Dag Madsen
We propose a new definition of Koszulity for graded algebras where the degree zero part has finite global dimension, but is not necessarily semi-simple. The standard Koszul duality theorems hold in this setting. We give an application to algebras arising from multiplicity free blocks of the BGG category $\mathcal O$.
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We establish a character formula for indecomposable tilting modules for connected reductive groups in characteristic p in terms of p-Kazhdan-Lusztig polynomials, for p>h the Coxeter number. Using results of Andersen, one may deduce a character formula for simple modules if p>2h-3. Our results are a consequence of an extension to modular coefficients of a monoidal Koszul duality equivalence established by Bezrukavnikov and Yun.
We give a complete picture of the interaction between Koszul and Ringel dualities for graded standardly stratified algebras (in the sense of Cline, Parshall and Scott) admitting linear tilting (co)resolutions of standard and proper costandard modules. We single out a certain class of graded standardly stratified algebras imposing the condition that standard filtrations of projective modules are finite, and develop the tilting theory for such algebras. Under the assumption of existence of linear tilting (co)resolutions we show that algebras from this class are Koszul, that both Ringel and Koszu…
We show that there exists a natural non-degenerate pairing of the homomorphism space between two neighbor standard modules over a quasi-hereditary algebra with the first extension space between the corresponding costandard modules and vise versa. Investigation of this phenomenon leads to a family of pairings involving standard, costandard and tilting modules. In the graded case, under some "Koszul-like" assumptions (which we prove are satisfied for example for the blocks of the category $\mathcal{O}$), we obtain a non-degenerate pairing between certain graded homomorphism and graded extension …
We review Koszul duality in representation theory of category $ \cal O $, especially we give a new presentation of the Koszul duality functor. Combining this with work of Backelin, we show that the translation and Zuckerman functors are Koszul dual to each other, thus verifying a conjecture of Bernstein, Frenkel and Khovanov. Finally we use Koszul duality to give a short proof of the Enright-Shelton equivalence.
Koszul duality for finite-dimensional absolutely Koszul algebras
本文通过Koszul对偶刻画有限维绝对Koszul代数，证明其等价于对偶代数的分次左余凝聚性，并得出模的Poincaré级数有理性和有限分次有限维数等推论。
Tilting sheaves for real groups and Koszul duality
本文发展了实解析簇上的倾斜层理论，构造了其到Soergel双模范畴实类似的完全忠实嵌入，并以此给出了Bezrukavnikov和Vilonen定理的纯几何证明。
In this paper we prove that the linear Koszul duality equivalence constructed in a previous paper provides a geometric realization of the Iwahori-Matsumoto involution of affine Hecke algebras.
Given an elementary simple-minded collection in the derived category of a non-positive dg algebra with finite-dimensional total cohomology, we construct a silting object via Koszul duality.
Let $X$ be a partial flag variety, stratified by orbits of the Borel. We give a criterion for the category of modular perverse sheaves to be equivalent to modules over a Koszul ring. This implies that modular category $\mathcal O$ is governed by a Koszul-algebra in small examples.
Quiver Schur algebras and Koszul duality
本文证明了循环 Quiver Schur 代数的分次有限生成表示范畴具有 Koszul 性质。
We give a proof of the parabolic/singular Koszul duality for the category O of affine Kac-Moody algebras. The main new tool is a relation between moment graphs and finite codimensional affine Schubert varieties. We apply this duality to q-Schur algebras and to cyclotomic rational double affine Hecke algebras.
We generalize the modular Koszul duality of Achar-Riche to the setting of Soergel bimodules associated to any finite Coxeter system. The key new tools are a functorial monodromy action and wall-crossing functors in the mixed modular derived category. In characteristic 0, this duality together with Soergel's conjecture (proved by Elias-Williamson) imply that our Soergel-theoretic graded category $\mathcal{O}$ is Koszul self-dual, generalizing the result of Beilinson-Ginzburg-Soergel.
We present an easily applicable sufficient condition for standard Koszul algebras to be Koszul with respect to $Δ$. If a quasi-hereditary algebra $Ł$ is Koszul with respect to $Δ$, then $Ł$ and the Yoneda extension algebra of $Δ$ are Koszul dual in a sense explained below, implying in particular that their bounded derived categories of finitely generated graded modules are equivalent. We also prove that the extension algebra of $Δ$ is Koszul in the classical sense.
We are concerned with relating derived categories of all modules of two dual Koszul algebras defined by a locally bounded quiver. We first generalize the well known Acyclic Assembly Lemma and formalize an old method of extending a functor from an additive category into a complex category to its complex category. Applying this to the Koszul functor associated with a Koszul algebra defined by a gradable quiver, we obtain a Koszul complex functor, that descends to an equivalence of a continuous family of pairs of triangulated subcategories of doubly unbounded complexes of the respective derived c…
Derived projective covers and Koszul duality of simple-minded and silting collections
本文研究了导出投射包与硅廷集合的关系，并证明了硅廷和简单心灵集合的科兹尔双子结果。
We give a complete picture of the interaction between Koszul and Ringel dualities for quasi-hereditary algebras admitting linear tilting (co)resolutions of standard and costandard modules. We show that such algebras are Koszul, that the class of these algebras is closed with respect to both dualities and that on this class these two dualities commute. All arguments reduce to short computations in the bounded derived category of graded modules.
We attempt to give a gentle (though ahistorical) introduction to Koszul duality phenomena for the Hecke category, focusing on the form of this duality studied in joint work of Achar, Riche, Williamson, and the author. We illustrate some key phenomena and constructions for the simplest nontrivial case of (finite) $\mathrm{SL}_2$ using Soergel bimodules, a concrete algebraic model of the Hecke category.
Contravariant Koszul duality between non-positive and positive dg algebras
研究非正与正dg代数的Koszul对偶性，刻画Koszul对偶局部有限的正dg代数，证明对偶函子诱导导出范畴反变等价并建立ST对应，在假设下得出共变有限有界heart为长度heart及Smalø对称性三角类比成立的结论。
In this paper we construct, for F_1 and F_2 subbundles of a vector bundle E, a "Koszul duality" equivalence between derived categories of G_m-equivariant coherent (dg-)sheaves on the derived intersection of F_1 and F_2 inside E, and the corresponding derived intersection of the orthogonals of F_1 and F_2 inside the dual vector bundle E^*. We also propose applications to Hecke algebras.
In this paper we prove that the linear Koszul duality isomorphism for convolution algebras in K-homology defined in a previous paper and the Fourier transform isomorphism for convolution algebras in Borel-Moore homology are related by the Chern character. So, Koszul duality appears as a categorical upgrade of Fourier transform of constructible sheaves. This result explains the connection between the categorification of the Iwahori-Matsumoto involution for graded affine Hecke algebras (due to Evens and the first author) and for usual affine Hecke algebras (obtained in a previous paper).
The Hecke category participates in an equivalence called monoidal Koszul duality, which exchanges it with the category of (Langlands-dual) "free-monodromic tilting sheaves." Motivated by a recent conjecture of Gorsky and the first-named author on HOMFLYPT link homology, we propose to enhance this duality with an additional grading. We provide evidence for this enhancement in the case of $\mathrm{GL}_2$, working in the language of the second-named author's joint work with Achar, Riche, and Williamson.
Let g be the Lie algebra of a connected, simply connected semisimple algebraic group over an algebraically closed field of sufficiently large positive characteristic. We study the compatibility between the Koszul grading on the restricted enveloping algebra (Ug)_0 of g constructed in a previous paper, and the structure of Frobenius algebra of (Ug)_0. This answers a question raised to the author by W. Soergel.
We prove an analogue of Koszul duality for category $\mathcal{O}$ of a reductive group $G$ in positive characteristic $\ell$ larger than 1 plus the number of roots of $G$. However there are no Koszul rings, and we do not prove an analogue of the Kazhdan--Lusztig conjectures in this context. The main technical result is the formality of the dg-algebra of extensions of parity sheaves on the flag variety if the characteristic of the coefficients is at least the number of roots of $G$ plus 2.
In this paper we prove that if G is a connected, simply-connected, semi-simple algebraic group over an algebraically closed field of sufficiently large characteristic, then all the blocks of the restricted enveloping algebra (Ug)_0 of the Lie algebra g of G can be endowed with a Koszul grading (extending results of Andersen, Jantzen and Soergel). We also give information about the Koszul dual rings. Our main tool is the localization theory in positive characteristic developed by Bezrukavnikov, Mirkovic and Rumynin.
We introduce the alternating Schur algebra $AS_F(n,d)$ as the commutant of the action of the alternating group $A_d$ on the $d$-fold tensor power of an $n$-dimensional $F$-vector space. When $F$ has characteristic different from $2$, we give a basis of $AS_F(n,d)$ in terms of bipartite graphs, and a graphical interpretation of the structure constants. We introduce the abstract Koszul duality functor on modules for the even part of any $\mathbf Z/2\mathbf Z$-graded algebra. The algebra $AS_F(n,d)$ is $\mathbf Z/2\mathbf Z$-graded, having the classical Schur algebra $S_F(n,d)$ as its even part. …
Building on the theory of parity sheaves due to Juteau-Mautner-Williamson, we develop a formalism of "mixed modular perverse sheaves" for varieties equipped with a stratification by affine spaces. We then give two applications: (1) a "Koszul-type" derived equivalence relating a given flag variety to the Langlands dual flag variety, and (2) a formality theorem for the modular derived category of a flag variety (extending a previous result of Riche-Soergel-Williamson).
Higher Koszul duality and connections with $n$-hereditary algebras
建立表示理论中 Koszul 对偶性与更高同调代数的联系，通过推广 T - Koszul 代数概念获更高版本对偶性，用于 n - 遗传代数，给出 n - T - Koszul 代数的刻画及相关结论。
We obtain Koszul-type dualities for categories of graded modules over a graded associative algebra which can be realized as the semidirect product of a bialgebra coinciding with its degree zero part and a graded module algebra for the latter. In particular, this applies to graded representations of the universal enveloping algebra of the Takiff Lie algebra (or the truncated current algebra) and its (super)analogues, and also to semidirect products of quantum groups with braided symmetric and exterior module algebras in case the latter are flat deformations of classical ones.
Koszul Duality for Quadratic Monomial Algebras
研究有限维二次单项式代数及其Koszul对偶的模结构，证明Koszul对偶是分级连贯且分级余连贯的，并推导出绝对Koszul性、全局线性缺陷至多为1等性质，进而细化导出Koszul对偶和Bernstein-Gelfand-Gelfand对应。
We show that the endomorphism ring of the projective generator in the category of Soergel modules (for dihedral groups) is Koszul self-dual.
Koszul duality for Coxeter groups
本文构造了Coxeter系统在给定实现与对偶实现下Hecke范畴之间的Koszul对偶等价，推广了几何语境下的相关工作，并证明倾斜反常层的组合学可由对偶实现下Hecke代数的典范基编码。
We show the existence of tilting objects in the singularity category $\mathsf{D}_{\mathsf{ Sg}}^{\mathsf{ gr}}(eAe)$ associated to certain noetherian AS-regular algebras $A$ and idempotents $e$. This gives a triangle equivalence between $\mathsf{D}_{\mathsf{ Sg}}^{\mathsf{ gr}}(eAe)$ and the derived category of a finite-dimensional algebra. In particular, we obtain a tilting object if the Beilinson algebra of $A$ is a levelled Koszul algebra. This generalises the existence of a tilting object in $\mathsf{D}_{\mathsf{ Sg}}^{\mathsf{ gr}}(S^G)$, where $S$ is a Koszul AS-regular algebra and $G$ i…
We use linear Koszul duality, a geometric version of the standard duality between modules over symmetric and exterior algebras studied in previous papers of the authors to give a geometric realization of the Iwahori-Matsumoto involution of affine Hecke algebras. More generally we prove that linear Koszul duality is compatible with convolution in a general context related to convolution algebras.
We introduce a cup-cap duality in the Koszul calculus of N-homogeneous algebras. As an application, we prove that the graded symmetry of the Koszul cap product is a consequence of the graded commutativity of the Koszul cup product. We propose a conceptual approach that may lead to a proof of the graded commutativity, based on derived categories in the framework of DG algebras and DG bimodules. Various enriched structures are developed in a weaker situation corresponding to N>2.
In this paper we continue the study (initiated in a previous article) of linear Koszul duality, a geometric version of the standard duality between modules over symmetric and exterior algebras. We construct this duality in a very general setting, and prove its compatibility with morphisms of vector bundles and base change.
We prove an analogon of Koszul duality for category O in positive characteristic. However, there are no Koszul rings, and we do not prove an analog of the Kazhdan-Lusztig conjectures in this context.
Koszul Duality for Quasi-split Real Groups
该研究构建了拟分裂实群的范畴化Vogan对偶，证明了Soergel猜想在拟分裂情形下的正确性。
In the first part of this paper the projective dimension of the structural modules in the BGG category $\mathcal{O}$ is studied. This dimension is computed for simple, standard and costandard modules. For tilting and injective modules an explicit conjecture relating the result to Lusztig's $\mathbf{a}$-function is formulated (and proved for type $A$). The second part deals with the extension algebra of Verma modules. It is shown that this algebra is in a natural way $\mathbb{Z}^2$-graded and that it has two $\mathbb{Z}$-graded Koszul subalgebras. The dimension of the space $\mathrm{Ext}^1$ int…
We prove that on a certain class of smooth complex varieties (those with "affine even stratifications"), the category of mixed Hodge modules is "almost" Koszul: it becomes Koszul after a few unwanted extensions are eliminated. We also give an equivalence between perverse sheaves on such a variety and modules for a certain graded ring, obtaining a formality result as a corollary. For flag varieties, these results were proved earlier by Beilinson-Ginzburg-Soergel using a rather different construction.
We construct an ungraded version of Beilinson-Ginzburg-Soergel's Koszul duality for Langlands dual flag varieties, inspired by Beilinson's construction of rational motivic cohomology in terms of $K$-theory. For this, we introduce and study categories $\operatorname{DK}_{\mathcal{S}}(X)$ of $\mathcal{S}$-constructible $K$-motivic sheaves on varieties $X$ with an affine stratification $\mathcal{S}$. We show that there is a natural and geometric functor, called Beilinson realisation, from $\mathcal{S}$-constructible mixed sheaves $\operatorname{D}^{mix}_{\mathcal{S}}(X)$ to $\operatorname{DK}_{\m…
A geometric realization of Koszul duality for graded gentle algebras
本文证明了同调光滑分次gentle代数的Koszul函子可由几何模型中的半旋转实现，并以此证明了涉及该函子的交维数公式。
We show that if $Λ$ is a $n$-Koszul algebra and $E=E(Λ)$ is its Yoneda algebra, then there is a full subcategory $\mathcal{L}_E$ of the category $Gr_E$ of graded $E$-modules, which contains all the graded $E$-modules presented in even degrees, that embeds fully faithfully into the category $C(Gr_Λ)$ of cochain complexes of graded $Λ$-modules. That extends the known equivalence, for $Λ$ Koszul (i.e. for $n=2$), between $Gr_E$ and the category of linear complexes of graded $Λ$-modules
A fundamental result of Beilinson-Ginzburg-Soergel states that on flag varieties and related spaces, a certain modified version of the category of l-adic perverse sheaves exhibits a phenomenon known as Koszul duality. The modification essentially consists of discarding objects whose stalks carry a nonsemisimple action of Frobenius. In this paper, we prove that a number of common sheaf functors (various pull-backs and push-forwards) induce corresponding functors on the modified category or its triangulated analogue. In particular, we show that these functors preserve semisimplicity of the Frobe…
Mixed modular perverse sheaves on affine flag varieties and Koszul duality
本文基于技术假设，证明了仿射旗流形上不可分解 tilting perverse sheaves 的重数公式，并解释了连接混合模 perverse sheaves 与普通 perverse sheaves 的“degrading functor”的构造。
The parabolic category $\mathcal{O}$ for affine ${\mathfrak{gl}}_N$ at level $-N-e$ admits a structure of a categorical representation of $\widetilde{\mathfrak{sl}}_e$ with respect to some endofunctors $E$ and $F$. This category contains a smaller category $\mathbf{A}$ that categorifies the higher level Fock space. We prove that the functors $E$ and $F$ in the category $\mathbf{A}$ are Koszul dual to Zuckerman functors. The key point of the proof is to show that the functor $F$ for the category $\mathbf{A}$ at level $-N-e$ can be decomposed in terms of the components of the functor $F$ for the…
Koszul duality for generalised Steinberg representations of $p$-adic groups
本文证明了 p-进半单群的广义 Steinberg 表示扩张代数模范畴与 Langlands 参数簇上等变反常层子范畴之间的 Koszul 对偶等价。
A Non-graded Koszul Duality and Its Applications
本文建立有限维Koszul代数的有界导出范畴在分次与非分次情形下的导出Koszul对偶，无需对Koszul对偶代数施加有限性条件，并应用于范畴O的块和几何表示论中的反常层。
We define and investigate a class of Koszul quasi-hereditary algebras for which there is a natural equivalence between the bounded derived category of graded modules and the bounded derived category of graded modules over (a proper version of) the extension algebra of standard modules. Examples of such algebras include, in particular, the multiplicity free blocks of the BGG category $\mathcal{O}$, and some quasi-hereditary algebras with Cartan decomposition in the sense of K{ö}nig.
Calabi-Yau property in derived Koszul calculus
本文证明了导出Koszul演算中导出函子的存在性，从而无需存在性假设即可定义强Kc-Calabi-Yau代数并得到对偶定理，并证明多项式代数是强Kc-Calabi-Yau的。
The paper studies quadratic and Koszul duality for modules over positively graded categories. Typical examples are modules over a path algebra, which is graded by the path length, of a not necessarily finite quiver with relations. We present a very general definition of quadratic and Koszul duality functors backed up by explicit examples. This generalises previous results in two substantial ways: We work in the setup of graded categories, i.e. we allow infinitely many idempotents and also define a ``Koszul'' duality functor for not necessarily Koszul categories. As an illustration of the techn…
This is a report on recent work of Chalupnik and Touze. We explain the Koszul duality for the category of strict polynomial functors and make explicit the underlying monoidal structure which seems to be of independent interest. Then we connect this to Ringel duality for Schur algebras and describe Serre duality for strict polynomial functors.
Given an affine hyperplane arrangement with some additional structure, we define two finite-dimensional, noncommutative algebras, both of which are motivated by the geometry of hypertoric varieties. We show that these algebras are Koszul dual to each other, and that the roles of the two algebras are reversed by Gale duality. We also study the centers and representation categories of our algebras, which are in many ways analogous to integral blocks of category O.
Koszul duality and a conjecture of Francis-Gaitsgory
本文研究Koszul对偶，证明其在零完备代数与余零完备余代数的最大可能子范畴间等价，否定Francis-Gaitsgory猜想，且可推导诸多已知部分结果。
In this paper, we define $A_{\infty}$-Koszul duals for directed $A_{\infty}$-categories in terms of twists in their $A_{\infty}$-derived categories. Then, we compute a concrete formula of $A_{\infty}$-Koszul duals for path algebras with directed $A_n$-type Gabriel quivers. To compute an $A_\infty$-Koszul dual of such an algebra $A$, we construct a directed subcategory of a Fukaya category which are $A_\infty$-derived equivalent to the category of $A$-modules and compute Dehn twists as twists. The formula unveils all the ext groups of simple modules of the parh algebras and their higher composi…
Non-homogeneous Koszul duality in representation theory
本文研究相伴分次为Koszul的滤过代数，证明其无界导出范畴等价于Positselski弯曲dg对偶代数上内射模同伦范畴的显式商范畴，推广并恢复了Positselski的结果。
We describe several general methods for calculating weights of mixed tilting sheaves. We introduce a notion called "non-cancellation property" which implies a strong uniqueness of mixed tilting sheaves and enables one to calculate their weights effectively. When we have a certain Radon transform, we prove a geometric analogue of Ringel duality which sends tilting objects to projective objects. We apply these methods to (partial) flag varieties and affine (partial) flag varieties and show that the weight polynomials of mixed tilting sheaves on flag and affine flag varieties are essentially give…
For any Kac-Moody group $G$ with Borel $B$, we give a monoidal equivalence between the derived category of $B$-equivariant mixed complexes on the flag variety $G/B$ and (a certain completion of) the derived category of $B^\vee$-monodromic mixed complexes on the enhanced flag variety $G^\vee/U^\vee$, here $G^\vee$ is the Langlands dual of $G$. We also prove variants of this equivalence, one of which is the equivalence between the derived category of $U$-equivariant mixed complexes on the partial flag variety $G/P$ and certain "Whittaker model" category of mixed complexes on $G^\vee/B^\vee$. In …
For any dg algebra $A$ we construct a closed model category structure on dg $A$-modules such that the corresponding homotopy category is compactly generated by dg $A$-modules that are finitely generated and free over $A$ (disregarding the differential). We prove that this closed model category is Quillen equivalent to the category of comodules over a certain, possibly nonconilpotent dg coalgebra, a so-called extended bar construction of $A$. This generalises and complements certain aspects of dg Koszul duality for associative algebras.
We study symplectic invariants of the open symplectic manifolds $X_Γ$ obtained by plumbing cotangent bundles of 2-spheres according to a plumbing tree $Γ$. For any tree $Γ$, we calculate (DG-)algebra models of the Fukaya category $\mathcal{F}(X_Γ)$ of closed exact Lagrangians in $X_Γ$ and the wrapped Fukaya category $\mathcal{W}(X_Γ)$. When $Γ$ is a Dynkin tree of type $A_n$ or $D_n$ (and conjecturally also for $E_6,E_7,E_8$), we prove that these models for the Fukaya category $\mathcal{F}(X_Γ)$ and $\mathcal{W}(X_Γ)$ are related by (derived) Koszul duality. As an application, we give explicit…
Equivariant Koszul Duality, Modular Category $\mathcal{O}$, and Periodic Kazhdan--Lusztig Polynomials
本文在假设卢斯滕堡猜想成立的条件下，推导了模范畴O中延伸群维数的公式，并利用环面等变科兹尔双模理论，通过周期性卡兹德-卢斯滕堡多项式系数来表达结果。
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