Universal averages in gauge actions
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.
 Publication:

arXiv eprints
 Pub Date:
 November 2019
 arXiv:
 arXiv:1911.03907
 Bibcode:
 2019arXiv191103907L
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematics  Differential Geometry;
 17B01;
 17B55;
 55P62
 EPrint:
 19 pages, 3 figures