Identities with coefficients in simple compact Lie groups

TL;DR

This paper investigates the behavior of word maps with coefficients in simple compact Lie groups, proving a conjecture for specific groups using a ping-pong argument, which enhances understanding of their algebraic structure.

Contribution

It proves a conjecture that non-constant word maps with coefficients are non-constant functions for certain simple compact Lie groups, expanding knowledge in group theory.

Findings

Proved the conjecture for groups A_r, B_r, E_6, and G_2

Used a ping-pong argument to establish non-constancy

Confirmed the conjecture for simple compact Lie groups with trivial center

Abstract

We conjecture that if $G$ is a simple compact Lie group with trivial center, then every $d$-variable non-constant word map with coefficients in $G$ defines a non-constant function on $G^d$. We prove the conjecture for $A_r$, $B_r$, $E_6$, and $G_2$ using a ping-pong argument.