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34
t–analogs of q–characters of quantum affine algebras of type An, Dn
, 2002
"... We give a tableaux sum expression of t–analog of q–characters of finite dimensional representations (standard modules) of quantum affine algebras Uq(Lg) when g is of type An, Dn. ..."
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We give a tableaux sum expression of t–analog of q–characters of finite dimensional representations (standard modules) of quantum affine algebras Uq(Lg) when g is of type An, Dn.
Extremal weight modules of quantum affine algebras
 Adv. Stud. in Pure Math
"... Let ̂g be an affine Lie algebra, and let Uq(̂g) be the quantum affine algebra introduced by Drinfeld and Jimbo. In [11] Kashiwara introduced a Uq(̂g)module V (λ), having a global crystal base for an integrable weight λ of level 0. We call it an extremal weight module. It is isomorphic to the Weyl m ..."
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Let ̂g be an affine Lie algebra, and let Uq(̂g) be the quantum affine algebra introduced by Drinfeld and Jimbo. In [11] Kashiwara introduced a Uq(̂g)module V (λ), having a global crystal base for an integrable weight λ of level 0. We call it an extremal weight module. It is isomorphic to the Weyl module introduced by ChariPressley [6]. In [12, §13] Kashiwara gave a conjecture on the structure of extremal weight modules. We prove his conjecture when ̂g is an untwisted affine Lie algebra of a simple Lie algebra g of type ADE, using a result of BeckChariPressley [5]. As a byproduct, we also show that the extremal weight module is isomorphic to a universal standard module, defined via quiver varieties by the author [16, 18]. This result was conjectured by VaragnoloVasserot [19] and ChariPressley [6] in a less precise form. Furthermore, we give a characterization of global crystal bases by an almost orthogonality propery, as in the case of global crystal base of highest weight modules.
Quiver Varieties and Branching
 SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS
, 2009
"... Braverman and Finkelberg recently proposed the geometric Satake correspondence for the affine Kac–Moody group Gaff [Braverman A., Finkelberg M., arXiv:0711.2083]. They conjecture that intersection cohomology sheaves on the Uhlenbeck compactification of the framed moduli space of Gcptinstantons on ..."
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Braverman and Finkelberg recently proposed the geometric Satake correspondence for the affine Kac–Moody group Gaff [Braverman A., Finkelberg M., arXiv:0711.2083]. They conjecture that intersection cohomology sheaves on the Uhlenbeck compactification of the framed moduli space of Gcptinstantons on R4 /Zr correspond to weight spaces of representations of the Langlands dual group G ∨ aff at level r. When G = SL(l), the Uhlenbeck compactification is the quiver variety of type sl(r)aff, and their conjecture follows from the author’s earlier result and I. Frenkel’s levelrank duality. They further introduce a convolution diagram which conjecturally gives the tensor product multiplicity [Braverman A., Finkelberg M., Private communication, 2008]. In this paper, we develop the theory for the branching in quiver varieties and check this conjecture for G = SL(l).
Geometric construction of representations of affine algebras
 Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), 423–438, Higher Ed
, 2002
"... Let Γ be a finite subgroup of SL2(C). We consider Γfixed point sets in Hilbert schemes of points on the affine plane C 2. The direct sum of homology groups of components has a structure of a representation of the affine Lie algebra ̂g corresponding to Γ. If we replace homology groups by equivariant ..."
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Let Γ be a finite subgroup of SL2(C). We consider Γfixed point sets in Hilbert schemes of points on the affine plane C 2. The direct sum of homology groups of components has a structure of a representation of the affine Lie algebra ̂g corresponding to Γ. If we replace homology groups by equivariant Khomology groups, we get a representation of the quantum toroidal algebra Uq(L̂g). We also discuss a higher rank generalization and character formulas in terms of intersection homology groups.
TENSOR PRODUCT VARIETIES AND CRYSTALS: THE Ade Case
, 2003
"... Let g be a simple simply laced Lie algebra. In this paper two families of varieties associated to the Dynkin graph of g are described: tensor product and multiplicity varieties. These varieties are closely related to Nakajima’s quiver varieties and should play an important role in the geometric cons ..."
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Let g be a simple simply laced Lie algebra. In this paper two families of varieties associated to the Dynkin graph of g are described: tensor product and multiplicity varieties. These varieties are closely related to Nakajima’s quiver varieties and should play an important role in the geometric constructions of tensor products and intertwining operators. In particular, it is shown that the set of irreducible components of a tensor product variety can be equipped with the structure of a gcrystal isomorphic to the crystal of the canonical basis of the tensor product of several simple finitedimensional representations of g, and that the number of irreducible components of a multiplicity variety is equal to the multiplicity of a certain representation in the tensor product of several others. Moreover, the decomposition of a tensor product into a
On two geometric constructions of U(sln) and its representations
, 2009
"... Ginzburg and Nakajima have given two different geometric constructions of quotients of the universal enveloping algebra of sln and its irreducible finitedimensional highest weight representations using the convolution product in the BorelMoore homology of flag varieties and quiver varieties resp ..."
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Ginzburg and Nakajima have given two different geometric constructions of quotients of the universal enveloping algebra of sln and its irreducible finitedimensional highest weight representations using the convolution product in the BorelMoore homology of flag varieties and quiver varieties respectively. The purpose of this paper is to explain the precise relationship between the two constructions. In particular, we show that while the two yield different quotients of the universal enveloping algebra, they produce the same representations and the natural bases which arise in both constructions are the same. We also examine how this relationship can be used to translate the crystal structure on irreducible components of quiver varieties, defined by Kashiwara and Saito, to a crystal structure on the varieties appearing in Ginzburg’s construction, thus recovering results of Malkin.
Finitedimensional algebras and quivers
 Encylopedia of Mathematical Physics
, 2006
"... Abstract. This is an overview article on finitedimensional algebras and quivers, written for the Encyclopedia of Mathematical Physics. We cover path algebras, RingelHall algebras and the quiver varieties of Lusztig and Nakajima. 1. ..."
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Abstract. This is an overview article on finitedimensional algebras and quivers, written for the Encyclopedia of Mathematical Physics. We cover path algebras, RingelHall algebras and the quiver varieties of Lusztig and Nakajima. 1.