Gunnar Floystad
Let A and A! be dual Koszul algebras. By Positselski a filtered algebra U with gr U = A is Koszul dual to differential graded algebra (A!,d). We relate the module categories of this dual pair by a tensor-Hom adjunction. This descends to give an equivalence of suitable quotient categories and generalizes work of Beilinson, Ginzburg and Soergel.
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Let $A$ be a Koszul Calabi-Yau algebra. We show that there exists an isomorphism of Batalin-Vilkovisky algebras between the Hochschild cohomology ring of $A$ and that of its Koszul dual algebra $A^!$. This confirms (a generalization of) a conjecture of R.~Rouquier.
We discuss the notion of Poincaré duality for graded algebras and its connections with the Koszul duality for quadratic Koszul algebras. The relevance of the Poincaré duality is pointed out for the existence of twisted potentials associated to Koszul algebras as well as for the extraction of a good generalization of Lie algebras among the quadratic-linear algebras.
In this paper we prove a version of curved Koszul duality for Z/2Z-graded curved coalgebras and their coBar differential graded algebras. A curved version of the homological perturbation lemma is also obtained as a useful technical tool for studying curved (co)algebras and precomplexes. The results of Koszul duality can be applied to study the category of matrix factorizations MF(R,W). We show how Dyckerhoff's generating results fit into the framework of curved Koszul duality theory. This enables us to clarify the relationship between the Borel-Moore Hochschild homology of curved (co)algebras …
We study the curved Koszul duality theory for associative algebras presented by quadratic-linear-constant (QLC) relations. As an application, we investigate the cyclic (co)homology of a QLC algebra and its Koszul dual curved DG algebra, and extend a result due to Feigin and Tsygan. We also study a commutative/Lie analog of this curved Koszul duality theory.
We identify two Batalin-Vilkovisky algebra structures, one obtained by Kowalzig and Krahmer on the Hochschild cohomology of an Artin-Schelter regular algebra with semisimple Nakayama automorphism and the other obtained by Lambre, Zhou and Zimmermann on the Hochschild cohomology of a Frobenius algebra also with semisimple Nakayama automorphism, provided that these two algebras are Koszul dual to each other.
Diagonalizable modular derivations on bigraded Poisson spaces and Koszul duality
本文证明在模导子可对角化条件下，Koszul对偶诱导双分次泊松空间泊松上同调间的Batalin--Vilkovisky代数同构，涵盖非幺模情形。
We present a unifying framework for the key concepts and results of higher Koszul duality theory for N-homogeneous algebras: the Koszul complex, the candidate for the space of syzygies, and the higher operations on the Yoneda algebra. We give a universal description of the Koszul dual algebra under a new algebraic structure. For that we introduce a general notion: Gröbner bases for algebras over non-symmetric operads.
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.
Koszul-Morita duality
本研究推广了Keller–Lefèvre意义下的Koszul对偶，使其适用于未必增广的代数，且该对偶与经典Morita对偶密切相关，特定情形下可退化为Morita对偶。
In the first part of this paper, we study Koszul property of directed graded categories. In the second part of this paper, we prove a general criterion for an infinite directed category to be Koszul. We show that infinite directed categories in the theory of representation stability are Koszul over a field of characteristic zero.
We discuss the consequences of the Poincaré duality, versus AS- Gorenstein property, for Koszul algebras (homogeneous and non homogeneous). For homogeneous Koszul algebras, the Poincaré duality property implies the existence of twisted potentials which characterize the corresponding algebras while in the case of quadratic linear Koszul algebras, the Poincaré duality is needed to get a good generalization of universal enveloping algebras of Lie algebras. In the latter case we describe and discuss the corresponding generalization of Lie algebras. We also give a short review of the notion of Kosz…
We study the noncommutative Poincaré duality between the Poisson homology and cohomology of unimodular Poisson algebras, and show that Kontsevich's deformation quantization as well as Koszul duality preserve the corresponding Poincaré duality. As a corollary, the Batalin-Vilkovisky algebra structures that naturally arise in these cases are all isomorphic.
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。
Associator dependent algebras and Koszul duality
本文解决了一个十年未解的问题，通过研究伴随依赖代数中的科兹尔操作符，确定了作用于八元数代数的科兹尔操作符。
Koszul Monoids in Quasi-abelian Categories
本文在具有足够平坦投射对象的双完备闭对称幺半拟阿贝尔范畴中，以无元素方式定义 Koszul 与二次幺半群，并借助左心嵌入证明其分次模导出范畴子范畴的对偶等价。
We prove a version of Koszul duality and the induced derived equivalence for Adams connected $A_\infty$-algebras that generalizes the classical Beilinson-Ginzburg-Soergel Koszul duality. As an immediate consequence, we give a version of the Bernšte{\uı}n-Gel'fand-Gel'fand correspondence for Adams connected $A_\infty$-algebras. We give various applications. For example, a connected graded algebra $A$ is Artin-Schelter regular if and only if its Ext-algebra $\Ext^\ast_A(k,k)$ is Frobenius. This generalizes a result of Smith in the Koszul case. If $A$ is Koszul and if both $A$ and its Koszul dual…
In this paper we study a category of trees TI and prove that it is a Koszul category. Consequences are the interpretation of the reduced bar construction of operads of Ginzburg and Kapranov as the Koszul complex of this category, and the interpretation of operads up to homotopy as a functor from the minimal resolution of TI to the category of graded vector spaces. We compare also three different bar constructions of operads. Two of them have already been compared by Shnider-Von Osdol and Fresse.
Koszul duality and the link surgery formula
本文描述了编码Heegaard Floer Dehn手术公式的结合代数的Koszul对偶，构造了对偶化双模并证明范畴等价，为链环手术公式的计算问题提供应用。
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
We define generalized Koszul modules and rings and develop a generalized Koszul theory for $\mathbb{N}$-graded rings with the degree zero part noetherian semiperfect. This theory specializes to the classical Koszul theory for graded rings with degree zero part artinian semisimple developed by Beilinson-Ginzburg-Soergel and the ungraded Koszul theory for noetherian semiperfect rings developed by Green and Martin{é}z-Villa. Let $A$ be a left finite $\mathbb{N}$-graded ring generated in degree $1$ with $A_0$ noetherian semiperfect, $J$ be its graded Jacobson radical and $S=A/J$. By the Koszul dua…
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…
In this paper we study the homology of 2 versions of the swiss-cheese operad. We prove that the zeroth homology of these two versions are Koszul operads and relate this to strong homotopy Lebiniz pairs and OCHA, defined by Kajiura and Stasheff.
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.
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 book contains a detailed exposition of the nonhomogeneous Koszul duality theory in the relative situation over a noncentral, noncommutative, nonsemisimple base ring, as announced in Section 0.4 of arXiv:0708.3398. We prove the Poincare-Birkhoff-Witt theorem in this context and construct the triangulated equivalences of derived Koszul duality. The duality between the ring of differential operators and the de Rham DG-algebra, with the ring of functions as the base ring, is the thematic example. The moderate generality level makes the exposition in this book more accessible than the very hea…
This paper can be thought of as an extended introduction to arXiv:0708.3398; nevertheless, most of its results are not covered by loc. cit. We consider the derived categories of DG-modules, DG-comodules, and DG-contramodules, the coderived and contraderived categories of CDG-modules, the coderived categories of CDG-comodules, and the contraderived categories of CDG-contramodules. The equivalence between the latter two categories (the comodule-contramodule correspondence) is established. Nonhomogeneous Koszul duality or "triality" (an equivalence between exotic derived categories corresponding …
We present a new calculus which is well-adapted to quadratic algebras. This calculus consists in Koszul (co)homology, together with Koszul cup and cap products. Some applications are given. Koszul duality for Koszul (co)homology is proved for any quadratic algebra.
Koszul property was generalized to homogeneous algebras of degree N>2 in [5], and related to N-complexes in [7]. We show that if the N-homogeneous algebra A is generalized Koszul, AS-Gorenstein and of finite global dimension, then one can apply the Van den Bergh duality theorem [23] to A, i.e., there is a Poincare duality between Hochschild homology and cohomology of A, as for N=2.
We generalize the theory of Koszul complexes and Koszul algebras (in particular, Koszul duality between symmetric and exterior algebras) to symmetric tensor categories. In characteristic $p\ge 5$, this theory exhibits peculiar effects, not observed in the classical theory. In particular, we show that the symmetric and exterior algebras of a non-invertible simple object in the Verlinde category ${\rm Ver}_p$ are almost Koszul (although not Koszul), and show how this gives examples of $(r,s)$-Koszul algebras with any $r,s\ge 2$. We also develop a theory of Lie algebras in symmetric tensor catego…
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。
Differential calculus on bigraded spaces and Koszul duality
本文研究了双格次空间上的泊松结构与科兹尔对偶，证明了两个代数在科兹尔对偶下保持二次泊松结构，并展示了相应的微分结构和巴塔林-维洛维克代数结构的同构性。
Koszul duality, minimal model and $L_\infty$-structure for differential algebras with weight
本文继续Loday的研究，为非零权情况下的微分代数operad建立了Koszul对偶，确定了极小模型与Koszul对偶同伦cooperad，并推导出带权同伦微分代数概念及其形变复形的$L_\infty$-代数结构。
Let $A$ be a graded algebra. In this paper we develop a generalized Koszul theory by assuming that $A_0$ is self-injective instead of semisimple and generalize many classical results. The application of this generalized theory to directed categories and finite EI categories is described.
There are many structures (algebras, categories, etc) with natural gradings such that the degree 0 components are not semisimple. Particular examples include tensor algebras with non-semisimple degree 0 parts, extension algebras of standard modules of standardly stratified algebras. In this thesis we develop a generalized Koszul theory for graded algebras (categories) whose degree 0 parts may be non-semisimple. Under some extra assumption, we show that this generalized Koszul theory preserves many classical results such as the Koszul duality. Moreover, it has some close relation to the classic…
Koszul duality between Higgs and Coulomb categories $\mathcal{O}$
本研究证明了量子化Coulomb分支权模范畴与$G$-等变D-模范畴的Koszul对偶，构造了广义范畴$\u0026#x27;O$间的对偶函子，在箭簇簇等情形下为等价，且与扭、混洗函子交错，证实了辛对偶猜想的核心部分。
Differential graded Koszul duality: an introductory survey
本文概述了域上的导出非齐次Koszul对偶，总结了DG-代数与弯曲DG-余代数间的协变对偶及相关三元性，旨在作为教学导论。
Deformation theory and Koszul duality for Rota-Baxter systems
本文从operad理论视角研究Rota-Baxter系统，构造其极小模型与Koszul对偶同伦cooperad，导出同伦Rota-Baxter系统及控制形变的$L_\infty$-代数，并引入infinity-Yang-Baxter pairs以推广经典对应关系。
A DG algebras $A$ over a field $k$ with $H(A)$ connected and $H_{<0}(A)=0$ has a unique up to isomorphism DG module $K$ with $H(K)\cong k$. It is proved that if $H(A)$ is degreewise finite, then $RHom_A(?,K): D^{df}_{+}(A)^{op} \equiv D_{df}^{+}}(RHom_A(K,K))$ is an exact equivalence of derived categories of DG modules with degreewise finite-dimensional homology. It induces an equivalences of $D^{df}_{b}(A)^{op}$ and the category of perfect DG $RHom_A(K,K)$-modules, and vice-versa. Corresponding statements are proved also when $H(A)$ is simply connected and $H^{<0}(A)=0$.
Compatible structures of operads by polarization, their Koszul duality and Manin products
该研究将不变量理论的极化推广到operad层面，统一了线性、匹配、全相容三类operad结构，揭示了其Koszul对偶性质与Manin乘积构造，且部分有限生成二元二次operad的Koszul性在相容构造中保持。
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