Post-Lie deformations of pre-Lie algebras and their applications in Regularity Structures

TL;DR

This paper develops a deformation theory for pre-Lie algebras into post-Lie algebras, constructs relevant cohomology, and applies it to Regularity Structures, revealing the algebraic structure of decorated trees.

Contribution

It introduces a cohomology framework for post-Lie deformations of pre-Lie algebras and applies this to understand algebraic structures in Regularity Structures.

Findings

Post-Lie algebra structures can be obtained as deformations of pre-Lie algebras.

The second cohomology group classifies infinitesimal deformations and rigidity.

Post-Lie algebraic structures on decorated trees are deformations of pre-Lie algebras.

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.