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Twisting of properads Merkoulov (merkulov), Serguei in Journal of Pure and Applied Algebra (2023), 227 We study Thomas Willwacher's twisting endofunctor tw in the category of dg properads P under the operad of (strongly homotopy) Lie algebras. It is proven that if P is a properad under properad Lieb of Lie ... [more ▼] We study Thomas Willwacher's twisting endofunctor tw in the category of dg properads P under the operad of (strongly homotopy) Lie algebras. It is proven that if P is a properad under properad Lieb of Lie bialgebras , then the associated twisted properad tw(P) becomes in general a properad under quasi-Lie bialgebras (rather than under Lieb). This result implies that the cyclic cohomology of any cyclic homotopy associative algebra has in general an induced structure of a quasi-Lie bialgebra. We show that the cohomology of the twisted properad tw(Lieb) is highly non-trivial -- it contains the cohomology of the so called haired graph complex introduced and studied recently in the context of the theory of long knots and the theory of moduli spaces of algebraic curves. Using a polydifferential functor from the category of props to the category of operads, we introduce the notion of a Maurer-Cartan element of a strongly homotopy Lie bialgebra, and use it to construct a new twisting endofunctor Tw in the category dg prop(erad)s P under HoLieb, the minimal resolution of Lieb. We prove that Tw(Holieb) is quasi-isomorphic to Lieb, and establish its relation to the homotopy theory of triangular Lie bialgebras. It is proven that the dg Lie algebra controlling deformations of the map from Lieb to P acts on Tw(P) by derivations. In some important examples this dg Lie algebra has a rich and interesting cohomology (containing, for example, the Grothendieck-Teichmueller Lie algebra). Finally, we introduce a diamond version of the endofunctor Tw which works in the category of dg properads under involutive (strongly homotopy) Lie bialgebras, and discuss its applications in string topology. [less ▲] Detailed reference viewed: 24 (1 UL)Prop of ribbon hypergraphs and strongly homotopy involutive Lie bialgebras Merkulov, Sergei in International Mathematics Research Notices (2022), rnac023 For any integer d we introduce a prop RHrad of d-oriented ribbon hypergraphs (in which "edges" can connect more than two vertices) and prove that there exists a canonical morphism Holieb⋄d⟶RHrad from the ... [more ▼] For any integer d we introduce a prop RHrad of d-oriented ribbon hypergraphs (in which "edges" can connect more than two vertices) and prove that there exists a canonical morphism Holieb⋄d⟶RHrad from the minimal resolution Holieb⋄d of the (degree shifted) prop of involutive Lie bialgebras into the prop of ribbon hypergraphs which is non-trivial on each generator of Holieb⋄d. As an application we show that for any graded vector space W equipped with a family of cyclically (skew)symmetric higher products the associated vector space of cyclic words in elements of W has a combinatorial Holieb⋄d-structure. As an illustration we construct for each natural number N≥1 an explicit combinatorial strongly homotopy involutive Lie bialgebra structure on the vector space of cyclic words in N graded letters which extends the well-known Schedler's necklace Lie bialgebra structure from the formality theory of the Goldman-Turaev Lie bialgebra in genus zero. [less ▲] Detailed reference viewed: 151 (1 UL)Quantizations of Lie bialgebras, duality involution and oriented graph complexes Merkoulov (merkulov), Serguei ; Zivkovic, Marko in Letters in Mathematical Physics (2022), DOI 10.1007(s11005-022-01505-6), We prove that the action of the Grothendieck-Teichmüller group on the genus completed properad of (homotopy) Lie bialgebras commutes with the reversing directions involution of the latter. We also prove ... [more ▼] We prove that the action of the Grothendieck-Teichmüller group on the genus completed properad of (homotopy) Lie bialgebras commutes with the reversing directions involution of the latter. We also prove that every universal quantization of Lie bialgebras is homotopy equivalent to the one which commutes with the duality involution exchanging Lie bracket and Lie cobracket. The proofs are based on a new result in the theory of oriented graph complexes (which can be of independent interest) saying that the involution on an oriented graph complex that changes all directions on edges induces the identity map on its cohomology. [less ▲] Detailed reference viewed: 76 (0 UL)From gravity to string topology Merkoulov (merkulov), Serguei E-print/Working paper (2022) The chain gravity properad introduced earlier by the author acts on the cyclic Hochschild of any cyclic A∞ algebra equipped with a scalar product of degree −d. In particular, it acts on the cyclic ... [more ▼] The chain gravity properad introduced earlier by the author acts on the cyclic Hochschild of any cyclic A∞ algebra equipped with a scalar product of degree −d. In particular, it acts on the cyclic Hochschild complex of any Poincare duality algebra of degree d, and that action factors through a quotient dg properad ST3−d of ribbon graphs which is in focus of this paper. We show that its cohomology properad H∙(ST3−d) is highly non-trivial and that it acts canonically on the reduced equivariant homology H¯S1∙(LM) of the loop space LM of any simply connected d-dimensional closed manifold M. By its very construction, the string topology properad H∙(ST3−d) comes equipped with a morphism from the gravity properad which is fully determined by the compactly supported cohomology of the moduli spaces Mg,n of stable algebraic curves of genus g with marked points. This result gives rise to new universal operations in string topology as well as reproduces in a unified way several known constructions: we show that (i) H∙(ST3−d) is also a properad under the properad of involutive Lie bialgebras in degree 3−d whose induced action on H¯S1∙(LM) agrees precisely with the famous purely geometric construction of M. Chas and D. Sullivan, (ii) H∙(ST3−d) is a properad under the properad of homotopy involutive Lie bialgebras in degree 2−d; (iii) E. Getzler's gravity operad injects into H∙(ST3−d) implying a purely algebraic counterpart of the geometric construction of C. Westerland establishing an action of the gravity operad on H¯S1∙(LM). [less ▲] Detailed reference viewed: 94 (3 UL)From deformation theory of wheeled props to classification of Kontsevich formality maps Andersson, Assar ; Merkulov, Sergei in International Mathematical Research Notices (2021), rnab012 We study homotopy theory of the wheeled prop controlling Poisson structures on arbitrary formal graded finite-dimensional manifolds and prove, in particular, that Grothendieck-Teichmueller group acts on ... [more ▼] We study homotopy theory of the wheeled prop controlling Poisson structures on arbitrary formal graded finite-dimensional manifolds and prove, in particular, that Grothendieck-Teichmueller group acts on that wheeled prop faithfully and homotopy non-trivially. Next we apply this homotopy theory to the study of the deformation complex of an arbitrary Maxim Kontsevich formality map and compute the full cohomology group of that deformation complex in terms of the cohomology of a certain graph complex introduced earlier by Maxim Kontsevich in [K1] and studied by Thomas Willwacher in [W1]. [less ▲] Detailed reference viewed: 114 (23 UL)Gravity prop and moduli spaces Mg,n Merkoulov (merkulov), Serguei E-print/Working paper (2021) Let Mg,n be the moduli space of algebraic curves of genus g with m+n marked points decomposed into the disjoint union of two sets of cardinalities m and n, and H∙c(Mm+n) its compactly supported cohomology ... [more ▼] Let Mg,n be the moduli space of algebraic curves of genus g with m+n marked points decomposed into the disjoint union of two sets of cardinalities m and n, and H∙c(Mm+n) its compactly supported cohomology group. We prove that the collection of S-bimodules {H∙−mc(Mg,m+n)} has the structure of a properad (called the gravity properad) such that it contains the (degree shifted) E. Getzler's gravity operad as the sub-collection {H∙−1c(M0,1+n)}n≥2. Moreover, we prove that the generators of the 1-dimensional cohomology groups H∙−1c(M0,1+2), H∙−2c(M0,2+1) and H∙−3c(M0,3+0) satisfy with respect to this properadic structure the relations of the (degree shifted) quasi-Lie bialgebra, a fact making the totality of cohomology groups ∏g,m,nH∙c(Mg,m+n)⊗Sopm×Sn(sgnm⊗Idn) into a complex with the differential fully determined by the just mentioned three cohomology classes . It is proven that this complex contains infinitely many cohomology classes, all coming from M. Kontsevich's odd graph complex. The gravity prop structure is established with the help of T. Willwacher's twisting endofunctor (in the category of properads under the operad of Lie algebras) and K. Costello's theory of moduli spaces of nodal disks with marked boundaries and internal marked points (such that each disk contains at most one internal marked point). [less ▲] Detailed reference viewed: 75 (2 UL)Grothendieck-Teichmueller group, operads and graph complexes: a survey Merkulov, Sergei in Integrability, Quantization, and Geometry: II. Quantum Theories and Algebraic Geometry (2021) This paper attempts to provide a more or less self-contained introduction into theory of the Grothendieck-Teichmueller group and Drinfeld associators using the theory of operads and graph complexes. Detailed reference viewed: 162 (2 UL)On deformation quantization of quadratic Poisson structures Merkoulov (merkulov), Serguei ; E-print/Working paper (2021) We study the deformation complex of the dg wheeled properad of Z-graded quadratic Poisson structures and prove that it is quasi-isomorphic to the even M. Kontsevich graph complex. As a first application ... [more ▼] We study the deformation complex of the dg wheeled properad of Z-graded quadratic Poisson structures and prove that it is quasi-isomorphic to the even M. Kontsevich graph complex. As a first application we show that the Grothendieck-Teichmüller group acts on the genus completion of that wheeled properad faithfully and essentially transitively. As a second application we classify all universal quantizations of Z-graded quadratic Poisson structures together with the underlying (so called) homogeneous formality maps. [less ▲] Detailed reference viewed: 97 (2 UL)Classification of universal formality maps for quantizations of Lie bialgebras Merkulov, Sergei ; in Compositio Mathematica (2020), 156(10), 2111-2148 We introduce an endofunctor D in the category of augmented props with the property that for any representation of a prop P in a vector space V the associated prop DP admits an induced representation on ... [more ▼] We introduce an endofunctor D in the category of augmented props with the property that for any representation of a prop P in a vector space V the associated prop DP admits an induced representation on the graded commutative tensor algebra S(V) given in terms of polydifferential operators. Applying this functor to the prop LieB of Lie bialgebras we show that universal formality maps for quantizations of Lie bialgebras are in in 1-1 correspondence with prop morphisms from the minimal resolution AssB_infty of the prop of associative bialgebras to the polydifferential prop DLieB_infty satisfying certain boundary conditions. We prove that the set of such formality morphisms (having an extra property of being Lie connected) is non-empty. The latter result is used in turn to give a short proof of the formality theorem for universal quantizations of arbitrary Lie bialgebras which says that for any Drinfeld associator there is an associated Lie_infty quasi-isomorphism between the Lie_infty algebras controlling, respectively, deformations of the standard bialgebra structure in S(V) and deformations of any given Lie bialgebra structure in V. We study the deformation complex of an arbitrary universal formality morphism and show that it is quasi-isomorphic (up to one class corresponding to the standard rescaling automorphism of the properad LieB) to the oriented graph complex GC or 3 studied earlier in \cite{Wi2}. This result gives a complete classification of the set of gauge equivalence classes of universal Lie connected formality maps --- it is a torsor over the Grothendieck-Teichm\"uller group GRT and can hence can be identified with the set of Drinfeld associators. [less ▲] Detailed reference viewed: 107 (0 UL)Multi-oriented props and homotopy algebras with branes Merkulov, Sergei in Letters in Mathematical Physics (2020), 110 We introduce a new category of differential graded {\em multi-oriented}\, props whose representations (called homotopy algebras with branes) in a graded vector space require a choice of a collection of $k ... [more ▼] We introduce a new category of differential graded {\em multi-oriented}\, props whose representations (called homotopy algebras with branes) in a graded vector space require a choice of a collection of $k$ linear subspaces in that space, $k$ being the number of extra directions (if $k=0$ this structure recovers an ordinary prop); symplectic vector spaces equipped with $k$ Lagrangian subspaces play a distinguished role in this theory. Manin triples is a classical example of an algebraic structure (concretely, a Lie bialgebra structure) given in terms of a vector space and its subspace; in the context of this paper Manin triples are precisely symplectic Lagrangian representations of the {\em 2-oriented} generalization of the classical operad of Lie algebras. In a sense, the theory of multi-oriented props provides us with a far reaching strong homotopy generalization of Manin triples type constructions. [less ▲] Detailed reference viewed: 159 (13 UL)Deformation theory of Lie bialgebra properads Merkulov, Sergei ; in Geometry and Physics: A Festschrift in honour of Nigel Hitchin (2018) We compute the homotopy derivations of the properads governing even and odd Lie bialgebras as well as involutive Lie bialgebras. The answer may be expressed in terms of the Kontsevich graph complexes. In ... [more ▼] We compute the homotopy derivations of the properads governing even and odd Lie bialgebras as well as involutive Lie bialgebras. The answer may be expressed in terms of the Kontsevich graph complexes. In particular, this shows that the Grothendieck-Teichm\"uller group acts faithfully (and essentially transitively) on the completions of the properads governing even Lie bialgebras and involutive Lie bialgebras, up to homotopy. This shows also that by contrast to the even case the properad governing odd Lie bialgebras admits precisely one non-trivial automorphism --- the standard rescaling automorphism, and that it has precisely one non-trivial deformation which we describe explicitly. [less ▲] Detailed reference viewed: 229 (11 UL)An explicit two step quantization of Poisson structures and Lie bialgebras Merkulov, Sergei ; in Communications in Mathematical Physics (2018), 364(2), 505578 We develop a new approach to deformation quantizations of Lie bialgebras and Poisson structures which goes in two steps. In the first step one associates to any Poisson (resp. Lie bialgebra) structure a ... [more ▼] We develop a new approach to deformation quantizations of Lie bialgebras and Poisson structures which goes in two steps. In the first step one associates to any Poisson (resp. Lie bialgebra) structure a so called quantizable Poisson (resp. Lie bialgebra) structure. We show explicit transcendental formulae for this correspondence. In the second step one deformation quantizes a quantizable Poisson (resp. Lie bialgebra) structure. We show again explicit transcendental formulae for this second step correspondence (as a byproduct we obtain configuration space models for biassociahedron and bipermutohedron). In the Poisson case the first step is the most non-trivial one and requires a choice of an associator while the second step quantization is essentially unique, it is independent of a choice of an associator and can be done by a trivial induction. We conjecture that similar statements hold true in the case of Lie bialgebras. The main new result is a surprisingly simple explicit universal formula (which uses only smooth differential forms) for universal quantizations of finite-dimensional Lie bialgebras. [less ▲] Detailed reference viewed: 191 (5 UL)Formality Theorem for Quantizations of Lie Bialgebras Merkulov, Sergei in Letters in Mathematical Physics (2016), 106(2), 169-195 Using the theory of props we prove a formality theorem associated with universal quantizations of Lie bialgebras. Detailed reference viewed: 217 (13 UL)The Frobenius operad is Koszul ; Merkulov, Sergei ; in Duke Mathematical Journal (2016), 165(15), 2921-2989 We show Koszulness of the prop governing involutive Lie bialgebras and also of the props governing non-unital and unital-counital Frobenius algebras, solving a long-standing problem. This gives us minimal ... [more ▼] We show Koszulness of the prop governing involutive Lie bialgebras and also of the props governing non-unital and unital-counital Frobenius algebras, solving a long-standing problem. This gives us minimal models for their deformation complexes, and for deformation complexes of their algebras which are discussed in detail. Using an operad of graph complexes we prove, with the help of an earlier result of one of the authors [W3], that there is a highly non-trivial action of the Grothendieck-Teichm¨uller group GRT on (completed versions of) the minimal models of the properads governing Lie bialgebras and involutive Lie bialgebras by automorphisms. As a corollary one obtains a large class of universal deformations of any (involutive) Lie bialgebra and any Frobenius algebra, parameterized by elements of the Grothendieck-Teichmueller Lie algebra. We also prove that, for any given homotopy involutive Lie bialgebra structure in a vector space, there is an associated homotopy Batalin-Vilkovisky algebra structure on the associated Chevalley-Eilenberg complex. [less ▲] Detailed reference viewed: 253 (14 UL)On quantizable odd Lie bialgebras ; Merkulov, Sergei ; in Letters in Mathematical Physics (2016), 106(9), 1199-1215 The notion of a quantizable odd Lie bialgebra is introduced. A minimal resolution of the properad governing such Lie bialgebras is constructed. Detailed reference viewed: 196 (5 UL)Props of ribbon graphs, involutive Lie bialgebras and moduli spaces of curves M_g,n Merkulov, Sergei ; E-print/Working paper (2015) We establish a new and surprisingly strong link between two previously unrelated theories: the theory of moduli spaces of curves $\cM_{g,n}$ (which, according to Penner, is controlled by the ribbon graph ... [more ▼] We establish a new and surprisingly strong link between two previously unrelated theories: the theory of moduli spaces of curves $\cM_{g,n}$ (which, according to Penner, is controlled by the ribbon graph complex) and the homotopy theory of $E_d$ operads (controlled by ordinary graph complexes with no ribbon structure, introduced first by Kontsevich). The link between the two goes through a new intermediate {\em stable}\, ribbon graph complex which has roots in the deformation theory of quantum $A_\infty$ algebras and the theory of Kontsevich compactifications of moduli spaces of curves $\overline{\cM}_{g,n}^K$. Using a new prop of ribbon graphs and the fact that it contains the prop of involutive Lie bialgebras as a subprop we find new algebraic structures on the classical ribbon graph complex computing $H^\bu(\cM_{g,n})$. We use them to prove Comparison Theorems, and in particular to construct a non-trivial map from the ordinary to the ribbon graph cohomology. On the technical side, we construct a functor $\f$ from the category of prop(erad)s to the category of operads. If a properad $\cP$ is in addition equipped with a map from the properad governing Lie bialgebras (or graded versions thereof), then we define a notion of $\cP$-``graph'' complex, of stable $\cP$-graph complex and a certain operad, that is in good cases an $E_d$ operad. In the ribbon case, this latter operad acts on the deformation complexes of any quantum $A_\infty$-algebra. We also prove that there is a highly non-trivial, in general, action of the Grothendieck-Teichm\"uller group $GRT_1$ on the space of so-called {\em non-commutative Poisson structures}\, on any vector space $W$ equipped with a degree $-1$ symplectic form (which interpolate between cyclic $A_\infty$ structures in $W$ and ordinary polynomial Poisson structures on $W$ as an affine space). [less ▲] Detailed reference viewed: 108 (15 UL)Grothendieck-Teichmueller group and Poisson cohomologies ; Merkulov, Sergei in Journal of Noncommutative Geometry (2015), 9(1), 185-214 We study actions of the Grothendieck–Teichmüller group GRT on Poisson cohomologies of Poisson manifolds, and prove some “go” and “no-go” theorems associated with these actions. Detailed reference viewed: 141 (13 UL)Grothendieck-Teichmueller and Batalin-Vilkovisky Merkulov, Sergei ; in Letters in Mathematical Physics (2014), 104(5), 625-634 It is proven that, for any affine supermanifold M equipped with a constant odd symplectic structure, there is a universal action (up to homotopy) of the Grothendieck-Teichmueller Lie algebra grt on the ... [more ▼] It is proven that, for any affine supermanifold M equipped with a constant odd symplectic structure, there is a universal action (up to homotopy) of the Grothendieck-Teichmueller Lie algebra grt on the set of quantum BV structures (i. e.\ solutions of the quantum master equation) on M. [less ▲] Detailed reference viewed: 184 (12 UL)Permutahedra, HKR isomorphism and polydifferential Gerstenhaber-Schack complex Merkulov, Sergei in Higher structures in geometry and physics (2011) Detailed reference viewed: 123 (3 UL)Operads, configuration spaces and quantization Merkulov, Sergei in Bulletin of the Brazilian Mathematical Society (2011), 42(4), 683781 We review several well-known operads of compactified configuration spaces and construct several new such operads, $\overline{C}$, in the category of smooth manifolds with corners whose complexes of ... [more ▼] We review several well-known operads of compactified configuration spaces and construct several new such operads, $\overline{C}$, in the category of smooth manifolds with corners whose complexes of fundamental chains give us (i) the 2-coloured operad of $A_\infty$-algebras and their homotopy morphisms, (ii) the 2-coloured operad of $L_\infty$-algebras and their homotopy morphisms, and (iii) the 4-coloured operad of open-closed homotopy algebras and their homotopy morphisms. Two gadgets --- a (coloured) operad of Feynman graphs and a de Rham field theory on $\overline{C}$ --- are introduced and used to construct quantized representations of the (fundamental) chain operad of $\overline{C}$ which are given by Feynman type sums over graphs and depend on choices of propagators. [less ▲] Detailed reference viewed: 232 (5 UL) |
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