Quantizations of Lie bialgebras, duality involution and oriented graph complexes

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 ▲]

Differentials on graph complexes III: hairy graphs and deleting a vertex

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 ▲]

On twisted N= 2 5D super Yang-Mills theory

in Letters in Mathematical Physics (2016), 106(1), 127

On quantizable odd Lie bialgebras

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.

Grothendieck-Teichmueller and Batalin-Vilkovisky

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 ▲]

Deformations of coisotropic submanifolds for fibrewise entire Poisson structures

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 ▲]

Almost Kähler deformation quantization

in Letters in Mathematical Physics (2001), 57(2), 135-148

Central extensions and semi-infinite wedge representations of Krichever-Novikov algebras for more than two points

in Letters in Mathematical Physics (1990), 20

Krichever-Novikov algebras for more than two points

in Letters in Mathematical Physics (1990), 19(2), 151--165