Post-Lie deformations of pre-Lie algebras and their applications in Regularity Structures
Yvain Bruned, Yunhe Sheng, Rong Tang
TL;DR
The paper develops a deformation theory that morphs a pre-Lie algebra into a post-Lie algebra by constructing a controlling differential graded Lie algebra $(C_{ mathsf{PL}}^*( mathfrak{g}, mathfrak{g}),[ cdot, cdot]_{ mathsf{PL}},d_ hd)$ and introducing post-Lie cohomology $ H^*_{ mathsf{PL}}( mathfrak{g}; mathfrak{g})$. Infinitesimal post-Lie deformations correspond to 2-cocycles in this cohomology, and the Maurer–Cartan equation $d_ hd Pi+ rac{1}{2}[ Pi, Pi]_{ mathsf{PL}}=0$ characterizes actual deformations via $ Pi=( pi, omega)$. A long exact sequence links post-Lie cohomology to the usual pre-Lie cohomology, illuminating derivations and rigidity criteria through $ H^2_{ mathsf{PL}}( mathfrak{g}; mathfrak{g})$, and the formal deformation theory provides a robust framework for classifying and controlling deformations. The theory is then instantiated in Regularity Structures, where the post-Lie structure on decorated trees is shown to be a deformation of a pre-Lie algebra, yielding a concrete formal deformation and clarifying the algebraic underpinnings of renormalization in stochastic PDEs.
Abstract
In this paper, we study post-Lie deformations of a pre-Lie algebra, namely deforming a pre-Lie algebra into a post-Lie algebra. We construct the differential graded Lie algebra that governs post-Lie deformations of a pre-Lie algebra. We also develop the post-Lie cohomology theory for a pre-Lie algebra, by which we classify infinitesimal post-Lie deformations of a pre-Lie algebra using the second cohomology group. The rigidity of such kind of deformations is also characterized using the second cohomology group. Finally, we apply this deformation theory to Regularity Structures. We prove that the post-Lie algebraic structure on the decorated trees which appears spontaneously in Regularity Structures is a post-Lie deformation of a pre-Lie algebra.