The notion of gluing of abelian categories was introduced by Kazhdan and Laumon in an attempt of another geometric construction of representations of finite Chevalley groups; the approach was later developed by Polishchuk and Braverman. We observe that this notion of gluing is a particular case of a general categorical construction (used also by Kontsevich and Rosenberg to define "noncommutative schemes"). We prove a conjecture of Kazhdan which says that the D-module counterpart of the Kazhdan-Laumon gluing construction produces a category equivalent to modules over the ring $\mathcal D$ of global differential operators on the basic affine space. As an application we show that $\mathcal D$ is Noetherian, and has finite injective dimension as a module over itself.
We provide an explicit procedure to glue (not necessarily compact) silting objects along recollements of triangulated categories with coproducts having a 'nice' set of generators, namely, well generated triangulated categories. This procedure is compatible with gluing co-t-structures and it generalizes a result by Liu, Vitória and Yang. We provide conditions for our procedure to restrict to tilting objects and to silting and tilting modules. As applications, we retrieve the classification of silting modules over the Kronecker algebra and the classification of non-compact tilting sheaves over a…
We construct a four-parameter family of affine Yangian algebras by gluing two copies of the affine Yangian of $\mathfrak{gl}_1$. Our construction allows for gluing operators with arbitrary (integer or half integer) conformal dimension and arbitrary (bosonic or fermionic) statistics, which is related to the relative framing. The resulting family of algebras is a two-parameter generalization of the $\mathcal{N}=2$ affine Yangian, which is isomorphic to the universal enveloping algebra of $\mathfrak{u}(1)\oplus \mathcal{W}^{\mathcal{N}=2}_{\infty}[λ]$. All algebras that we construct have natural …
We discuss gluing of objects and gluing of morphisms in tensor triangulated categories. We illustrate the results by producing, among other things, a Mayer-Vietoris exact sequence involving Picard groups.
Recently, Wang, Wei and Zhang define the recollement of extriangulated categories, which is a generalization of both recollement of abelian categories and recollement of triangulated categories. For a recollement $(\mathcal A ,\mathcal B,\mathcal C)$ of extriangulated categories, we show that $n$-tilting (resp. $n$-cotilting) subcategories in $\mathcal A$ and $\mathcal C$ can be glued to get $n$-tilting (resp. $n$-cotilting) subcategories in $\mathcal B$ under certain conditions.
The Grothendieck construction of a diagram $X$ of categories can be seen as a process to construct a single category $\Gr(X)$ by gluing categories in the diagram together. Here we formulate diagrams of categories as colax functors from a small category $I$ to the 2-category $\kCat$ of small $\k$-categories for a fixed commutative ring $\k$. In our previous paper we defined derived equivalences of those colax functors. Roughly speaking two colax functors $X, X' \colon I \to \kCat$ are derived equivalent if there is a derived equivalence from $X(i)$ to $X'(i)$ for all objects $i$ in $I$ satisfyi…
We present a new way to construct $n$-cluster tilting subcategories of abelian categories. Our method takes as input a direct system of abelian categories $\mathcal{A}_i$ with certain subcategories and, under reasonable conditions, outputs an $n$-cluster tilting subcategory of an admissible target $\mathcal{A}$ of the direct system. We apply this general method to a direct system of module categories $\text{mod}Λ_i$ of representation-directed algebras $Λ_i$ and obtain an $n$-cluster tilting subcategory $\mathcal{M}$ of a module category $\text{mod}\mathcal{C}$ of a locally bounded Krull-Schmid…
We give a description of certain categories of equivariant coherent sheaves on Grothendieck's resolution in terms of the categorical affine Hecke algebra of Soergel. As an application, we deduce a relationship of these coherent sheaf categories to the categories of perverse sheaves considered in the work of Bezrukavnikov-Yun, generalizing results of Arkhipov-Bezrukavnikov. In addition, we deduce that the weak braid group action on sheaves of Riche and Bezrukavnikov-Riche can be upgraded to a strict braid group action.
We introduce subquotient categories of the restricted affine category O at the critical level and show that some of them have a realization in terms of moment graph sheaves.
The geometric Satake isomorphism is an equivalence between the categories of spherical perverse sheaves on affine Grassmanian and the category of representations of the Langlands dual group. We provide a similar description for derived categories of l-adic sheaves on an affine flag variety which are geometric counterparts of a maximal commutative subalgebra in the Iwahori Hecke algebra; of the anti-spherical module over this algebra; and of the space of Iwahori-invariant Whitakker functions.
We prove a weak version of Lusztig's conjecture on explicit description of the asymptotic Hecke algebras (both finite and affine), and explain its relation to Lusztig's classification of character sheaves.
In this paper we construct and study an action of the affine braid group associated to a semi-simple algebraic group on derived categories of coherent sheaves on various varieties related to the Springer resolution of the nilpotent cone. In particular, we describe explicitly the action of the Artin braid group. This action is a "categorical version" of Kazhdan--Lusztig--Ginzburg's construction of the affine Hecke algebra, and is used in particular by the first author and Ivan Mirkovic in the course of the proof of Lusztig's conjectures on equivariant K-theory of Springer fibers.
George Lusztig conjectured that asymptotic affine Hecke algebra of a simply connected group can be explicitly described in terms of convolution algebras. Main Theorem of this note (which is a continuation of RT/0010089) is a weak version of this Conjecture. This version is strong enough to reprove all previously known results (due to Nanhua Xi) in this direction, for example the case of type $\tilde A_n$, see QA/0010159.
We classify integrable irreducible $\hat{g}[σ]$-modules in categories E and C, where E is proved to contain the well known evaluation modules and C to unify highest weight modules, evaluation modules and their tensor product modules.
Recently, Wang, Wei and Zhang introduced the notion of recollements of extriangulated categories. In this paper, let $(\mathcal{A},\mathcal{B},\mathcal{C})$ be a recollement of extriangulated categories. We provide some methods to construct resolving subcategories in $(\mathcal{A},\mathcal{B},\mathcal{C})$. As applications of the Auslander-Reiten correspondence, we get the gluing of cotilting modules in a recollement of module categories for artin algebras. We also give some bounds of resolution dimensions of the categories involved in $(\mathcal{A},\mathcal{B},\mathcal{C})$ with respect to re…
We study the Wakimoto modules over the affine Kac-Moody algebras at the critical level from the point of view of the equivalences of categories proposed in our previous works, relating categories of representations and certain categories of sheaves. In particular, we describe explicitly geometric realizations of the Wakimoto modules as Hecke eigen-D-modules on the affine Grassmannian and as quasi-coherent sheaves on the flag variety of the Langlands dual group.
We define an affine Jacquet functor and use it to describe the structure of induced affine Harish-Chandra modules at noncritical levels, extending the theorem of Kac and Kazhdan [KK] on the structure of Verma modules in the Bernstein-Gelfand-Gelfand categories O for Kac-Moody algebras. This is combined with a vanishing result for certain extension groups to construct a block decomposition of the categories of affine Harish-Chandra modules of Lian and Zuckerman [LZ]. The latter provides an extension of the works of Rocha-Caridi, Wallach [RW] and Deodhar, Gabber, Kac [DGK] on block decomposition…
In this paper, we introduce a rewriting theory of linear monoidal categories. Those categories are a particular case of what we will define as linear (n, p)-categories. We will also define linear (n, p)-polygraphs, a linear adapation of n-polygraphs, to present linear (n -- 1, p)-categories. We focus then on linear (3, 2)-polygraphs to give presentations of linear monoidal categories. We finally give an application of this theory in linear (3, 2)-polygraphs to prove a basis theorem on the category AOB with a new method using a rewriting property defined by van Ostroom: decreasingness.
We study monoidal categorifications of certain monoidal subcategories $\mathcal{C}_J$ of finite-dimensional modules over quantum affine algebras, whose cluster algebra structures coincide and arise from the category of finite-dimensional modules over quiver Hecke algebra of type A${}_\infty$. In particular, when the quantum affine algebra is of type A or B, the subcategory coincides with the monoidal category $\mathcal{C}_{\mathfrak{g}}^0$ introduced by Hernandez-Leclerc. As a consequence, the modules corresponding to cluster monomials are real simple modules over quantum affine algebras.
We relate commutative algebras in braided tensor categories to braid-reversed tensor equivalences, motivated by vertex algebra representation theory. First, for $\mathcal{C}$ a braided tensor category, we give a detailed construction of the canonical algebra in $\mathcal{C}\boxtimes\mathcal{C}^\text{rev}$: if $\mathcal{C}$ is semisimple but not necessarily finite or rigid, then $\bigoplus_{X\in\text{Irr}(\mathcal{C})}X'\boxtimes X$ is a commutative algebra, with $X'$ a representing object for $\text{Hom}_\mathcal{C}(\bullet\otimes_\mathcal{C}X,\mathbf{1}_{\mathcal{C}})$. Conversely, let $A=\bi…
We prove a conjecture by Lusztig, which describes the tensor categories of perverse sheaves on affine flag manifolds, with tensor structure provided by truncated convolution, in terms of the Langlands dual group. We also give a geometric (categorical) description of Lusztig's bijection between two-sided cells in an affine Weyl group, and unipotent conjugacy classes in the Langlands dual group. The main tool is the sheaf-theoretic construction of the center of the affine Hecke algebra due to Gaitsgory (based on ideas of Beilinson and Kottwitz), see math.AG/9912074.
By a local geometric Langlands correspondence for a complex reductive group G we understand a construction which assigns to a local system on the punctured disc for the Langlands dual group of G, a category equipped with an action of the formal loop group G((t)). We propose a conjectural description of these categories as categories of representations of the corresponding affine Kac-Moody algebra of critical level, and, in some cases, as categories of D-modules on the ind-schemes G((t))/K. We describe in detail these categories and interrelations between them and provide supporting evidence fo…
For a Dynkin quiver $Q$ (of type ADE), we consider a central completion of the convolution algebra of the equivariant K-group of a certain Steinberg type graded quiver variety. We observe that it is affine quasi-hereditary and prove that its category of finite-dimensional modules is identified with a block of Hernandez-Leclerc's monoidal category $\mathcal{C}_Q$ of modules over the quantum loop algebra $U_q(L\mathfrak{g})$ via Nakajima's homomorphism. As an application, we show that Kang-Kashiwara-Kim's generalized quantum affine Schur-Weyl duality functor gives an equivalence between the cate…
We consider the category of modules over the affine Kac-Moody algebra g^ of critical level with regular central character. In our previous paper math.RT/0508382 we conjectured that this category is equivalent to the category of Hecke eigen-D-modules on the affine Grassmannian G((t))/G[[t]]. This conjecture was motivated by our proposal for a local geometric Langlands correspondence. In this paper we prove this conjecture for the corresponding I^0 equivariant categories, where I^0 is the radical of the Iwahori subgroup of G((t)). Our result may be viewed as an affine analogue of the equivalence…
Koenig and Xi introduced {\em affine cellular algebras}. Kleshchev and Loubert showed that an important class of {\em infinite dimensional} algebras, the KLR algebras $R(Γ)$ of finite Lie type $Γ$, are (graded) affine cellular; in fact, the corresponding affine cell ideals are idempotent. This additional property is reminiscent of the properties of {\em quasihereditary algebras} of Cline-Parshall-Scott in a {\em finite dimensional} situation. A fundamental result of Cline-Parshall-Scott says that a finite dimensional algebra $A$ is quasihereditary if and only if the category of finite dimensio…
The main result is that the category of ordinary modules of an affine vertex operator algebra of a simply laced Lie algebra at admissible level is rigid and thus a braided fusion category. If the level satisfies a certain coprime property then it is even a modular tensor category. In all cases open Hopf links coincide with the corresponding normalized S-matrix entries of torus one-point functions. This is interpreted as a Verlinde formula beyond rational vertex operator algebras. A preparatory Theorem is a convenient formula for the fusion rules of rational principal W-algebras of any type.
We construct an exact tensor functor from the category $\mathcal{A}$ of finite-dimensional graded modules over the quiver Hecke algebra of type $A_\infty$ to the category $\mathscr C_{B^{(1)}_n}$ of finite-dimensional integrable modules over the quantum affine algebra of type $B^{(1)}_n$. It factors through the category $\mathcal T_{2n}$, which is a localization of $\mathcal{A}$. As a result, this functor induces a ring isomorphism from the Grothendieck ring of $\mathcal T_{2n}$ (ignoring the gradings) to the Grothendieck ring of a subcategory $\mathscr C^{0}_{B^{(1)}_n}$ of $\mathscr C_{B^{(1…
We review briefly the existing vertex-operator-algebraic constructions of various tensor category structures on module categories for affine Lie algebras. We discuss the results first conjectured in the work of Moore and Seiberg that led us to the construction of the modular tensor category structure in the positive integral level case. Then we review the existing constructions and results in the following three cases: (i) the level plus the dual Coxeter number is not a nonnegative rational number, (ii) the level is a positive integer and (iii) the level is an admissible number. We also presen…
Let Y be a smooth algebraic stack exhausted by quotient stacks. Given a Kirwan-Ness stratification of the cotangent stack T^*Y, we establish a recollement package for twisted D-modules on Y, gluing the category from subquotients described via modules microsupported on the Kirwan-Ness strata of T^*Y. The package includes unusual existence and "preservation-of-finiteness'' properties for functors of the full category of twisted D-modules, extending the standard functorialities for holonomic modules. In the case that Y = X/G is a quotient stack, our results provide a higher categorical analogue o…
We show that the equivariant hypertoric convolution algebras introduced by Braden-Licata-Proudfoot-Webster are affine quasi hereditary in the sense of Kleshchev and compute the Ext groups between standard modules. Together with the main result of arXiv:2009.03981, this implies a number of new homological results about the bordered Floer algebras of Ozsvath-Szabo, including the existence of standard modules over these algebras. We prove that the Ext groups between standard modules are isomorphic to the homology of a variant of the Lipshitz-Ozsvath-Thurston bordered strands dg algebras.
In this paper we express certain multiplicities in modular representation-theoretic categories of type A in terms of affine p-Kazhdan-Lusztig polynomials. The representation-theoretic categories we deal with include the categories of rational representations of GL(n), representations of the quantum group for gl(n), and representations of (degenerate) cyclotomic Hecke and Schur algebras, where the base field is an algebraically closed field of arbitrary prime characteristic. In order to approach this problem we define Soergel-theoretic versions of parabolic categories O in characteristic p. We …
We generalize the construction of reflection functors from classical representation theory of quivers to arbitrary small categories with freely attached sinks or sources. These reflection morphisms are shown to induce equivalences between the corresponding representation theories with values in arbitrary stable homotopy theories, including representations over fields, rings or schemes as well as differential-graded and spectral representations. Specializing to representations over a field and to specific shapes, this recovers derived equivalences of Happel for finite, acyclic quivers. However,…
