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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: 53 (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: 150 (13 UL)Differentials on graph complexes III: hairy graphs and deleting a vertex Zivkovic, Marko in Letters in Mathematical Physics (2018) We continue studying the cohomology of the hairy graph complexes which compute the rational homotopy of embedding spaces, generalizing the Vassiliev invariants of knot theory, after the second part in ... [more ▼] We continue studying the cohomology of the hairy graph complexes which compute the rational homotopy of embedding spaces, generalizing the Vassiliev invariants of knot theory, after the second part in this series. In that part we have proven that the hairy graph complex HGC_{m,n} with the extra differential is almost acyclic for even m. In this paper, we give the expected same result for odd m. As in the previous part, our results yield a way to construct many hairy graph cohomology classes by the waterfall mechanism also for odd m. However, the techniques are quite different. The main tool used in this paper is a new differential, deleting a vertex in non-hairy Kontsevich’s graphs, and a similar map for hairy vertices. We hope that the new differential can have further applications in the study of Kontsevich’s graph cohomology. Namely it is conjectured that the Kontsevich’s graph complex with deleting a vertex as an extra differential is acyclic. [less ▲] Detailed reference viewed: 157 (12 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)On twisted N= 2 5D super Yang-Mills theory Qiu, Jian ; in Letters in Mathematical Physics (2016), 106(1), 127 Detailed reference viewed: 83 (0 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: 203 (13 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: 187 (5 UL)Warped products and Yang-Mills equations on noncommutative spaces Zampini, Alessandro in Letters in Mathematical Physics (2015), 105(2), 221243 Detailed reference viewed: 79 (2 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)Hyperbolic Systems and Propagation on Causal Manifolds Schapira, Pierre in Letters in Mathematical Physics (2013), 103(10), 1149-1164 Detailed reference viewed: 94 (1 UL)Deformations of coisotropic submanifolds for fibrewise entire Poisson structures Schatz, Florian ; in Letters in Mathematical Physics (2013), 103(7), 777-791 We show that deformations of a coisotropic submanifold inside a fibrewise entire Poisson manifold are controlled by the L-infinity-algebra introduced by Oh-Park (for symplectic manifolds) and Cattaneo ... [more ▼] We show that deformations of a coisotropic submanifold inside a fibrewise entire Poisson manifold are controlled by the L-infinity-algebra introduced by Oh-Park (for symplectic manifolds) and Cattaneo-Felder. In the symplectic case, we recover results previously obtained by Oh-Park. Moreover we consider the extended deformation problem and prove its obstructedness. [less ▲] Detailed reference viewed: 92 (1 UL)Equivariant symbol calculus for differential operators acting on forms ; ; Mathonet, Pierre et al in Letters in Mathematical Physics (2002), 62(3), 219-232 Detailed reference viewed: 52 (2 UL)Almost Kähler deformation quantization ; Schlichenmaier, Martin in Letters in Mathematical Physics (2001), 57(2), 135-148 Detailed reference viewed: 59 (0 UL)String branchings on complex tori and algebraic representations of generalized Krichever-Novikov algebras ; ; Schlichenmaier, Martin in Letters in Mathematical Physics (1992), 26(1), 23-32 Detailed reference viewed: 88 (0 UL)Central extensions and semi-infinite wedge representations of Krichever-Novikov algebras for more than two points Schlichenmaier, Martin in Letters in Mathematical Physics (1990), 20 Detailed reference viewed: 92 (0 UL)Krichever-Novikov algebras for more than two points: explicit generators Schlichenmaier, Martin in Letters in Mathematical Physics (1990), 19(4), 327-336 Detailed reference viewed: 93 (1 UL)Krichever-Novikov algebras for more than two points Schlichenmaier, Martin in Letters in Mathematical Physics (1990), 19(2), 151--165 Detailed reference viewed: 142 (2 UL) |
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