What do we call the set of elements fixed by an involution of the second kind?

Question

If $A$ is an algebra over a field $F$, and $\sigma:A\rightarrow A$ is an involution of the second kind, then it seems natural to talk about the set $S=\{a\in A\mid\sigma(a)=a\}$. I am not finding any standard notation or terminology for this. Admittedly I just got my copy of The Book of Involutions, but I'm giving a talk tomorrow and would love to have a concise and known way of referring to that set.

I realize that when $F$ is of the form $K(\sqrt{-d})$ where $d\in K^+$ and $K\subset\mathbb{R}$, $S$ is often just $K$, but this is not the case in the examples I'm working with.

So what do we call it? Also is there a common notation for it?

Answer 1

Not sure why this was a comment rather than an answer so...

"I believe the standard notation in the book is $Sym(A,\sigma)$. You can call them symmetric elements, or simply elements fixed by $\sigma$. – @dbluesk"

Answer 2

If $G$ is a group acting on a set $X$, one notation you can use for the set of fixed points of the action is $X^G$. And if $f$ is a function acting on a set $X$, one notation you can use for the set of fixed points of $f$ is $\text{Fix}(f)$. So I think either $A^{\sigma}$ or $\text{Fix}(\sigma)$ is fine, depending on whether you want to emphasize $A$ or $\sigma$.