“Nonlinear pullbacks” of functions and L∞-morphisms for homotopy Poisson structures

@article{Voronov2017NonlinearPO,
  title={“Nonlinear pullbacks” of functions and L∞-morphisms for homotopy Poisson structures},
  author={Theodore Th. Voronov},
  journal={Journal of Geometry and Physics},
  year={2017},
  volume={111},
  pages={94-110}
}
  • T. Voronov
  • Published 23 September 2014
  • Mathematics
  • Journal of Geometry and Physics
Microformal Geometry and Homotopy Algebras
  • T. Voronov
  • Mathematics
    Proceedings of the Steklov Institute of Mathematics
  • 2018
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References

SHOWING 1-10 OF 29 REFERENCES
THICK MORPHISMS OF SUPERMANIFOLDS AND OSCILLATORY INTEGRAL OPERATORS
We show that thick morphisms (or microformal morphisms) between smooth (super)manifolds, introduced by us before, are classical limits of 'quantum thick morphisms' defined here as particular
Higher derived brackets and homotopy algebras
Deformation Quantization of Poisson Manifolds
I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the
Quantum microformal morphisms of supermanifolds: an explicit formula and further properties
We give an explicit formula, as a formal differential operator, for quantum microformal morphisms of (super)manifolds that we introduced earlier. Such quantum microformal morphisms are essentially
Noncommutative Differential Forms and Quantization of the Odd Symplectic Category
There is a simple and natural quantization of differential forms on odd Poisson supermanifolds, given by the relation [f,dg]={f,g} for all functions f and g. We notice that this noncommutative
Symplectic Categories
Quantization problems suggest that the category of symplectic manifolds and symplectomorphisms be augmented by the inclusion of canonical relations as morphisms. These relations compose well when a
Graded Manifolds and Drinfeld Doubles for Lie Bialgebroids
We define graded manifolds as a version of supermanifolds endowed with an extra Z-grading in the structure sheaf, called weight (not linked with parity). Examples are ordinary supermanifolds, vector
On Odd Laplace Operators
We consider odd Laplace operators acting on densities of various weights on an odd Poisson (= Schouten) manifold M. We prove that the case of densities of weight 1/2 (half-densities) is distinguished
EXTENSION OF REPRESENTATIONS OF THE CLASSICAL GROUPS TO REPRESENT A nONS OF CA TEGORIES
It is shown that with each series An, Bn, Cn, Dn ofthe complex classical Lie groups a certain category GA, B, C, D is connected in a natural fashion. A finitedimensional representation of any
Introduction to SH Lie algebras for physicists
UNC-MATH-92/2originally April 27, 1990, revised September 24, 1992INTRODUCTION TO SH LIE ALGEBRAS FOR PHYSICISTSTom LadaJim StasheffMuch of point particle physics can be described in terms of Lie
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