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 formalism of `non-commutative Symplectic geometry'. We show that this formalism acquires its most adequate and natural formulation in the much more general framework of P-geometry, a `non-commutative geometry' for an algebra over an arbitrary cyclic Koszul operad.
We introduce a construction that associates, to each finite dimensional k-vector space V, a family of projective k-varieties that comes equipped with the structure of a operad in the category of k-schemes. When dim V = 1, this operad contains, as a suboperad, the family of moduli spaces of stably marked rational curves. For general V, our operad contains the family of Chen-Gibney-Krashen moduli spaces of stably marked trees of projective spaces as a suboperad. We realize our operad as part of a larger theory that describes how to construct operads from suitable functors. Given a category C tha…
Modular operads relevant to string theory can be equipped with an additional structure, coming from the connected sum of surfaces. Motivated by this example, we introduce a notion of connected sum for general modular operads. We show that a connected sum induces a commutative product on the space of functions associated to the modular operad. Moreover, we combine this product with Barannikov's non-commutative Batalin-Vilkovisky structure present on this space of functions, obtaining a Beilinson-Drinfeld algebra. Finally, we study the quantum master equation using the exponential defined using …
We prove that a connected commutator (or NC) complete associative algebra can be recovered in the derived setting from its abelianization together with its natural induced structure. Specifically, we prove an equivalence between connected derived commutator complete associative algebras and connected commutative algebras equipped with a coaction of the comonad arising from the adjunction between associative and commutative algebras. This provides a Koszul dual description of connected derived commutator (or NC) complete associative algebras and furthermore may be interpreted as a theory of rel…
Using the combinatorial species setting, we propose two new operad structures on multigraphs and on pointed oriented multigraphs. The former can be considered as a canonical operad on multigraphs, directly generalizing the Kontsevich-Willwacher operad, and has many interesting suboperads. The latter is a natural extension of the pre-Lie operad in a sense developed here and related to the multigraph operad. We study some of the finitely generated suboperads of the multigraph operad and establish links between them and the commutative operad and the commutative magmatic operad.
In this paper, we develop a differential-graded symplectic (Batalin-Vilkovisky) version of the framework of Crawley-Boevey, Etingof and Ginzburg on noncommutative differential geometry based on double derivations to construct non-commutative analogues of the Courant algebroids introduced by Liu, Weinstein and Xu. Adapting geometric constructions of Ševera and Roytenberg for (commutative) graded symplectic supermanifolds, we express the BRST charge, given in our framework by a `homological double derivation', in terms of Van den Bergh's double Poisson algebras for graded bi-symplectic non-commu…
The classical Hochschild--Kostant--Rosenberg (HKR) theorem computes the Hochschild homology and cohomology of smooth commutative algebras. In this paper, we generalise this result to other kinds of algebraic structures. Our main insight is that producing HKR isomorphisms for other types of algebras is directly related to computing quasi-free resolutions in the category of left modules over an operad; we establish that an HKR-type result follows as soon as this resolution is diagonally pure. As examples we obtain a permutative and a pre-Lie HKR theorem for smooth commutative and smooth brace al…
Let $A$ be an algebra over an operad in a cocomplete closed symmetric monoidal category. We study the category of $A$-modules. We define certain symmetric product functors of such modules generalising the tensor product of modules over commutative algebras, which we use to define the notion of a jet module. This in turn generalises the notion of a jet module over a module over a classical commutative algebra. We are able to define Atiyah classes (i.e. obstructions to the existence of connections) in this generalised context. We use certain model structures on the category of $A$-modules to stu…
Using methods of computer algebra, especially Gröbner bases for submodules of free modules over polynomial rings, we solve a classification problem in theory of algebraic operads: we show that the only nontrivial (possibly inhomogeneous) distributive law between the operad of Lie algebras and the operad of commutative associative algebras is given by the Livernet-Loday formula deforming the Poisson operad into the associative operad.
We construct small cylinders for cellular non-symmetric DG-operads over an arbitrary commutative ring by using the basic perturbation lemma from homological algebra. We show that our construction, applied to the A-infinity operad, yields the operad parametrizing A-infinity maps whose linear part is the identity. We also compute some other examples with non-trivial operations in arities 1 and 0.
We prove that the category of algebras over a cofibrant operad admits a closed model category structure. This leads to the notion of "virtual operad algebra" - the algebra over a cofibrant resolution of the given operad. In particular, virtual commutative algebras can serve to an algebraic description of homotopy p-tipes as in the recent preprint of Mandell (electronic Hopf topology archive, october '98, see http://www.hopf.math.purdue.edu. Our main result allows one to significally simplify the proof of Mandell's theorem.
This paper introduces a chain model for the Deligne-Mumford operad formed by homotopically trivializing the circle in a chain model for the framed little disks. We then show that under degeneration of the Hochschild to cyclic cohomology spectral sequence, a known action of the framed little disks on Hochschild cochains lifts to an action of this new chain model. We thus establish homotopy hyper-commutative algebra structures on both Hochschild and cyclic cochain complexes, and we interpret the gravity brackets on cyclic cohomology as obstructions to degeneration of this spectral sequence. Our …
To any finite group G in SL_2(C), and each `t' in the center of the group algebra of G, we associate a category, Coh_t. It is defined as a suitable quotient of the category of graded modules over (a graded version of) the deformed preprojective algebra introduced by Crawley-Boevey and Holland. The category Coh_t should be thought of as the category of coherent sheaves on a `noncommutative projective 2-space', equipped with a framing at the line at infinity. Our first result establishes an isomorphism between the moduli space of torsion free objects of Coh_t and the Nakajima quiver variety aris…
An operad describes a category of algebras and a (co)homology theory for these algebras may be formulated using the homological algebra of operads. A morphism of operads $f:\mathcal{O}\rightarrow\mathcal{P}$ describes a functor allowing a $\mathcal{P}$-algebra to be viewed as an $\mathcal{O}$-algebra. We show that the $\mathcal{O}$-algebra (co)homology of a $\mathcal{P}$-algebra may be represented by a certain operadic comodule. Thus filtrations of this comodule result in spectral sequences computing the (co)homology. As a demonstration we study operads with a filtered distributive law; for th…
We will consider P-graph complexes, where P is a cyclic operad. P-graph complexes are natural generalizations of Kontsevich's graph complexes -- for P = the operad for associative algebras it is the complex of ribbon graphs, for P = the operad for commutative associative algebras, the complex of all graphs. We construct a `universal class' in the cohomology of the graph complex with coefficients in a theory. The Kontsevich-type invariant is then an evaluation, on a concrete cyclic algebra, of this class. We also explain some results of M. Penkava and A. Schwarz on the construction of an invari…
We establish a highly flexible condition that guarantees that all colored symmetric operads in a symmetric monoidal model category are admissible, i.e., the category of algebras over any operad admits a model structure transferred from the original model category. We also give a necessary and sufficient criterion that ensures that a given weak equivalence of admissible operads admits rectification, i.e., the corresponding Quillen adjunction between the categories of algebras is a Quillen equivalence. In addition, we show that Quillen equivalences of underlying symmetric monoidal model categori…
In two seminal papers M. Kontsevich introduced graph homology as a tool to compute the homology of three infinite dimensional Lie algebras, associated to the three operads `commutative,' `associative' and `Lie.' We generalize his theorem to all cyclic operads, in the process giving a more careful treatment of the construction than in Kontsevich's original papers. We also give a more explicit treatment of the isomorphisms of graph homologies with the homology of moduli space and Out(F_r) outlined by Kontsevich. In [`Infinitesimal operations on chain complexes of graphs', Mathematische Annalen, …
In this paper, we prove that there is a canonical homotopy $(n+1)$-algebra structure on the shifted operadic deformation complex $Def(e_n\to\mathcal{P})[-n]$ for any operad $\mathcal{P}$ and a map of operads $f\colon e_n\to\mathcal{P}$. This result generalizes the result of [T2], where the case $\mathcal{P}=\mathrm{End}_{Op}(X)$ was considered. Another more computational proof of the same statement was recently sketched in [CW]. Our method combines the one of [T2] with the categorical algebra on the category of symmetric sequences, introduced in [R] and further developed in [KM] and [Fr1]. We …
We develop a theory of minimal models for algebras over an operad defined over a commutative ring, not necessarily a field, extending and supplementing the work of Sagave in the associative case.
We define a dg-category of looped diagrams which we use to construct operations on the Hochschild complex of commutative Frobenius dg-algebras. We show that we recover the operations known for symmetric Frobenius dg-algebras constructed using Sullivan chord diagrams as well as all formal operations for commutative algebras (including Loday's lambda operations) and prove that there is a chain level version of a suspended Cactus operad inside the complex of looped diagrams. This recovers the suspended BV algebra structure on the Hochschild homology of commutative Frobenius algebras defined by Ab…
In this paper we show that a strongly homotopy commutative (or $C_\infty$-) algebra with an invariant inner product on its cohomology can be uniquely extended to a symplectic $C_\infty$-algebra (an $\infty$-generalisation of a commutative Frobenius algebra introduced by Kontsevich). This result relies on the algebraic Hodge decomposition of the cyclic Hochschild cohomology of a $\ci$-algebra and does not generalize to algebras over other operads.
We study the operadic and categorical formulations of (conformal) full field algebras. In particular, we show that a grading-restricted $\R\times \R$-graded full field algebra is equivalent to an algebra over a partial operad constructed from spheres with punctures and local coordinates. This result is generalized to conformal full field algebras over $V^L\otimes V^R$, where V^L and V^R are two vertex operator algebras satisfying certain finiteness and reductivity conditions. We also study the geometry interpretation of conformal full field algebras over $V^L\otimes V^R$ equipped with a nondeg…
In the paper \cite{KS}, Kontsevich and Soibelman in particular associate to each finite quiver $Q$ with a set of vertices $I$ the so-called Cohomological Hall algebra $\cH,$ which is $\Z_{\geq 0}^I$-graded. Its graded component $\cH_γ$ is defined as cohomology of Artin moduli stack of representations with dimension vector $γ.$ The product comes from natural correspondences which parameterize extensions of representations. In the case of symmetric quiver, one can refine the grading to $\Z_{\geq 0}^I\times\Z,$ and modify the product by a sign to get a super-commutative algebra $(\cH,\star)$ (wit…
We classify the twisted tensor products of a finite set algebra with a two elements set algebra using colored quivers obtained through considerations analogous to Ore extensions. This provides also a classification of entwining structures between a finite set algebra and the grouplike coalgebra on two elements. The resulting 2-nilpotent algebras have particular features with respect to Hochschild (co)homology and cyclic homology.
We propose an explicit relation between the cohomology of compactified and noncompactified moduli spaces of algebraic curves with punctures. This relationship generalizes one between commutative algebras and Lie algebras proposed by Lazard, Kontsevich and Ginzburg and Kapranov. More explicitly, we show that the cohomology of one moduli space is the graph complex decorated with the cohomology of the other, which in its turn generalizes Getzler's result for the genus zero moduli. We also show that a certain class of algebras over the chain operad of compactified moduli spaces has a natural struc…
We construct a functor that associates to any dg cooperad of dg commutative algebras (satisfying some conditions) an augmented commutative algebra. When applied to the cohomology operad of Francis Brown's moduli spaces it produces an algebra that formally models the algebra of multiple zeta values. We prove that there is an injection from the graded dual of the Grothendieck-Teichmueller Lie algebra into the indecomposables of the algebra associated to the Gerstenhaber cooperad, and that there is a morphism from the algebra associated to Brown's moduli spaces to the algebra associated to the Ge…
For any commutative algebra $R$ the shuffle product on the tensor module $T(R)$ can be deformed to a new product. It is called the quasi-shuffle algebra, or stuffle algebra, and denoted $T^q(R)$. We show that if $R$ is the polynomial algebra, then $T^q(R)$ is free for some algebraic structure called Commutative TriDendriform (CTD-algebras). This result is part of a structure theorem for CTD-bialgebras which are associative as coalgebras and whose primitive part is commutative. In other words, there is a good triple of operads $(As, CTD, Com)$ analogous to $(Com, As, Lie)$.
Using non-commutative differential forms, we construct a complex called singular Hochschild cochain complex for any associative algebra over a field. The cohomology of this complex is isomorphic to the Tate-Hochschild cohomology in the sense of Buchweitz. By a natural action of the cellular chain operad of the spineless cacti operad, introduced by R. Kaufmann, on the singular Hochschild cochain complex, we provide a proof of the Deligne's conjecture for this complex. More concretely, the complex is an algebra over the (dg) operad of chains of the little $2$-discs operad. By this action, we als…
I establish the relation of the non-commutative BV-formalism with super-invariant matrix integration. In particular, the non-commutative BV-equation, defining the quantum A-infinity-algebras, introduced in "Modular operads and Batalin-Vilkovisky geometry" IMRN, Vol. 2007, doi: 10.1093/imrn/rnm075, is represented via de Rham differential acting on the matrix spaces related with Bernstein-Leites simple associative algebras with odd trace q(N), and with gl(N|N). I also show that the Lagrangians of the matrix integrals from "Noncommmutative Batalin-Vilkovisky geometry and Matrix integrals", Compte…
