The calculus of multivectors on noncommutative jet spaces

Abstract

The Leibniz rule for derivations is invariant under cyclic permutations of co-multiples within the arguments of derivations. We explore the implications of this principle: in effect, we construct a class of noncommutative bundles in which the sheaves of algebras of walks along a tesselated affine manifold form the base, whereas the fibres are free associative algebras or, at a later stage, such algebras quotients over the linear relation of equivalence under cyclic shifts. The calculus of variations is developed on the infinite jet spaces over such noncommutative bundles. In the frames of such field-theoretic extension of the Kontsevich formal noncommutative symplectic (super)geometry, we prove the main properties of the Batalin--Vilkovisky Laplacian and Schouten bracket. We show as by-product that the structures which arise in the classical variational Poisson geometry of infinite-dimensional integrable systems do actually not refer to the graded commutativity assumption.

Talks given at Mathematics seminar (IHES, 25.11.2016) and Oberseminar (MPIM Bonn, 2.02.2017), 23 figures, 60 pages

References in corpus (18)

• Modular operads and Batalin-Vilkovisky geometry
• The variational Poisson cohomology
• Bihamiltonian Cohomologies and Integrable Hierarchies I: A Special Case
• The Kontsevich tetrahedral flow revisited
• Essential variational Poisson cohomology
• Poisson brackets symmetry from the pentagon-wheel cocycle in the graph complex
• The geometry of variations in Batalin-Vilkovisky formalism
• The heptagon-wheel cocycle in the Kontsevich graph complex
• Variational Lie algebroids and homological evolutionary vector fields
• The Jacobi identity for graded-commutative variational Schouten bracket revisited
• On the variational noncommutative Poisson geometry
• Non-Abelian Lie algebroids over jet spaces
• The deformation quantization mapping of Poisson- to associative structures in field theory
• The right-hand side of the Jacobi identity: to be naught or not to be?
• Homological evolutionary vector fields in Korteweg-de Vries, Liouville, Maxwell, and several other models
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• Towards an axiomatic model of fundamental interactions at Planck scale
• On the associative homotopy Lie algebras and the Wronskians

Cited by in corpus (9)

• The Kontsevich tetrahedral flow revisited
• Do the Kontsevich tetrahedral flows preserve or destroy the space of Poisson bi-vectors?
• The geometry of variations in Batalin-Vilkovisky formalism
• The deformation quantization mapping of Poisson- to associative structures in field theory
• Open problems in the Kontsevich graph construction of Poisson bracket symmetries
• The expansion $\star$ mod $\bar{o}({\hslash }^4)$ and computer-assisted proof schemes in the Kontsevich deformation quantization
• On the geometry of the Batalin-Vilkovisky Laplacian
• Universal cocycles and the graph complex action on homogeneous Poisson brackets by diffeomorphisms
• On the (non)removability of spectral parameters in $Z_2$-graded zero-curvature representations and its applications