Dr. Jean-David Jacques

Given a commutative algebra A, we exhibit a canonical structure of post-Lie algebra on the space A⊗Der(A) where Der(A) is the space of derivations on A, in order to use the machinery given in [Guin & Oudom 2008] and [Ebrahimi-Fard & Lundervold & Munthe-Kaas 2015] and to define a Hopf algebra structure on the associated enveloping algebra with a natural action on A. We apply these results to the setting of [Linares & Otto & Tempelmayr 2023], giving a simpler and more efficient construction of their action and extending the recent work [Bruned & Katsetsiadis]. This approach gives an optimal setting to perform explicit computations in the associated structure group.

In order to derive a class of geometric-type deformations of post-Lie algebras, we first extend the geometrical notions of torsion and curvature for a general bilinear operation on a Lie algebra, then we derive compatibility conditions which will ensure that the post-Lie structure remains preserved. This type of deformation applies in particular to the post-Lie algebra introduced in arXiv:2306.02484v3 in the context of regularity structures theory. We use this deformation to derive a pre-Lie structure for the regularity structures approach given in arXiv:2103.04187v4, which is isomorphic to the post-Lie algebra studied in arXiv:2306.02484v3 at the level of their associated Hopf algebras. In the case of sections of smooth vector bundles of a finite-dimensional manifold, this deformed structure contains also, as a subalgebra, the post-Lie algebra structure introduced in arXiv:1203.4738v3 in the geometrical context of moving frames.