I have seen that Lie superalgebras are classified by some algebras like $\mathfrak{osp}(m|2n)$, but I don't know how to fit super Poincar\'e algebras into this. Especially what are the fundamental representations of them for various $\mathcal{N}$ and when massive/massless respectively?

Edit: I just realized that the super Poincar\'e algebras are not semisimple, so they are definitely different from $\mathfrak{osp}(m|2n)$, but I guess it is still possible to represent them by matrices.


I just found a good reference:

Freund, P. (1986). Introduction to Supersymmetry (Cambridge Monographs on Mathematical Physics). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511564017

The relevant part is chapter 4. Let me summarize it in the following:

(Note that I did not indicate real and complex algebras explicitly, but I think they should be clear from the context.)

The basic idea is that a non semisimple Lie (super)algebra can be constructed from a semisimple one by Wigner-Inönü contraction. Then using some 'accidental' isomorphism between low dimensional simple lie algebras, we can find a correct way to embed the (uncontracted simple) Poincaré algebra into the bosonic part of $\mathfrak{osp}(m|2n)$.

  1. Wigner-Inönü contraction

The idea of Wigner-Inönü contraction is that a curved space locally looks like a flat one. A baby example would be that a 2-sphere (with the usual metric) locally looks like a (flat) plane. Therefore the isometry group $SO(3)$ becomes to $ISO(2)$ in this limit. Now let's see how this can be done in a more systematical way. The Lie algebra $\mathfrak{so}(3)$ is well known: $$[J_1, J_2]=J_3\\ [J_2, J_3]=J_1\\ [J_3, J_1]=J_2$$ Now let's redefine $\bar{J_1}=\lambda J_1, \bar{J_2}=\lambda J_2,\bar{J_3}=J_3$, then the brackets become to: $$[\bar{J_1}, \bar{J_2}]=\lambda^2\bar{J_3}\\ [\bar{J_2}, \bar{J_3}]=\bar{J_1}\\ [\bar{J_3}, \bar{J_1}]=\bar{J_2}$$ In the limit $\lambda\rightarrow 0$, this gives the Lie algebra $\mathfrak{iso}(2)$.

With the same idea one can get the Poincaré algebra by contracting either $\mathfrak{so}(1,4)$ or $\mathfrak{so}(2,3)$, which generate the isometry of de Sitter space and anti de Sitter space.

  1. Embedding $\mathfrak{so}(1,4)$ or $\mathfrak{so}(2,3)$ into a simple Lie superalgebra.

The bosonic part of $\mathfrak{osp}(m|2n)$ is $\mathfrak{so}(m)\oplus\mathfrak{sp}(2n)$. However, we cannot embed $\mathfrak{so}(1,4)$ or $\mathfrak{so}(2,3)$ into the $\mathfrak{so}(m)$ part because this includes the scalar and vector representation of the Lorentz algebra (therefore the supercharges would transform in these two representations rather than the spinor representation). It turns out that $\mathfrak{so}(2,3)$ is isomorphic $\mathfrak{sp}(4,\mathbb{R})$ (which makes sense by looking at the Dynkin diagrams of $\mathfrak{so}(5,\mathbb{C})$ and $\mathfrak{sp}(4,\mathbb{C})$). Besides, the natural representation of $\mathfrak{sp}(4,\mathbb{R})$ includes the Lorentz algebra as its (Dirac) spin representation, which makes sense by looking at $$\mathfrak{spin}(1,3)_{\mathbb{C}}\cong\mathfrak{sl}(2,\mathbb{C})\oplus\mathfrak{sl}(2,\mathbb{C})\cong\mathfrak{sp}(2,\mathbb{C})\oplus\mathfrak{sp}(2,\mathbb{C})\subseteq\mathfrak{sp}(4,\mathbb{C}).$$

A careful Wigner-Inönü contraction of $\mathfrak{osp}(m|4)$ will give us an $\mathcal{N}=m$ super Poincaré algebra with internal symmetry and cental charges as the other bosonic part $\mathfrak{so}(m)$.


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