657 research outputs found
On the quantization of Poisson brackets
In this paper we introduce two classes of Poisson brackets on algebras (or on
sheaves of algebras). We call them locally free and nonsingular Poisson
brackets. Using the Fedosov's method we prove that any locally free nonsingular
Poisson bracket can be quantized. In particular, it follows from this that all
Poisson brackets on an arbitrary field of characteristic zero can be quantized.
The well known theorem about the quantization of nondegenerate Poisson brackets
on smooth manifolds follows from the main result of this paper as well.Comment: Latex, 24 pp., essentially corrected versio
Cohomology and Deformation of Leibniz Pairs
Cohomology and deformation theories are developed for Poisson algebras
starting with the more general concept of a Leibniz pair, namely of an
associative algebra together with a Lie algebra mapped into the
derivations of . A bicomplex (with both Hochschild and Chevalley-Eilenberg
cohomologies) is essential.Comment: 15 page
Additive Deformations of Hopf Algebras
Additive deformations of bialgebras in the sense of Wirth are deformations of
the multiplication map of the bialgebra fulfilling a compatibility condition
with the coalgebra structure and a continuity condition. Two problems
concerning additive deformations are considered. With a deformation theory a
cohomology theory should be developed. Here a variant of the Hochschild
cohomology is used. The main result in the first part of this paper is the
characterization of the trivial deformations, i.e. deformations generated by a
coboundary. When one starts with a Hopf algebra, one would expect the deformed
multiplications to have some analogue to the antipode, which we call deformed
antipodes. We prove, that deformed antipodes always exist, explore their
properties, give a formula to calculate them given the deformation and the
antipode of the original Hopf algebra and show in the cocommutative case, that
each deformation splits into a trivial part and into a part with constant
antipodes.Comment: 18 page
Note on operadic harmonic oscillator
It is explained how the time evolution of the operadic variables may be
introduced. As an example, an operadic Lax representation of the harmonic
oscillator is considered.Comment: LaTeX2e, 6 pages, no figure
- …