arXiv subjects
Publications and source records attributed to Ludwig Rahm.
This paper defines the algebraic structure of tracial post-Lie-Rinehart algebras and describes the free object in this category. Post-Lie-Rinehart algebras is a generalisation of pre-Lie-Rinehart algebras, and of post-Lie algebroids.
We survey the Munthe-Kaas--Wright Hopf algebra defined on planar rooted trees. This algebra serves a role akin to that of the Butcher--Connes--Kreimer Hopf algebra on non-planar rooted trees within the domain of numerical methods for ordinary differential equations. In the course of our presentation, we revisit Foissy's work on finite-dimensional comodules over the Butcher--Connes--Kreimer Hopf algebra and expand on his findings to include the Munthe-Kaas--Wright Hopf algebra. This involves detailing its endomorphisms and recursively constructing its primitive elements. These results are applied within the context of rough paths, where we describe an isomorphism between planarly branched and geometric rough paths. Our approach hinges on the extension of the Guin--Oudom construction to post-Lie algebras. In the case of the free post-Lie algebra defined on planar rooted trees, it yields the dual of the Munthe-Kaas--Wright Hopf algebra. Surprisingly, we uncover a natural connection between the concept of bialgebras in cointeraction and the Guin--Oudom construction. Prompted by the rough path perspective, we explore this finding through the lens of translations on rough paths. Additionally, we investigate the geometric embedding for planar regularity structures.
Branched rough paths, used to solve ODEs on $\mathbb{R}$, have been generalised in two different directions. In one direction, there are regularity structures aimed at solving SPDEs on $\mathbb{R}$. In the other direction, there are planarly branched rough paths to solve ODEs on homogeneous spaces. This paper combines these two directions to construct planar regularity structures, for (S)PDEs on homogeneous spaces.
We generalize Bruned et.al.'s notion of translation in geometric and branched rough paths to a notion of translation in rough paths over any combinatorial Hopf algebra. We show that this notion of translation is equivalent to two bialgebras being in cointeraction, subject to certain additional conditions. We argue that reformulating translations in terms of substitutions, provides simpler conditions for the cointeraction formulation. For the special case where the translation can be obtained from a product, we show how to obtain a description of the dual coaction. As a concrete example, we describe translations in planarly branched rough paths.
The paper follows an operadic approach to provide a bialgebraic description of substitution for Lie-Butcher series. We first show how the well-known bialgebraic description for substitution in Butcher's $B$-series can be obtained from the pre-Lie operad. We then apply the same construction to the post-Lie operad to arrive at a bialgebra $\mathcal{Q}$. By considering a module over the post-Lie operad, we get a cointeraction between $\mathcal{Q}$ and the Hopf algebra $\mathcal{H}_N$ that describes composition for Lie-Butcher series. We use this coaction to describe substitution for Lie-Butcher series.