Related papers: Post-Lie algebras in Regularity Structures
The set of primitive elements of a Hopf algebra in the braided category of group graded vector spaces (with a commutative group) carry the structure of a generalized Lie algebra. In particular the graded derivations of an associative…
We construct an universal enveloping algebra associated to the ternary extension of Lie (super)algebras called Lie algebra of order three. A Poincar\'e-Birkhoff-Witt theorem is proven is this context. It this then shown that this universal…
For a Poisson algebra $A$, by exploring its relation with Lie-Rinehart algebras, we prove a Poincar\'e-Birkoff-Witt theorem for its universal enveloping algebra $A^e$. Some general properties of the universal enveloping algebras of Poisson…
We consider the approach of replacing trees by multi-indices as an index set of the abstract model space $\mathsf{T}$ introduced by Otto, Sauer, Smith and Weber to tackle quasi-linear singular SPDEs. We show that this approach is consistent…
A formalism of lattice supersymmetry based on a lattice-deformed superalgebra which was originally introduced in the link approach formulation is presented. We propose that the superalgebra can in fact be identified as a Hopf algebra,…
Typed decorated trees are used by Bruned, Hairer and Zambotti to give a description of a renormalisation processon stochastic PDEs. We here study the algebraic structures on these objects: multiple prelie algebrasand related operads…
In this paper, we introduce the notion of post-Hopf algebroids, generalizing the pre-Hopf algebroids introduced in [Bronasco, Laurent, 2025] in the study of exotic aromatic S-series. We construct action post-Hopf algebroids through actions…
We show how the Hopf algebra structure of renormalization discovered by Kreimer can be found in the Epstein-Glaser framework without using an analogue of the forest formula of Zimmermann.
In this paper we focus on a certain self-distributive multiplication on coalgebras, which leads to so-called rack bialgebra. Inspired by semi-group theory (adapting the Suschkewitsch theorem), we do some structure theory for rack bialgebras…
Recent advances in stochastic PDEs, Hopf algebras of typed trees and integral equations have inspired the study of algebraic structures with replicating operations. To understand their algebraic and combinatorial nature, we first use rooted…
The Hopf algebra and the Rota-Baxter algebra are the two algebraic structures underlying the algebraic approach of Connes and Kreimer to renormalization of perturbative quantum field theory. In particular the Hopf algebra of rooted trees…
The PostLie algebra is an enriched structure of the Lie algebra that has recently arisen from operadic study. It is closely related to pre-Lie algebra, Rota-Baxter algebra, dendriform trialgebra, modified classical Yang-Baxter equations and…
A Hom-type algebra is called involutive if its Hom map is multiplicative and involutive. In this paper, we obtain an explicit construction of the free involutive Hom-associative algebra on a Hom-module. We then apply this construction to…
The notion of quantum algebras is merged with that of Lie systems in order to establish a new formalism called Poisson-Hopf algebra deformations of Lie systems. The procedure can be naturally applied to Lie systems endowed with a symplectic…
We propose several constructions of commutative or cocommutative Hopf algebras based on various combinatorial structures, and investigate the relations between them. A commutative Hopf algebra of permutations is obtained by a general…
We give a universal construction of families of Hopf $P$-algebras for any Hopf operad $P$. As a special case, we recover the Connes-Kreimer Hopf algebra of rooted trees.
Since the Connes--Kreimer Hopf algebra was proposed, revisiting present quantum field theory has become meaningful and important from algebraic points. In this paper, the Hopf algebra in the cutting rules is constructed. Its coproduct…
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…
We introduce a notion of Hopf-Lie-Rinehart algebra and show that the universal algebra of a Hopf-Lie-Rinehart algebra acquires an ordinary Hopf algebra structure.
Using the categorical approach to Poincar\'e-Birkhoff-Witt type theorems from our previous work with Tamaroff, we prove three such theorems: for universal enveloping Rota-Baxter algebras of tridendriform algebras, for universal enveloping…