Nonassociative analogs of Lie groupoids

Abstract

We introduce nonassociative geometric objects generalising naturally Lie groupoids, called (smooth) quasiloopoids and loopoids. We prove that the tangent bundles of smooth loopoids is canonically a smooth loopoid (which is nontrivial in case of loopoids) and this is untrue for the cotangent bundles. After providing a few natural constructions, we show how the Lie-like functor associates with these objects skew-algebroids and almost Lie algebroids, respectively, and how discrete mechanics on Lie groupoids can be reformulated in the nonassociative case.

On Some Aspects of the Courant-Type Algebroids, the Related Coadjoint Orbits and Integrable Systems

“…Motivated by these results, the authors of [13] analyzed many interesting properties of Courant algebroids and the related Poisson-Lie T duality; in particular, they extended the known results to a much wider class of dualities, including cases with gauging, in addition to presenting an illustration of the use of the formalism to provide new classes of special solutions to modified type-two supergravity equations in symmetric spaces. Other interesting properties of Courant algebroids were studied in [12,[14][15][16][17][18][19] within which the authors proposed a Lie algebroid on the loop space pinned down to the Lie algebroid on the manifold. The authors conjectured that this construction, as applied to the Dirac structure, should give rise to the Lie algebroid of symmetries specifying special σ models.…”

On Lie semiheaps and ternary principal bundles

“…Another source of motivation comes from the applications of nonassociative algebras in geometry and geometric mechanics, for example smooth loopoids of Grabowski & Ravanpak (see [11,12]). Ternary operations are, by definition, nonassociative.…”