Continuous Inverse Ambiguous Functions on Lie Groups

TL;DR

This work classifies when continuous inverse ambiguous functions exist on classical Lie groups by translating the problem into the existence of a square root of the inversion map within the homeomorphism group. It provides both obstructions from topology—via fundamental groups and orientation—and explicit constructions across families including tori, elliptic curves, vector spaces, and matrix groups. The results include nonexistence on $S^1$ and on odd-dimensional tori, existence on even tori, and a nuanced landscape for elliptic curves (with existence tied to the discriminant in real cases and cubic roots over finite fields), as well as parallel conclusions for vector spaces and matrix groups depending on field and dimension. Together, the findings illuminate when a “square root of inversion” can be realized as a continuous automorphism, connecting topology, algebraic geometry, and Lie group theory with potential implications for symmetry and automorphism structures in geometric contexts.

Abstract

In a previous study, the first author defines an inverse ambiguous function on a group $G$ to be a bijective function $f : G \to G$ satisfying the functional equation $f^{-1}(x) = f(x^{-1})$ for all $x \in G$. In this paper, we investigate the existence of continuous inverse ambiguous functions on classical Lie groups. In particular, we look at tori, elliptic curves over various fields, vector spaces, additive matrix groups, and multiplicative matrix groups.

Theorems & Definitions (35)

• Theorem 1.1
• Lemma 2.1
• proof
• Theorem 2.2
• proof
• Proposition 2.3
• Theorem 2.4
• Remark 2.5
• Theorem 3.1
• proof