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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: 101 (0 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: 181 (5 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: 218 (11 UL)Differentials on graph complexes II: hairy graphs ; ; Zivkovic, Marko in Letters in Mathematical Physics (2017), 107(10), 17811797 We study the cohomology of the hairy graph complexes which compute the rational homotopy of embedding spaces, generalizing the Vassiliev invariants of knot theory. We provide spectral sequences converging ... [more ▼] We study the cohomology of the hairy graph complexes which compute the rational homotopy of embedding spaces, generalizing the Vassiliev invariants of knot theory. We provide spectral sequences converging to zero whose first pages contain the hairy graph cohomology. Our results yield a way to construct many nonzero hairy graph cohomology classes out of (known) non-hairy classes by studying the cancellations in those sequences. This provide a first glimpse at the tentative global structure of the hairy graph cohomology. [less ▲] Detailed reference viewed: 134 (5 UL)Differentials on graph complexes ; ; Zivkovic, Marko in Advances in Mathematics (2017), 307 We study the cohomology of complexes of ordinary (non-decorated) graphs, introduced by M. Kontsevich. We construct spectral sequences converging to zero whose first page contains the graph cohomology. In ... [more ▼] We study the cohomology of complexes of ordinary (non-decorated) graphs, introduced by M. Kontsevich. We construct spectral sequences converging to zero whose first page contains the graph cohomology. In particular, these spectral sequences may be used to show the existence of an infinite series of previously unknown and provably non-trivial cohomology classes, and put constraints on the structure of the graph cohomology as a whole. [less ▲] Detailed reference viewed: 171 (12 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: 243 (14 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: 95 (13 UL)Multiple edges in M. Kontsevich's graph complexes and computations of the dimensions and Euler characteristics ; Zivkovic, Marko in Advances in Mathematics (2015), 272 We study the cohomology of complexes of ordinary (non- decorated) graphs, introduced by M. Kontsevich. We construct spectral sequences converging to zero whose first page contains the graph cohomology. In ... [more ▼] We study the cohomology of complexes of ordinary (non- decorated) graphs, introduced by M. Kontsevich. We construct spectral sequences converging to zero whose first page contains the graph cohomology. In particular, these spectral sequences may be used to show the existence of an infinite series of previously unknown and provably non-trivial cohomology classes, and put constraints on the structure of the graph cohomology as a whole. [less ▲] Detailed reference viewed: 137 (5 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: 171 (12 UL) |
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