The calculus of multivectors on noncommutative jet spaces

TL;DR

This paper develops a calculus of multivectors on noncommutative jet spaces, extending variational calculus and symplectic geometry to noncommutative bundles, with implications for integrable systems.

Contribution

It introduces a novel framework for noncommutative calculus on jet spaces, including properties of BV Laplacian and Schouten bracket without relying on graded commutativity.

Findings

Established properties of BV Laplacian and Schouten bracket in noncommutative setting

Extended variational calculus to noncommutative bundles

Showed classical Poisson structures do not require graded commutativity

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.