with Michael Polyak:
Diassociative algebras and Milnor's invariants for tangles
to appear in Lett. Math. Phys. arXiv:1011.0117
We extend Milnor's mu-invariants of link homotopy to ordered
(classical or virtual) tangles. Simple combinatorial formulas
for mu-invariants are given in terms of counting trees in Gauss diagrams.
Invariance under Reidemeister moves corresponds to axioms of Loday's diassociative algebra.
The relation of tangles to diassociative algebras
is formulated in terms of a morphism of corresponding operads.
Strongly homotopy Lie bialgebras
and Lie quasi-bialgebras
Lett. Math. Phys. 81 (2007), no. 1, 19--40.
Structures of Lie algebras, Lie
coalgebras, Lie bialgebras and Lie quasibialgebras are presented as
solutions of Maurer-Cartan equations on corresponding governing
differential graded Lie algebras. Cohomology theories of all these
structures are described in a concise way using the big bracket
construction of Kosmann-Schwarzbach. This approach provides a
definition of an L_\infty-(quasi)bialgebra (strong homotopy Lie
(quasi)bialgebra). We recover an L_\infty-algebra structure as a
particular case of our construction. The formal geometry
interpretation leads to a definition of an L_\infty
(quasi)bialgebra structure on V as a differential operator Q on
V, self-commuting with respect to the Poisson bracket. Finally,
we establish an L_\infty-version of a Manin (quasi) triple and
get a correspondence theorem with L_\infty-(quasi) bialgebra.
of Batalin-Vilkovisky algebras Banach Center
Publications v. 51, (2000)
Poisson geometry, J.Grabowski and P.Urbanski (eds.) 131-139. math.QA/9903191
We show that a commutative algebra $A$ with any square zero odd differential
operator is a natural generalization of the Batalin-Vilkovisky algebra.
While such an operator of order $2$ defines the Lie structure on $A$, an
operator of an order higher than $2$ (Koszul-Akman definition ) leads to
the structure of a Lie up-to homotopy algebra on $A$. This allows us to
propose a definition of a Batalin-Vilkovisky algebra up-to homotopy. We
also make an important conjecture generalizing Kontsevich formality theorem
to the BV-algebra level.
to calculate the Fedosov connection (Exercice de style)
February 99, Glanon
-I I (1998) proceedings. 11pp. math.SG/0008157
This is an expository note on Fedosov's construction of deformation
quantization. Given a symplectic manifold and a connection on it, we show
how to calculate the star-product step by step. We draw simple
diagrams to solve the recursive equations for the Fedosov connection and
for flat sections of the Weyl algebra bundle corresponding to functions.
We also reflect on the differences of symplectic and Riemannian geometries.
Quantization of Symplectic Fibrations
Compositio Mathematica vol. 123: 131-165 (2000)
A symplectic fibration is a fibre bundle in the symplectic category.
We find the relation between deformation quantization of the base and the
fibre, and the total space. We use the weak coupling form of Guillemin,
Lerman, Sternberg and find the characteristic class of deformation of symplectic
fibration. We also prove that the classical moment map could be quantized
if there exists an equivariant connection. Along the way we touch upon
the general question of quantization with values in a bundle of algebras.
Moment Map in Deformation
Quantization February 98,
-I (1997) proceedings. 8pp.
We consider a Hamiltonian action of a group on a symplectic manifold
with a given moment map. Provided with an equivariant connection on this
manifold we show that the action in the deformed algebra is the same,
namely by the Poisson bracket. We employ Fedosov's construction of deformation
Operator Formalism and Tau-function
from Supersymmetric Ghosts in Higher Genus.
Phys. Lett. B 231 (1989), no. 2, 85 -- 93.
Central Extension of the Algebra of Pseudodifferential Symbols.
Functional Analysis and Applications 26 (1991) no. 2, 77 -- 79.