Relationship between Symmetric Brace Algebras and Pre-Lie Algebras

Abstract

In this paper, we review the definitions of brace algebras, symmetric brace algebras, and pre-Lie algebras. We also look at a few examples of the calculations used in the brace algebra relations. We discuss the results of other mathematicians in these fields and where the topic of symmetric brace algebras is used. We then show a direct proof that a symmetric brace algebra is isomorphic to a pre-Lie algebra, using only the definitions. Showing that a symmetric brace algebra yields a pre-Lie algebra is very straightforward. However, the converse, a pre-Lie algebra yields a symmetric brace algebra, is not obvious. This paper uses an inductive proof to show the symmetry holds and that the brace is well-defined. Combinatorics plays a large factor in showing the brace is well-defined, along with identifying and classifying terms in a relation with m+n variables.