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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-directed graph complexes and quasi-isomorphisms between them I: oriented graphs Zivkovic, Marko in Higher Structures (2020), 4(1), 266283 We construct a direct quasi-isomorphism from Kontsevich's graph complex GC_n to the oriented graph complex OGC_{n+1}, thus providing an alternative proof that the two complexes are quasi-isomorphic ... [more ▼] We construct a direct quasi-isomorphism from Kontsevich's graph complex GC_n to the oriented graph complex OGC_{n+1}, thus providing an alternative proof that the two complexes are quasi-isomorphic. Moreover, the result is extended to the sequence of multi-oriented graph complexes, where GC_n and OGC_{n+1} are the first two members. These complexes play a key role in the deformation theory of multi-oriented props recently invented by Sergei Merkulov. [less ▲] Detailed reference viewed: 93 (7 UL)Multi-directed Graph Complexes and Quasi-isomorphisms Between Them II: Sourced Graphs Zivkovic, Marko in International Mathematics Research Notices (2019), 00(0), 1-57 We prove that the projection from graph complex with at least one source to oriented graph complex is a quasi-isomorphism, showing that homology of the “sourced” graph complex is also equal to the ... [more ▼] We prove that the projection from graph complex with at least one source to oriented graph complex is a quasi-isomorphism, showing that homology of the “sourced” graph complex is also equal to the homology of standard Kontsevich’s graph complex. This result may have applications in theory of multi-vector fields T≥1poly of degree at least one, and to the hairy graph complex that computes the rational homotopy of the space of long knots. The result is generalized to multi-directed graph complexes, showing that all such graph complexes are quasi-isomorphic. These complexes play a key role in the deformation theory of multi-oriented props recently invented by Sergei Merkulov. We also develop a theory of graph complexes with arbitrary edge types. [less ▲] Detailed reference viewed: 129 (6 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)Trace decategorification of categorified quantum sl2 ; ; et al in Mathematische Annalen (2017), 367(1), 397440 The trace or the 0th Hochschild–Mitchell homology of a linear category C may be regarded as a kind of decategorification of C. We compute the traces of the two versions U˙ and U∗ of categorified quantum ... [more ▼] The trace or the 0th Hochschild–Mitchell homology of a linear category C may be regarded as a kind of decategorification of C. We compute the traces of the two versions U˙ and U∗ of categorified quantum sl2 introduced by the third author. The trace of U is isomorphic to the split Grothendieck group K_0(U˙), and the higher Hochschild–Mitchell homology of U˙ is zero. The trace of U∗ is isomorphic to the idempotented integral form of the current algebra U(sl2[t]). [less ▲] Detailed reference viewed: 93 (9 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)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) |
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