On the variational noncommutative Poisson geometry

TL;DR

This paper develops a framework for variational multivectors in noncommutative Poisson geometry, connecting jet calculus, algebraic structures, and quantum string theory, and provides criteria for Hamiltonian operators.

Contribution

It introduces a variational calculus for noncommutative Poisson structures and establishes properties of the Schouten bracket in this setting, linking to quantum string theory.

Findings

Derived a criterion for noncommutative Hamiltonian operators.

Established properties of the variational Schouten bracket.

Connected noncommutative jet calculus to quantum string theory.

Abstract

We outline the notions and concepts of the calculus of variational multivectors within the Poisson formalism over the spaces of infinite jets of mappings from commutative (non)graded smooth manifolds to the factors of noncommutative associative algebras over the equivalence under cyclic permutations of the letters in the associative words. We state the basic properties of the variational Schouten bracket and derive an interesting criterion for (non)commutative differential operators to be Hamiltonian (and thus determine the (non)commutative Poisson structures). We place the noncommutative jet-bundle construction at hand in the context of the quantum string theory.