Heisenberg-type groups over rings with involution

Question

Hello everyone!

In a paper "Coverings of twisted Chevalley groups over commutative rings" Eiichi Abe intoroduced the following construction:

Let $R$ be a commutative ring and $x\mapsto\overline{x}$ an involution on it. Set $$A=\left\{(a,b)\ |\ a\overline{a}=b+\overline{b}\right\}\subset R\times R.$$ The set $A$ has binary operation on it, namely $(a,b)\cdot(c,d)=(a+c,b+d+\overline{a}c)$.

This is connected with what's called Heisenberg group $H$ of a form $B$, where $B$ is a sesquilinear antihermitian form on a right $R$-module $V$. As a set, it coincides with $V\times R$, an operation is defined by $(u,a)\cdot(v,b)=(u+v,a+b-B(u,v))$. Abe's group $A$ coincides with so-called maximal form parameter, that is $$\Lambda_{max}=\left\{ \xi\in H\ |\ tr(\xi)=0\right\}$$ where $tr((u,a))=a-\overline{a}+B(u,u)$.

I'm interested if those groups has been studied by someone, at least over some classes of fields? In case of finite field with non-trivial involution and complex numbers it's easy to study them, but in other cases it isn't. I tried to google it, but I failed.

Answer 1

I wasn't aware of the paper by Abe that you mention, but I have used the group $A$ that you described in the case where $R$ is an octonion division algebra, in order to describe the rank one forms of groups of type $F_4$ (over arbitrary fields), in my paper "Moufang sets of type $F_4$" (with H. Van Maldeghem), Math. Z. 265 (2010), no. 3, 511-527. (See p. 513 of that paper; there is an unimportant sign difference compared to the group that you describe.)

In some cases, it is more convenient to construct the same group in a slightly different fashion, as $$ B = R \times S ,$$ where $S$ is the set of skew-symmetric elements $S = \{ r \in R \mid \bar{r} = -r \}$, with $$ (a,b) + (c,d) = (b + d + \bar{a}c - \bar{c}a).$$ This construction makes sense in a very general situation (namely when $R$ is a structurable algebra, which is a certain kind of (not necessarily commutative nor associative) algebra with involution, and is related to the Tits-Kantor-Koecher construction of Lie algebras from structurable algebras. (I think you have to assume that $2 \in R^\times$ in order to make $A$ and $B$ isomorphic.)

I'm not sure what kind of properties of these groups you are interested in, though, so depending on that my answer might or might not be useful to you.