Nonassociative analogs of Lie groupoids
arXiv:2101.07258 · doi:10.1016/j.difgeo.2022.101887
Abstract
We introduce nonassociative geometric objects generalising naturally Lie groupoids and called (smooth) quasiloopoids and loopoids. We prove that the tangent bundles of smooth loopoids are canonically smooth loopoids again (it is nontrivial in the case of loopoids). We show also that this is not true if the cotangent bundles are concerned. After providing a few natural constructions, we show how the Lie-like functor associates with loopoids skew-algebroids and almost Lie algebroids and how discrete mechanics on Lie groupoids can be reformulated in the nonassociative case.
arXiv comment31 pages
References in corpus (16)
- Lie algebroids and Poisson-Nijenhuis structures
- Lagrangian submanifolds and dynamics on Lie algebroids
- Dirac Algebroids in Lagrangian and Hamiltonian Mechanics
- Algebroids - general differential calculi on vector bundles
- Discrete Lagrangian and Hamiltonian Mechanics on Lie groupoids
- Variational calculus with constraints on general algebroids
- Nonholonomic Constraints: a New Viewpoint
- Discrete Nonholonomic Lagrangian Systems on Lie Groupoids
- Discrete Nonholonomic LL Systems on Lie Groups
- G2-structures and octonion bundles
- Modular classes of skew algebroid relations
- Lie groupoids in information geometry
- Symplectic groupoids and discrete constrained Lagrangian mechanics
- Discrete Hamilton-Pontryagin mechanics and generating functions on Lie groupoids
- Discrete mechanics on unitary octonions
- Smooth Loops and Fiber Bundles: Theory of Principal Q-bundles