Symmetrization of Brace Algebras

We show that the symmetrization of a brace algebra structure yields the structure of a symmetric brace algebra

Symmetrization of brace algebra

summary:Summary: We show that the symmetrization of a brace algebra structure yields the structure of a symmetric brace algebra. We also show that the symmetrization of the natural brace structure on $\bigoplus_{k\ge 1}\operatorname{Hom}(V^{\otimes k},V)$ coincides with the natural symmetric brace structure on $\bigoplus_{k\ge 1}\operatorname{Hom}(V^{\otimes k},V)^{as}$, the direct sum of spaces of antisymmetric maps $V^{\otimes k}\to V$

The pre-Lie structure of the time-ordered exponential

The usual time-ordering operation and the corresponding time-ordered exponential play a fundamental role in physics and applied mathematics. In this work we study a new approach to the understanding of time-ordering relying on recent progress made in the context of enveloping algebras of pre-Lie algebras. Various general formulas for pre-Lie and Rota-Baxter algebras are obtained in the process. Among others, we recover the noncommutative analog of the classical Bohnenblust-Spitzer formula, and get explicit formulae for operator products of time-ordered exponentials

Right-handed Hopf algebras and the preLie forest formula

Three equivalent methods allow to compute the antipode of the Hopf algebras of Feynman diagrams in perturbative quantum field theory (QFT): the Dyson-Salam formula, the Bogoliubov formula, and the Zimmermann forest formula. Whereas the first two hold generally for arbitrary connected graded Hopf algebras, the third one requires extra structure properties of the underlying Hopf algebra but has the nice property to reduce drastically the number of terms in the expression of the antipode (it is optimal in that sense).The present article is concerned with the forest formula: we show that it generalizes to arbitrary right-handed polynomial Hopf algebras. These Hopf algebras are dual to the enveloping algebras of preLie algebras -a structure common to many combinatorial Hopf algebras which is carried in particular by the Hopf algebras of Feynman diagrams