A. S. Dzhumadil'daev
An algebra with identities $a\circ(b\circ c-c\circ b)=(a\circ b)\circ c-(a\circ c)\circ b$ and $a\circ(b\circ c)=b\circ(a\circ c)$ is called Novikov. We show that Novikov operad is not Koszul.
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We describe the Koszul dual of two quadratic operads on planar forests introduced to study the infinitesimal Hopf algebra of planar rooted trees and prove that these operads are Koszul.
$δ$-Mock-Novikov Algebras: Structural Theory, Operad and Poisson-Type Constructions
本文引入参数依赖的 $\delta$-mock-Novikov 代数，研究其结构、operad 性质及 Poisson 型构造，证明有限维情形在特征零下幂零并完成低维分类。
Varieties of initial dialgebras and some of their Koszul dual operads
本文提出了一种通用算法，用于从给定的变种Var中构造一个子变种，从而可以恢复属于Var的代数。同时，本文还构建了自由初等李和结合二代数的基础。
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.
The purpose of this paper is to prove the koszulity of the operad ${\mathcal Quad}$, governing quadri-algebras. That ${\mathcal Quad}$ is Koszul was conjectured by Aguiar and Loday. The operad ${\mathcal Dend}$, governing dendriform algebras, is known to be Koszul, by work of Loday, and ${\mathcal Quad}$ is its second black square power. We find a new complex, based on the associahedron, which captures the structure of ${\mathcal Dend}$. This complex behaves well with respect to the black squaring process, and allows us to conclude. Also, this proves koszulity of higher powers of ${\mathcal De…
This paper studies the operad of linearly compatible di-algebras, denoted by $As^{2}$, which is a nonsymmetric operad encoding the algebras with two binary operations that satisfy individual and sum associativity conditions. We also prove that the operad $As^{2}$ is exactly the Koszul dual operad of the operad $^{2}As$ encoding totally compatible di-algebras. We show that the operads $As^{2}$ and $^{2}As$ are Koszul by rewriting method. We make explicit the Homotopy Transfer Theorem for $As^{2}$-algebras.
The operad of anti-pre-Lie algebras and its Lie elements
我们开启了针对反预李代数的系统 operadic 研究方法。研究了控制反预李代数的 operad、其与 Novikov 代数 operad 的关系及其循环结构。证明反预李代数的 Lie 括号满足五次标准恒等式，还证明反预李 operad 相对于该恒等式具有二次极大性。随后表明反预李 operad 及其 Koszul 对偶均非 Koszul，并完整描述了后者。最后，将 Koszul 对偶 operad 中的 Lie 元素与 metabelian Lie 代数的 operad 对应起来。
Using computer calculations, we prove the statement in the title.
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。
Operad and cohomology of associative algebras with generalized derivations
本文研究带广义导子的结合代数（AsGDer 三元组），引入并证明其编码 operad 为 Koszul operad，进而构造同伦版本与 AsGDer 上同调理论，并证明该上同调控制 AsGDer 三元组的形式形变。
On the differential Novikov algebras
本文研究带有导子的 Novikov 代数及其对偶 operad，发现该对偶 operad 与双交换代数存在联系，研究动机源于 Novikov operad 的白与黑 Manin 乘积。
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。
We present a study of quadratic operads for n-ary algebras and their dual for n odd. We will focus on the ternary case (i.e n=3). The aim is to underline the problem of computing the dual operad and the fact that this last is in general defined in the graded differential operad framework. We prove that the operad associated to 3-ary partially associative algebra is not Koszul. Recall that, if n is even, the operad of n-ary partially associative algebras is Koszul.
The present article exploits the fact that permutads (aka shuffle algebras) are algebras over a terminal operad in a certain operadic category Per. In the first, classical part we formulate and prove a claim envisaged by Loday and Ronco that the cellular chains of the permutohedra form the minimal model of the terminal permutad which is moreover, in the sense we define, self-dual and Koszul. In the second part we study Koszulity of Per-operads. Among other things we prove that the terminal Per-operad is Koszul self-dual. We then describe strongly homotopy permutads as algebras of its minimal m…
$n\text{-}Lie_d$ Operad and its Koszul Dual
本文研究 $n\text{-}Lie_d$ 算子及其 Koszul 对偶 $n\text{-}Com_{-d+n-2}$，通过 Specht 模和奇图谱揭示两者联系。
Distributive lattices of varieties of Novikov algebras
本文证明诺维科夫代数变种的可分配格当且仅当其三度恒等式定义的子变种格可分配，回答了布库特五十年前关于诺维科夫代数的问题。
On Pre-Novikov Algebras and Derived Zinbiel Variety
本文研究代数簇及其二元 operad 的导出构造，给出导出簇代数可嵌入相应微分代数的充分条件，并证明 Zinbiel 导出簇即 pre-Novikov 代数类中存在不可嵌入带导子 Zinbiel 代数的例子。
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.
On the Dong Property for a binary quadratic operad
本文研究了二元非结合代数满足Dong引理的条件，证明Novikov和Novikov-Poisson代数满足该引理，并通过操作代数语言确定了Dong引理成立的操作代数类型。
Induced structures of operated algebras with applications to multi-Novikov algebras
本文提出操作代数诱导结构的概念，统一了各类构造，并应用于Novikov代数和多Novikov代数的定义。
We study diverse parametrized versions of the operad of associative algebra, where the parameter are taken in an associative semigroup $Ω$ (generalization of matching or family associative algebras) or in its cartesian square (two-parameters associative algebras). We give a description of the free algebras on these operads, study their formal series and prove that they are Koszul when the set of parameters is finite. We also study operadic morphisms between the operad of classical associative algebras and these objects, and links with other types of algebras (diassociative, dendriform, post-Li…
Higher order Massey products for algebras over algebraic operads
本文为代数operad上的代数引入高阶Massey积，推广了Muro的次级积，研究其性质、形式性联系，并证明其表示operadic Eilenberg--Moore谱序列的微分，同时探讨与P∞结构的关系，涵盖Koszul及非Koszul情形。
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。
In this paper we study symmetric Leibniz and related algebras, namely symmetric dialgebras and symmetric Perm-algebras. We also calculate their Koszul duals, if not known. This will give us Lie-admissible algebras and new types of algebras, which we call commutative and associative admissible algebras. Finally we prove that all operads describing the considered algebras are Koszul.
White Manin product and Hadamard product
本文证明 alternative、assosymmetric 和 bicommutative 三种 operad 与 Novikov operad 的 Hadamard 积等于 white Manin 积，并据此识别出一类所有代数均为特殊的代数簇。
WWe describe the Koszul dual of the operad Quad of quadri-algebras, show the koszularity of Quad and give the formal series of Quad and its dual, which proves a conjecture due to Aguiar and Loday. A notion of quadri-bialgebra is also introduced, with applications to the Hopf algebras FQSym and WQSym.
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.
Cohomology theory of Novikov algebras and applications
本文通过 operad 方法刻画 Novikov 代数上同调，给出显式链复形，并证明其与底层 pre-Lie 代数上同调存在长正合序列，且第二上同调群分类无穷小形变与 Abel 扩张。
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 Pre-Lie algebra is a vector space L endowed with a bilinear product * : L \times L to L satisfying the relation (x*y)*z-x*(y*z)= (x*z)*y-x*(z*y), for all x,y,z in L. We give an explicit combinatorial description in terms of rooted trees of the operad associated to this type of algebras and prove that it is a Koszul operad.
Anti-successors and anti-replicators, Manin products and Koszul duality
本文针对二元 operad 发展反分裂与反复制的 operadic 理论，定义反双后继、反三后继等结构，建立其与曼宁积、Koszul 对偶的关系，恢复已知反分裂结构并生成新实例。
The cup product in the cohomology of algebras over quadratic operads has been studied in the general setting of Koszul duality for operads. We study the cup product on the cohomology of n-ary totally associative algebras with an operation of even (homological) degree. This cup product endows the cohomology with the structure of an n-ary partially associative algebra with an operation of even or odd degree depending on the parity of n. In the cases n = 3 and n = 4, we provide an explicit definition of this cup product and prove its basis properties.
$δ$-Novikov and $δ$-Novikov--Poisson algebras
本文研究 $δ$-Novikov 代数及其 Poisson 推广，刻画交换子结构、嵌入关系与相关构造，并证明对应 operad 对任意 $δ$ 均非 Koszul。
A Lie-admissible algebra gives by anticommutativity a Lie algebra. In this work we study remarkable classes of Lie-admissible algebras such as Vinberg, PreLie algebras. We compute the corresponding binary quadratic operads and study their Koszul duality. Considering Lie algebras as Lie-admissible algebras we can define for each Lie algebra a cohomology with values in a Lie-admissible module. This permits to study some deformations of Lie algebras in the category of Lie-admissible algebras. Lastly we study the tensor product between these operads and their dual operads. As application we constr…
The minimal model of Rota-Baxter operad with arbitrary weight
本文从operad视角研究任意权重Rota-Baxter结合代数，构造同伦余operad作为Koszul对偶，证明其cobar构造即最小模型，并给出同伦Rota-Baxter结合代数及形变复形。
The Deligne-Getzler-Hinich--$\infty$-groupoid or Maurer-Cartan simplicial set of an $L_\infty$-algebra plays an important role in deformation theory and many other areas of mathematics. Unfortunately, this construction only works over a field of characteristic $0$. The goal of this paper is to show that the notions of Maurer-Cartan equation and Maurer-Cartan simplicial set can be defined for a much larger number of operads than just the $L_\infty$-operad. More precisely, we show that the Koszul dual of every unital Hopf cooperad (a cooperad in the category of unital associative algebras) with …
The Weight Criterion, Dong Property, and Manin Products of Operads
本文研究二元 operad 的 Hadamard 积与 Manin 白积相等的条件，等价于二次部分控制 2-簇及 Dong 性质，并区分白积与其二次近似。
Unifying Koszul dualities via point-set models
本文通过点集模型构建了一个统一的框架，将经典链级构造与更高范畴的构造联系起来，恢复了Lurie的bar-cobar对偶性，并解释了与Francis-Gaitsgory和Heuts的构造的关系。
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