Universal averages in gauge actions

TL;DR

The paper introduces a universal symmetric averaging method for Lie algebra elements that generalizes geometric means of Lie group elements, with applications to differential graded Lie algebra models.

Contribution

It constructs a universal, symmetric average in free Lie algebras that extends geometric means without requiring a Lie group structure.

Findings

Explicit expansion of the average up to three brackets

Properties of iterated averages analyzed

Applications to symmetric differential graded Lie algebra models

Abstract

We give a construction of a universal average of Lie algebra elements whose exponentiation gives (when there is an associated Lie group) a totally symmetric geometric mean of Lie group elements (sufficiently closed to the identity) with the property that in an action of the group on a space $X$ for which $n$ elements all take a particular point $a\in{}X$ to a common point $b\in{}X$, also the mean will take $a$ to $b$. The construction holds without the necessity for the existence of a Lie group and the universal average $\mu_n(x_1,\ldots,x_n)$ is a totally symmetric universal expression in the free Lie algebra generated by $x_1,\ldots,x_n$. Its expansion up to three brackets is found explicitly and various properties of iterated averages are given. There are applications to the construction of explicit symmetric differential graded Lie algebra models. This work is based on the second author's minor thesis.