Generalizing a concept of Lipshitz, Ozsváth and Thurs-ton from Bordered Floer homology, we define $D$-structures on algebras of unital operads, which can also be interpreted as a generalization of a seemingly unrelated concept of Getzler and Jones. This construction gives rise to an equivalence of derived categories, which can be thought of as a unital version of Koszul duality using non-unital Quillen homology. We also discuss a multi-sorted version of the construction, which provides a framework for unifying the known algebraic contexts of Koszul duality.
We introduce here the notion of Koszul duality for monoids in the monoidal category of species with respect to the ordinary product. To each Koszul monoid we associate a class of Koszul algebras in the sense of Priddy, by taking the corresponding analytic functor. The operad $\mathscr{A}_M$ of rooted trees enriched with a monoid $M$ was introduced by the author many years ago. One special case of that is the operad of ordinary rooted trees, called in the recent literature the permutative non associative operad. We prove here that $\mathscr{A}_M$ is Koszul if and only if the corresponding monoi…
We prove that the operad of mock partially associative $n$-ary algebras is not Koszul, as conjectured by the second and the third author in 2009, and utilise the Zeilberger's algorithm for hypergeometric summation to demonstrate that non-Koszulness of that operad cannot be established by hunting for negative coefficients in the inverse of its Poincaré series.
We define, for a somewhat standard forgetful functor from nonsymmetric operads to weight graded associative algebras, two functorial "enveloping operad" functors, the right inverse and the left adjoint of the forgetful functor. Those functors turn out to be related by operadic Koszul duality, and that relationship can be utilised to provide examples showing limitations of two standard tools of the Koszul duality theory. We also apply these functors to get a homotopical algebra proof of the Lagrange inversion formula.
We investigate operads for various n-ary algebras. As a useful tool we introduce galgalim - analogs of the Lie-hedra for n-ary algebras. We then focus to algebras with one anti-associative operation. We describe the relevant part of the deformation cohomology for this type of algebras using the minimal model for the anti-associative operad. We also discuss free partially associative algebras and formulate some open problems.
In our earlier work we studied Koszulity of a family of operads depending on a natural number n and on the degree d of the generating operation. While we proved that, for n < 8, this operad is Koszul if and only if d is even, and while it follows from a work of Hoffbeck that it is Koszul for d even and arbitrary n, its (non)Koszulity for d odd and n > 7 remains an open problem. In this note, we describe some related numerical experiments, and formulate a conjecture suggested by the results of these computations.
We provide bar and cobar constructions as functors acting between various categories of curved operads and curved cooperads. Cobar and bar constructions are adjoint to each other. Given a twisting cochain between a curved augmented cooperad C with an extra grading and a curved operad O we construct a couple of adjoint functors between the category of curved O-modules and the category of curved C-comodules. The important feature is that the curved operad O is not necessarily augmented.
We extend the Koszul duality theory of associative algebras to algebras over an operad. Recall that in the classical case, this Koszul duality theory relies on an important chain complex: the Koszul complex. We show that the cotangent complex, involved in the cohomology theory of algebras over an operad, generalizes the Koszul complex.
It is well known that the differential graded operad of A_infinity-algebras is a cofibrant replacement (a dg-resolution) of the operad of associative differential graded algebras without units. In this article we find a cofibrant replacement of the operad of associative differential graded algebras with units. Algebras over it are called homotopy unital A_infinity-algebras. We prove that the operad bimodule of A_infinity-morphisms is a cofibrant replacement of the operad bimodule of morphisms of dg-algebras without units. Similarly we show that the operad bimodule of homotopy unital A_infinity…
We extend the bar-cobar adjunction to operads and properads, not necessarily augmented. Due to the default of augmentation, the objects of the dual category are endowed with a curvature. We handle the lack of augmentation by extending the category of coproperads to include objects endowed with a curvature. As usual, the bar-cobar construction gives a (large) cofibrant resolution for any properad, such as the properad encoding unital and counital Frobenius algebras, a notion which appears in 2d-TQFT. We also define a curved Koszul duality theory for operads or properads presented with quadratic…
We define the $m$th Veronese power of a weight graded operad $\mathcal{P}$ to be its suboperad $\mathcal{P}^{[m]}$ generated by operations of weight $m$. It turns out that, unlike Veronese powers of associative algebras, homological properties of operads are, in general, not improved by this construction. However, under some technical conditions, Veronese powers of quadratic Koszul operads are meaningful in the context of the Koszul duality theory. Indeed, we show that in many important cases the operads $\mathcal{P}^{[m]}$ are related by Koszul duality to operads describing strongly homotopy …
This paper gives a systematic study of matching dialgebras corresponding to the operad $As^{(2)}$ in \cite{Zi} as the only Koszul self dual operad there other than the operads of associative algebras and Poisson algebras. The close relationship of matching dialgebras with semi-homomorphisms and matched pairs of associative algebras are established. By anti-symmetrizing, matching dialgerbas are also shown to give compatible Lie algebras, pre-Lie algebras and PostLie algebras. By the rewriting method, the operad of matching dialgebras is shown to be Koszul and the free objects are constructed in…
In this paper, we study weakly unital dg categories as they were defined by Kontsevich and Soibelman [KS, Sect.4]. We construct a cofibrantly generated Quillen model structure on the category $\mathrm{Cat}_{\mathrm{dgwu}}(\Bbbk)$ of small weakly unital dg categories over a field $\Bbbk$. Our model structure can be thought of as an extension of the model structure on the category $\mathrm{Cat}_{\mathrm{dg}}(\Bbbk)$ of (strictly unital) small dg categories over $\Bbbk$, due to Tabuada [Tab]. More precisely, we show that the imbedding of $\mathrm{Cat}_{\mathrm{dg}}(\Bbbk)$ to $\mathrm{Cat}_{\math…
We show that the family of chain modules over the standard simplices can be equipped with an operad structure. Similarly, the family of cochain modules of the Stasheff polytopes can be equipped with an operad structure. We first show that these operads are Koszul dual to each other, and second that they are both Koszul operads. The algebras over the standard simplices operad, called associative trialgebras, are defined by 3 operations and 11 relations. The algebras over the Stasheff polytopes operad, called dendriform trialgebras, are defined by 3 operations and 7 relations.
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 define a generalization of Massey products for algebras over a Koszul operad in characteristic zero, extending Massey's and Allday's and Retah's in the associative and Lie cases, respectively. We establish connections with minimal models and with Dimitrova's universal operadic cohomology class. We compute a Gerstenhaber algebra example and a hypercommutative algebra example related to the Chevalley-Eilenberg complex of the Heisenberg Lie algebra.
The operads of Poisson and Gerstenhaber algebras are generated by a single binary element if we consider them as Hopf operads (i.e. as operads in the category of cocommutative coalgebras). In this note we discuss in details the Hopf operads generated by a single element of arbitrary arity. We explain why the dual space to the space of $n$-ary operations in this operads are quadratic and Koszul algebras. We give the detailed description of generators, relations and a certain monomial basis in these algebras.
This paper shows that the operad encoding modular operads is Koszul. Using this result we construct higher composition operations on (hairy) graph homology which characterize its rational homotopy type.
We study the André-Quillen cohomology with coefficients of an algebra over an operad. Using resolutions of algebras coming from Koszul duality theory, we make this cohomology theory explicit and we give a Lie theoretic interpretation. For which operads is the associated André-Quillen cohomology equal to an Ext-functor ? We give several criterion, based on the cotangent complex, to characterize this property. We apply it to homotopy algebras, which gives a new homotopy stable property for algebras over cofibrant operads.
We define a strong homotopy derivation of (cohomological) degree k of a strong homotopy algebra over an operad P. This involves resolving the operad obtained from P by adding a generator with "derivation relations". For a wide class of Koszul operads P, in particular Ass and Lie, we describe the strong homotopy derivations by coderivations and show that they are closed under the Lie bracket. We show that symmetrization of a strong homotopy derivation of an A-infinity algebra yields a strong homotopy derivation of the symmetrized L-infinity algebra. We give examples of strong homotopy derivatio…
We investigate algebras with one operation. We study when these algebras form a monoidal category and analyze Koszulness and cyclicity of the corresponding operads. We also introduce a new kind of symmetry for operads, the dihedrality, responsible for the existence of the dihedral cohomology. The main trick, which we call the polarization, will be to represent an algebra with one operation without any specific symmetry as an algebra with one commutative and one anticommutative operation. We will try to convince the reader that this change of perspective might sometimes lead to new insights and…
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…
