Classification of N-(super)-extended Poincar\'e algebras and bilinear invariants of the spinor representation of Spin(p,q)

We classify extended Poincar\'e Lie super algebras and Lie algebras of any signature (p,q), that is Lie super algebras and Z_2-graded Lie algebras g = g_0 + g_1, where g_0 = so(V) + V is the (generalized) Poincar\'e Lie algebra of the pseudo Euclidean vector space V = R^{p,q} of signature (p,q) and g_1 = S is the spinor so(V)-module extended to a g_0-module with kernel V. The remaining super commutators {g_1,g_1} (respectively, commutators [g_1, g_1]) are defined by an so(V)-equivariant linear mapping vee^2 g_1 -> V (respectively, wedge^2 g_1 -> V). Denote by P^+(n,s) (respectively, 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 P^+-(n,s) reduces to the construction of all so(V)-invariant bilinear forms on S and to the calculation of three 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^+-(n,s) = \dim P^+-(n,s) of independent Lie super algebras and algebras, which take values 0,1,2,3,4 or 6. Due to Bott periodicity, L^+-(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) -> (-n,s) with respect to the axis s=0 interchanges L^+ and L^- : L^+(-n,s) = L^-(n,s).