## 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 tessellated 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. (C) 2018 Elsevier B.V. All rights reserved.

Original language | English |
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Pages (from-to) | 130-167 |

Number of pages | 38 |

Journal | Journal of geometry and physics |

Volume | 130 |

DOIs | |

Publication status | Published - Aug-2018 |

## Keywords

- Noncommutative geometry
- Associative algebra
- (Quasi)crystal structure
- Cyclic invariance
- BV Laplacian
- Variational Poisson bi-vector
- LIE ALGEBROIDS
- ASSOCIATIVE ALGEBRAS
- VECTOR-FIELDS
- QUANTIZATION
- GEOMETRY
- ANTIBRACKET
- FORMALISM
- MANIFOLDS
- EQUATION