Classification of $N$-(super)-extended Poincaré algebras and bilinear invariants of the spinor representation of ${\rm Spin}(p,q)$
Dmitry V. Alekseevsky and Vicente Cortés
We classify extended Poincaré Lie super algebras and Lie algebras of any signature $(p, q)$, that is Lie super algebras and ${\Bbb Z}_2$-graded Lie algebras ${\mathfrak g} = {\mathfrak g}_0 + {\mathfrak g}_1$, where ${\mathfrak g}_0 = {\mathfrak{so}}(V) + V$ is the (generalized) Poincar\'e Lie algebra of the pseudo Euclidean vector space $ V = {\Bbb R}^{p,q}$ of signature $(p,q)$ and ${\mathfrak g}_1 = S$ is the spinor ${\mathfrak{so}}(V)$-module extended to a ${\mathfrak g}_0$-module with kernel $V$. The remaining super commutators $\{{\mathfrak g}_1,{\mathfrak g}_1\}$ (respectively, commutators $[{\mathfrak g}_1, {\mathfrak g}_1]$) are defined by an ${\mathfrak {so}}(V)$-equivariant linear mapping $$ \vee^2{\mathfrak g}_1 \to V \quad (\mbox{respectively},\quad \wedge^2{\mathfrak g}_1 \to V)\, .\hskip1in $$ Denote by ${\cal P}^+(n,s)$ (respectively, ${\cal P}^-(n,s)$) the vector space of all such Lie super algebras (respectively, Lie algebras), where $n = p + q = \dim V$ and $s = p-q $ is the signature. The description of ${\cal P}^{\pm}(n,s)$ reduces to the construction of all ${\mathfrak{so}}(V)$-invariant bilinear forms on $S$ and to the calculation of three ${\Bbb Z}_2$-valued invariants for some of them.
This calculation is based on a simple explicit model of an irreducible Clifford module $S$ for the Clifford algebra $Cl_{p,q}$ of arbitrary signature $(p,q)$. As a result of the classification, we obtain the numbers $L^{\pm}(n,s) = \dim {\cal P}^{\pm}(n,s)$ of independent Lie super algebras and algebras, which take values 0,1,2,3,4 or 6. Due to Bott periodicity, $L^{\pm}(n,s)$ may be considered as periodic functions with period 8 in each argument. They are invariant under the group $\Gamma$ generated by the four reflections with respect to the axes $n=-2$, $n=2$, $s-1 = -2$ and $s-1 =2$. Moreover, the reflection $(n,s) \to (-n,s)$ with respect to the axis $s=0$ interchanges $L^+$ and $L^-$ : $$ L^+(-n,s) = L^-(n,s)\, .\hskip1in $$