I'm interested in a classification of the real forms of the general linear Lie superalgebra $\mathfrak{gl}\_{m|m}(\mathbb{C})$.


The real forms of the *simple* complex Lie superalgebras were classified by Serganova in the paper *Classification of real simple Lie superalgebras and symmetric spaces*. For $\mathfrak{gl}\_{m|n}(\mathbb{C})$, with $m \ne n$, we have a decomposition of Lie superalgebras $\mathfrak{gl}\_{m|n}(\mathbb{C}) = \mathfrak{sl}\_{m|n}(\mathbb{C}) \oplus \mathbb{C}$. Since $\mathfrak{sl}\_{m|n}(\mathbb{C})$ is simple, and the only real form of the abelian complex Lie algebra $\mathbb{C}$ is the abelian real Lie algebra $\mathbb{R}$, we can use Serganova's classification to get all the real forms of $\mathfrak{gl}\_{m|n}(\mathbb{C})$.


The situation for $\mathfrak{gl}\_{m|m}(\mathbb{C})$ is more subtle, since it does not decompose as a direct sum of the corresponding simple Lie superalgebra $\mathfrak{psl}\_{m|m}(\mathbb{C})$ and a center. There are natural enlargements of the real forms of $\mathfrak{psl}\_{m|m}(\mathbb{C})$ to real forms of $\mathfrak{gl}\_{m|m}(\mathbb{C})$. But it's not clear to me if one obtains *all* real forms of $\mathfrak{gl}\_{m|m}(\mathbb{C})$ in this way.


This seems a natural enough question that I expected to find an answer in the literature that somehow bootstraps off the known classification of real forms for the simple Lie superalgebras. However, I haven't been able to locate it. I'm also interested in the analogous question for the Lie superalgebras of types $P$ and $Q$.



Real forms of the general linear Lie superalgebra
It turns out that one *does* get all real forms of $\mathfrak{gl}\_{m|m}(\mathbb{C})$ in the way described in the question. To see this, one can show that a real form of $\mathfrak{gl}\_{m|m}(\mathbb{C})$ is uniquely determined by the induced real form of $\mathfrak{psl}\_{m|m}(\mathbb{C})$. This same method also works for the Lie superalgebras of type $P$ and $Q$. I've written up the details in Appendix B of [this paper](https://arxiv.org/abs/2301.01414).


