Invariant Connections, Lie Algebra Actions, and Foundations of Numerical Integration on Manifolds

Abstract

Motivated by numerical integration on manifolds, we relate the algebraic properties of invariant connections to their geometric properties. Using this perspective, we generalize some classical results of Cartan and Nomizu to invariant connections on algebroids. This has fundamental consequences for the theory of numerical integrators, giving a characterization of the spaces on which Butcher and Lie-Butcher series methods, which generalize Runge-Kutta methods, may be applied.

The universal pre-Lie-Rinehart algebras of aromatic trees

“…More precisely the multi-aromas identify as orbits of such maps by the action of the symmetric group S n . Now consider the linear map (11) τ :…”

“…The derivation of a scalar field by a vector field is modelled by grafting the tree on the aromas. A suitable geometric model for this is pre-Lie algebroids, defined as Lie algebroids with a flat and torsion free connection [11]. Lie algebroids are vector bundles on a domain together with an 'anchor map', associating sections of the vector bundle with derivations of the ring of smooth scalar functions.…”

Numerical Simulations of Stochastic Differential Equations with Multiple Conserved Quantities by Conservative Methods

“…[2,3,11,13,25,[28][29][30]. In the last decades, the numerical methods preserving the geometric invariants along the flows, such as symplectic and Lie group structures, and quantities showing that the exact solution evolves on a manifold of the dimension smaller than that of n , have been widely studied [5,8,9,27,32]. Such numerical approaches are called the structure-preserving or geometric numerical integration (GNI) methods.…”

Integration and geometrization of Rota-Baxter Lie algebras

“…See [7,8,9,10] for more details about pre-Lie algebras and post-Lie algebras. Recently, the notion of post-Lie algebroids was introduced and applied to numerical integrations [28,29].…”