In this article, we give geometric constructions of tensor products in various categories using quiver varieties. More precisely, we introduce a lagrangian subvariety $\Zl$ in a quiver variety, and show the following results:
(1) The homology group of $\Zl$ is a representation of a symmetric Kac-Moody Lie algebra $\mathfrak g$, isomorphic to the tensor product $V(λ_1)\otimes...\otimes V(λ_N)$ of integrable highest weight modules.
(2) The set of irreducible components of $\Zl$ has a structure of a crystal, isomorphic to that of the $q$-analogue of $V(λ_1)\otimes...\otimes V(λ_N)$.
(3) The equivariant $K$-homology group of $\Zl$ is isomorphic to the tensor product of universal standard modules of the quantum loop algebra $\Ul$, when $\mathfrak g$ is of type $ADE$.
We also give a purely combinatorial description of the crystal of (2). This result is new even when N=1.
We study finite dimensional representations of the quantum affine algebra, using geometry of quiver varieties introduced by the author. As an application, we obtain character formulas expressed in terms of intersection cohomologies of quiver varieties.
We review the definition of quiver varieties and their relation to representation theory of Kac-Moody Lie algebras. Target readers are ring and representation theorists. We emphasize important roles of first extension groups of the preprojective algebra associated with a quiver.
In this note we give an explicit geometric description of some of the Nakajima's quiver varieties. More precisely, we show that the $Γ$-equivariant Hilbert scheme $X^{Γ[n]}$ and the Hilbert scheme $X_Γ^{[n]}$ (where $X=\C^2$, $Γ\subset SL(\C^2)$ is a finite subgroup, and $X_Γ$ is a minimal resolution of $X/Γ$) are quiver varieties for the affine Dynkin graph, corresponding to $Γ$ via the McKay correspondence, the same dimension vectors, but different parameters $ζ$ (for earlier results in this direction see [4, 12, 13]). In particular, it follows that the varieties $X^{Γ[n]}$ and $X_Γ^{[n]}$ a…
Let us consider a specialization of an untwisted quantum affine algebra of type $ADE$ at a nonzero complex number, which may or may not be a root of unity. The Grothendieck ring of its finite dimensional representations has two bases, simple modules and standard modules. We identify entries of the transition matrix with special values of ``computable'' polynomials, similar to Kazhdan-Lusztig polynomials. At the same time we ``compute'' $q$-characters for all simple modules. The result is based on ``computations'' of Betti numbers of graded/cyclic quiver varieties. (The reason why we put `` '' …
Using subvarieties, which we call Demazure quiver varieties, of the quiver varieties of Nakajima, we give a geometric realization of Demazure modules of Kac-Moody algebras with symmetric Cartan data. We give a natural geometric characterization of the extremal weights of a representation and show that Lusztig's semicanonical basis is compatible with the filtration of a representation by Demazure modules. For the case of affine sl_2, we give a characterization of the Demazure quiver variety in terms of a nilpotency condition on quiver representations and an explicit combinatorial description of…
Beginning with the data of a quiver Q, and its dimension vector d, we construct an algebra D_q=D_q(Mat_d(Q)), which is a flat q-deformation of the algebra of differential operators on the affine space Mat_d(Q). The algebra D_q is equivariant for an action by a product of quantum general linear groups, acting by conjugation at each vertex. We construct a quantum moment map for this action, and subsequently define the Hamiltonian reduction A^lambda_d(Q) of D_q with moment parameter λ. We show that A^λ_d(Q) is a flat formal deformation of Lusztig's quiver varieties, and their multiplicative count…
We give a crystal structure on the set of all irreducible components of Lagrangian subvarieties of quiver varieties. One con show that, as a crystal, it is isomorphic to the crystal base of an irreducible highest weight representation of a quantized universal enveloping algebra.
We prove a formula relating the fermionic forms and the Poincare polynomials of quiver varieties associated to a finite quiver. Applied to quivers of type ADE, our result implies a version of the fermionic conjecture of Lusztig.
Associated to each finite group $Γ$ in $SL_2(C)$ there is a family of noncommutative algebras which deforms the coordinate ring of the Kleinian singularity corresponding to that group. These algebras were defined by W. Crawley-Boevey and M. Holland, who also suggested a conjectural correspondence between the set of isomorphism classes of rank one projective modules over these algebras and associated Nakajima quiver varieties. In \cite{BGK}, V.Baranovski, V.Ginzburg and A.Kuznetsov proved the Crawley-Boevey-Holland conjecture using the methods of noncommutative projective geometry. In this pape…
We prove that quantized multiplicative quiver varieties and quantum character varieties define sheaves of Azumaya algebras over the corresponding classical moduli spaces, and we prove that the Azumaya locus of the Kauffman bracket skein algebras contains the smooth locus, proving a strong form of the Unicity Conjecture of Bonahon and Wong. The proofs exploit a strong compatibility between quantum Hamiltonian reduction and the quantum Frobenius homomorphism as it arises in each setting. We therefore introduce the concepts of Frobenius quantum moment maps and their Hamiltonian reduction, and of …
We show how equivariant volumes of tensor product quiver varieties of type A are given by matrix elements of vertex operators of centrally extended doubles of Yangians, and how they satisfy in some cases the rational, level 1, quantum Knizhnik--Zamolodchikov equation.
The theory of cluster algebras of S. Fomin and A. Zelevinsky has assigned a fan to each Dynkin diagram. Then A. Buan, R. Marsh, M. Reineke, I. Reiten and G. Todorov have generalized this construction using arbitrary quivers on Dynkin diagrams. In the special case of the unioriented quiver of type A, we describe the cohomology ring of the toric variety associated to this fan. A natural base is obtained and an explicit rule is given for the product of any two generators.
We prove a conjecture of Nakajima (for type A the result was announced by Ginzburg- 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 tthe characters of the simple finite dimensional representations of the quantized affine algebra and that of the yangian coincide.
In this paper we prove the rationality of the capped vertex function with descendents for arbitrary Nakajima quiver varieties with generic stability conditions. We generalise the proof given by Smirnov to the general case, which requires to use techniques of tautological classes rather than the fixed-point basis. This result confirms that the "monodromy" of the capped vertex function is trivial, which gives a strong constraint for the monodromy of the capping operators. We will also provide a GIT wall-crossing formula for the capped vertex function in terms of the quantum difference operators …
Associated with each finite subgroup $Γ$ of $\rm{SL}_2(\mathbb{C})$ there is a family of noncommutative algebras $O_τ(Γ)$ quantizing $\mathbb{C}^2/\!\!/Γ$. Let $G_Γ$ be the group of $Γ$-equivariant automorphisms of $O_τ$. One of the authors earlier defined and studied a natural action of $G_Γ$ on certain quiver varieties associated with $Γ$. He established a $G_Γ$-equivariant bijective correspondence between quiver varieties and the space of isomorphism classes of $O_τ$-ideals. The main theorem in this paper states that when $Γ$ is a cyclic group, the action of $G_Γ$ on each quiver variety is …
In order to extend the geometrization of Yangian $R$-matrices from Lie algebras $gl(n)$ to superalgebras $gl(M|N)$, we introduce new quiver-related varieties which are associated with representations of $gl(M|N)$. In order to define them similarly to the Nakajima-Cherkis varieties, we reformulate the construction of the latter by replacing the Hamiltonian reduction with the intersection of generalized Lagrangian subvarieties in the cotangent bundles of Lie algebras sitting at the vertices of the quiver. The new varieties come from replacing some Lagrangian subvarieties with their Legendre tran…
The theory of representations of quivers and of their preprojective algebras are reviewed. In particular, moduli spaces of representations of these algebras, quiver varieties and reflection functor are described. The proof that the bijection between moduli spaces induced by the reflection functor is an isomorphism of symplectic affine varieties is presented. Hamiltonian systems with complete flows on the quiver varieties are introduced. An application of the reflection functor to these systems is described. The results on the case of the cyclic quiver are reviewed and a role of the reflection …
We present a simple unified formula expressing the denominators of the normalized R-matrices between the fundamental modules over the quantum loop algebras of type ADE. It has an interpretation in terms of representations of the Dynkin quivers and can be proved in a unified way using the geometry of graded quiver varieties. As a by-product, we obtain a geometric interpretation of Kang-Kashiwara-Kim's generalized quantum affine Schur-Weyl duality functor when it arises from a family of fundamental modules. We also study several cases when the graded quiver varieties are isomorphic to the graded…
Quiver varieties have recently appeared in various different areas of Mathematics such as representation theory of Kac-Moody algebras and quantum groups, instantons on 4-manifolds, and resolutions Kleinian singularities. In this paper, we show that many important affine quiver varieties, e.g., the Calogero-Moser space, can be imbedded as coadjoint orbits in the dual of an appropriate infinite dimensional Lie algebra. In particular, there is an infinitesimally transitive action of the Lie algebra in question on the quiver variety. Our construction is based on an extension of Kontsevich's formal…
Kashiwara and Saito have a geometric construction of the infinity crystal for any symmetric Kac-Moody algebra. The underlying set consists of the irreducible components of Lusztig's quiver varieties, which are varieties of nilpotent representations of a pre-projective algebra. We generalize this to symmetrizable Kac-Moody algebras by replacing Lusztig's preprojective algebra with a more general one due to Dlab and Ringel. In non-symmetric types we are forced to work over non-algebraically-closed fields.
Henriques and Kamnitzer have defined a commutor for the category of crystals of a finite-dimensional complex reductive Lie algebra that gives it the structure of a coboundary category (somewhat analogous to a braided monoidal category). Kamnitzer and Tingley then gave an alternative definition of the crystal commutor, using Kashiwara's involution on Verma crystals, that generalizes to the setting of symmetrizable Kac-Moody algebras. In the current paper, we give a geometric interpretation of the crystal commutor using quiver varieties. Equipped with this interpretation we show that the commuto…
We define a family of homomorphisms on a collection of convolution algebras associated with quiver varieties, which gives a kind of coproduct on the Yangian associated with a symmetric Kac-Moody Lie algebra. We study its property using perverse sheaves
We study quantum geometry of Nakajima quiver varieties of two different types - framed A-type quivers and ADHM quivers. While these spaces look completely different we find a surprising connection between equivariant K-theories thereof with a nontrivial match between their equivariant parameters. In particular, we demonstrate that quantum equivariant K-theory of $A_n$ quiver varieties in a certain $n\to\infty$ limit reproduces equivariant K-theory of the Hilbert scheme of points on $\mathbb{C}^2$. We analyze the correspondence from the point of view of enumerative geometry, representation theo…
The purpose of this thesis is to present certain viewpoints on the geometric representation theory of Nakajima cyclic quiver varieties, in relation to the Maulik-Okounkov stable basis. Our main technical tool is the shuffle algebra, which arises as the K-theoretic Hall algebra of the double cyclic quiver. We prove the isomorphism between the shuffle algebra and the quantum toroidal algebra U_{q,t}(sl_n^^), and identify the quotients of Verma modules for the shuffle algebra with the K-theory groups of Nakajima cyclic quiver varieties, which were studied by Nakajima and Varagnolo-Vasserot. The s…
Braverman and Finkelberg recently proposed the geometric Satake correspondence for the affine Kac-Moody group $G_\aff$ [Braverman A., Finkelberg M., arXiv:0711.2083]. They conjecture that intersection cohomology sheaves on the Uhlenbeck compactification of the framed moduli space of $G_{\mathrm{cpt}}$-instantons on $\R^4/\Z_r$ correspond to weight spaces of representations of the Langlands dual group $G_\aff^\vee$ at level $r$. When $G = \SL(l)$, the Uhlenbeck compactification is the quiver variety of type $\algsl(r)_\aff$, and their conjecture follows from the author's earlier result and I. F…
We construct the fusion product of finite-dimensional sl_2-modules in the homology of (or in the space of constructible functions on) a certain subvariety L_l(w_1, ..., w_r) of Nakajima's tensor product variety L(w_1,..., w_r). We also give a combinatorial description of the irreducible components of this subvariety using the notions of graphical calculus and crossingless matches for sl_2.
Using methods of homological algebra, we obtain an explicit crystal isomorphism between two realizations of crystal bases of the lower part of the quantized enveloping algebra of (almost all) finite dimensional simply-laced Lie algebras. The first realization we consider is a geometric construction in terms of irreducible components of certain quiver varieties established by Kashiwara and Saito. The second is a realization in terms of isomorphism classes of quiver representations obtained by Reineke using Ringel's Hall algebra approach to quantum groups. We show that these two constructions ar…
We prove that a Weyl module for the current Lie algebra associated with a simple Lie algebra of type $ADE$ is rigid, that is, it has a unique Loewy series. Further we use this result to prove that the grading on a Weyl module defined by the degree of currents coincides with another grading which comes from the degree of the homology group of the quiver variety. As a corollary we obtain a formula for the Poincaré polynomials of quiver varieties of type $ADE$ in terms of the energy functions defined on the crystals for tensor products of level-zero fundamental representations of the correspondin…
We have two constructions of the level-$(0,1)$ irreducible representation of the quantum toroidal algebra of type $A$. One is due to Nakajima and Varagnolo-Vasserot. They constructed the representation on the direct sum of the equivariant K-groups of the quiver varieties of type $\hat{A}$. The other is due to Saito-Takemura-Uglov and Varagnolo-Vasserot. They constructed the representation on the q-deformed Fock space introduced by Kashiwara-Miwa-Stern. In this paper we give an explicit isomorphism between these two constructions. For this purpose we construct simultaneous eigenvectors on the q…
We study some classical integrable systems naturally associated with multiplicative quiver varieties for the (extended) cyclic quiver with $m$ vertices. The phase space of our integrable systems is obtained by quasi-Hamiltonian reduction from the space of representations of the quiver. Three families of Poisson-commuting functions are constructed and written explicitly in suitable Darboux coordinates. The case $m=1$ corresponds to the tadpole quiver and the Ruijsenaars-Schneider system and its variants, while for $m>1$ we obtain new integrable systems that generalise the Ruijsenaars-Schneider …
Motivated by a recent conjecture by Hernandez and Leclerc [arXiv:0903.1452], we embed a Fomin-Zelevinsky cluster algebra [arXiv:math/0104151] into the Grothendieck ring R of the category of representations of quantum loop algebras U_q(Lg) of a symmetric Kac-Moody Lie algebra g, studied earlier by the author via perverse sheaves on graded quiver varieties [arXiv:math/9912158]. 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 R, …
We consider a generalization of the quiver varieties of Lusztig and Nakajima to the case of all symmetrizable Kac-Moody Lie algebras. To deal with the non-simply laced case one considers admissible automorphisms of a quiver and the irreducible components of the quiver varieties fixed by this automorphism. We define a crystal structure on these irreducible components and show that the crystals obtained are isomorphic to those associated to the crystal bases of the lower half of the universal enveloping algebra and the irreducible highest weight representations of the non-simply laced Kac-Moody …
We construct the quantized enveloping algebra of any simple Lie algebra of type ADE as the quotient of a Grothendieck ring arising from certain cyclic quiver varieties. In particular, the dual canonical basis of a one-half quantum group with respect to Lusztig's bilinear form is contained in the natural basis of the Grothendieck ring up to rescaling. This paper expands the categorification established by Hernandez and Leclerc to the whole quantum groups. It can be viewed as a geometric counterpart of Bridgeland's recent work for type ADE.
These notes reflect the contents of three lectures given at the workshop of the 14th International Conference on Representations of Algebras (ICRA XIV), held in August 2010 in Tokyo. We first provide an introduction to quantum loop algebras and their finite-dimensional representations. We explain in particular Nakajima's geometric description of the irreducible q-characters in terms of graded quiver varieties. We then present a recent attempt to understand the tensor structure of the category of finite- dimensional representations by means of cluster algebras.
Following Braverman-Finkelberg-Feigin-Rybnikov (arXiv:1008.3655), we study the convolution algebra of a handsaw quiver variety, a.k.a. a parabolic Laumon space, and a finite W-algebra of type A. This is a finite analog of the AGT conjecture on 4-dimensional supersymmetric Yang-Mills theory with surface operators. Our new observation is that the C^*-fixed point set of a handsaw quiver variety is isomorphic to a graded quiver variety of type A, which was introduced by the author in connection with the representation theory of a quantum affine algebra. As an application, simple modules of the W-a…
Inspired by a previous work of Nakajima, we consider perverse sheaves over acyclic graded quiver varieties and study the Fourier-Sato-Deligne transform from a representation theoretic point of view. We obtain deformed monoidal categorifications of acyclic quantum cluster algebras with specific coefficients. In particular, the (quantum) positivity conjecture is verified whenever there is an acyclic seed in the (quantum) cluster algebra. In the second part of the paper, we introduce new quantizations and show that all quantum cluster monomials in our setting belong to the dual canonical basis of…
We will exhibit a group of symmetries of the simply-laced quantum connections, generalising the quantum/Howe duality relating KZ and the Casimir connection. These symmetries arise as a quantisation of the classical symmetries of the simply-laced isomonodromy systems, which in turn generalise the Harnad duality. The quantisation of the classical symmetries involves constructing the quantum Hamiltonian reduction of the representation variety of any simply-laced quiver, both in filtered and in deformation quantisation.
The geometric small property (Borho-MacPherson) of projective morphisms implies a description of their singularities in terms of intersection homology. In this paper we solve the smallness problem raised by Nakajima (math.QA/0105173) for certain resolutions of quiver varieties (analogs of the Springer resolution) : for Kirillov-Reshetikhin modules of simply-laced quantum affine algebras, we characterize explicitly the Drinfeld polynomials corresponding to the small resolutions. We use an elimination theorem for monomials of Frenkel-Reshetikhin q-characters that we establish for non necessarily…
Consider Kashiwara's crystal associated to a highest weight representation of a symmetric Kac-Moody algebra. There is a geometric realization of this object using Nakajima's quiver varieties, but in many particular cases it can also be realized by elementary combinatorial methods. Here we propose a framework for extracting combinatorial realizations from the geometric picture: We construct certain torus actions on the quiver varieties and use Morse theory to index the irreducible components by connected components of the subvariety of torus fixed points. We then discuss the case of affine sl(n…
