Groebner-Shirshov bases for brace algebras

TL;DR

This paper develops a Composition-Diamond lemma for brace algebras, enabling the embedding of any pre-Lie algebra into a brace algebra and providing an explicit basis for such constructions.

Contribution

It introduces a Composition-Diamond lemma for brace algebras and constructs explicit embeddings of pre-Lie algebras into brace algebras with a basis.

Findings

Established Composition-Diamond lemma for brace algebras

Proved every pre-Lie algebra can be embedded into a brace algebra

Determined an explicit linear basis for the constructed brace algebra

Abstract

Let $A$ be a brace algebra. This structure implies that $A$ is also a pre-Lie algebra. In this paper, we establish Composition-Diamond lemma for brace algebras. Using this Composition-Diamond lemma we prove that each pre-Lie algebra $L$ can be embedded into a brace algebra $A_L$, i.e., $L$ is a pre-Lie subalgebra of $A_L$ up to isomorphism. We also determine an explicit linear basis for the brace algebra $A_{L}$.