# Osp, USp, SU(,) and PSU

I would be glad if someone can give me some (hopefully easy to understand!) references for learning about these groups Osp, USp and PSU and their representations.

I run into these mostly while studying about supersymmetry and I wonder if PSU groups are the same as what are called supergroups".

I frequently run into $Osp(3\vert4)$, $USp(2N)$, $PSU(2,2\vert 4)$, $SU(2,2)$ and $SU(2\vert 4)$.

(I don't understand this notation of having a "$\vert$" and I don't know the definition of any of these groups)

Some of the concepts about them that I face are,

• The commuting subalgebra of the lie algebra of $SU(2,2,\vert m)$ is that of $SU(2,1\vert m-1)$. The generators of $SU(2,1\vert m-1)$ are related to those of $SU(2,2,\vert m)$ via some "obvious reduction". I don't know what that means.

• The "bosonic subgroup" of $SU(2,1\vert m-1)$ is apparently $SU(2,1) \times U(m-1)$. I wonder what this means.

• The supersymmetry generators are supposed to transform as "bifundamentals" under the bosonic subalgebra of $SU(2,2)$ and $SU(m)$. (I don't know what is a "bifundamental")

I am very confused about these concepts and would be grateful if someone one help traverse this subject.

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shouldn't this be on physics? –  yoyo Mar 2 '11 at 16:47
Well. I came to these groups from physics literature but I guess at the end of the day it is a question of group/representation theory (probably of some exotic kind!) –  Anirbit Mar 3 '11 at 5:52

Supergroups are a little tricky. It may be useful to learn about the corresponding Lie superalgebras instead. A possible place to start with learning the basics and notation is Kac's article, "A sketch of Lie superalgebra theory": http://projecteuclid.org/euclid.cmp/1103900590

In particular, you may be interested in the section on p.58 on real Lie superalgebras where you'll find the lowercase (Lie superalgebra) versions of what you want.

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Thanks for the reference! I will take a look at your suggestion. Ever since posting this question I have also come across two other large-review-ish writings by Kac on such topics. But it seems you have a shorter reference! :P –  Anirbit Jun 1 '12 at 19:48