2012/10/31 by Arthemy V. Kiselev
Mathematics · Physics and Astronomy · #Algebra over a field #Algebraic structures and combinatorial models #Commutative property #Homotopy and Cohomology in Algebraic Topology #Injective function #Integrable system #Invariant (physics) #Manifold (fluid mechanics) #Mathematical physics #Mathematics #Noncommutative algebraic geometry #Noncommutative geometry #Noncommutative quantum field theory #Nonlinear Waves and Solitons #Poisson manifold #Pure mathematics #Quotient #Symplectic geometry #Vector bundle #hep-th #math-ph #math.DG #math.MP #msc:05C38 #msc:16S10 #msc:58A20 #msc:70S05 #msc:81R60 #msc:81T45 #nlin.SI
paper · pdf · doi:10.1016/j.geomphys.2018.03.022
published as Journal of Geometry and Physics, Vol.130 (2018) 130--167 · Talks given at Mathematics seminar (IHES, 25.11.2016) and Oberseminar (MPIM Bonn, 2.02.2017), 23 figures, 60 pages
arxiv created 2017/12/25 · openalex publication_date 2018/04/06 · arxiv updated 2018/04/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.