# notation of super Lie algebras

The classic super Lie algebras are of type A: $\operatorname{sl}(m+1 \mid n+1)$, $m\neq n$, $\operatorname{psl}(n+1 \mid n+1)$, type B: $\operatorname{osp}(2m+1 \mid 2n)$, type C: $\operatorname{osp}(2 \mid 2n)$, type D: $\operatorname{osp}(2m \mid 2n)$.

But I find some paper they study $\operatorname{psu}(2, 2\mid 4)$. What type is this super Lie algebra? Thank you very much.

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## 1 Answer

The classic Lie super-algebras listed are complex Lie algebras, i.e. no operation of conjugation is defined.

The $\mathfrak{psu}(2,2|4)$ is a real form of $\mathfrak{psl}(4,4)$, i.e. there exists an involution $\sigma: \mathfrak{psl}(4,4) \mapsto \mathfrak{psl}(4,4)$ (i.e. $\sigma^2 = \mathrm{id}$) and generators of $\mathfrak{psu}(2,2|4)$ are those linear combinations of generators, that are fixed by $\sigma$.

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thank you very much. Why it is called psu? Are there some reference about this algebra? –  LJR Aug 16 '12 at 16:23
@user9791 I think it goes by the analogy with classical Lie algebras. $\mathfrak{su}$ is a real form of $\mathfrak{sl}$, and $\mathfrak{psu}$ is a real form $\mathfrak{psl}$. The "p" in $\mathfrak{psl}$ stands for projective, I think. –  Sasha Aug 16 '12 at 16:25
thank you very much. –  LJR Aug 16 '12 at 16:52