CLASSIFICATION OF N-(SUPER)-EXTENDED POINCARÉ ALGEBRAS AND BILINEAR INVARIANTS OF THE SPINOR

REPRESENTATION OF Spin(p, q)

DMITRY V. ALEKSEEVSKY VICENTE CORTÉS

Abstract. We classify extended Poincaré 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) Poincaré 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

∨^{2}𝔤_{1} → V (respectively, ∧^{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 Γ 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) .