonical Bases of Quantum Groups (2000, background): “. Actually, Molev considers representations of the Yangian of glN, but it is known that the ﬁnite dimensional representation theory of the quantized aﬃne algebra and that of the Yangian coincide, see [Va]. 8.2.3 Proof of Theorem 2. For each k= 1,...,rconsider the evaluation module Sα(k)(zk). This is”
Quantum Integrability and Supersymmetric Vacua (2009, background): “Nakajima and others of the action of the Yangians [19], [20] and quantum affine algebras on the classical cohomology and K-theory respectively of certain quiver varieties.”
Imaginary vectors in the dual canonical basis of Uq(n) (2002, result): “Hence we have an example of a pair (V,W) of irreducible representations such that the R-matrixRV W(z) has no singularity at some z0 ∈ C∗, although the tensor product V ⊗W(z0)is not irreducible. Using [V], we can derive a similar example for the Yangian of slN, thus showing that an observation of Cha”
Quantum Groups and Quantum Cohomology (2012, methodology): “d an analysis of its basic structure theory is another objective of this paper and occupies the bulk of its rst half. In the case when Qhas no loops, this construction is related to work of Varagnolo [121] and Nakajima [89], who construct a certain subalgebra of Y Q via generators and relations. In ”
Coproduct for Yangians of affine Kac–Moody algebras (2017, background): “including the gl1-case), there is a geometric construction of the Yangian using quiver varieties [MO12]. This construction gives a coproduct, as well as the universal R-matrix. By the construction in [Var00], we have a homomorphism from Y(g) to the Yangian in [MO12]. Our formula (4.8) implies that i”
Perverse sheaves and quantum Grothendieck rings (2001, methodology): “is obvious. Claim (c) is proved in Theorem 4.3. ⊓⊔ For a future use we introduce the following sets V (γ) = η∈Y+ +W(γ) γ−η6= {0} , V (γ) = η∈ V (γ) γη . Remark. The map gch appeared ﬁrst in [N3]. By [Vr], the same construction holds for Yangians. The specialization of gch at v= 1 ﬁrst appeared in [K”
Supersymmetric vacua and Bethe ansatz (2009, background): “Nakajima and others of the action of the Yangians [21],[22] and quantum affine algebras on the classical cohomology and K-theory respectively of certain quiver varieties.”
Yangians, quantum loop algebras and abelian difference equations (2013, background): “The analogous action of the Yangian Y (cid:2) ( g ) in cohomology due to Varagnolo [51] allows one to relate irreducible representations of U q ( L g ) and Y (cid:2) ( g ) via the Chern character, and shows in particular that the map E xp preserves dimensions.”
On cohomological Hall algebras of quivers: Generators (2017, methodology): “A cohomological version of this construction, due to Varagnolo [41], yields an action of the Yangian of gQ on the space”
Elliptic stable envelopes (2016, result): “This extends and generalizes earlier work of Nakajima [48], Varagnolo [59], and others, in which certain smaller algebras Y(gKac-Moody) ⊂ Y(g) were made to act by an explicit assignment on generators.”