Continuous inverse ambiguous functions on Lie groups

In Schmitz (Aequ Math 91:373–389, 2017), the first author defines an inverse ambiguous function on a group G to be a bijective function \(f: G \rightarrow G\) $f:G\to G$ satisfying the functional equation \(f^{-1}(x) = f(x^{-1})\) $f^{-1}(x)=f(x^{-1})$ for all \(x \in G\) $x\in G$ . In this paper, we investigate the existence of continuous inverse ambiguous functions on various Lie groups. In particular, we look at tori, elliptic curves over various fields, vector spaces, additive matrix groups, and multiplicative matrix groups.