CLASSIFICATION OF N-(SUPER)-EXTENDED POINCARE ALGEBRAS AND BILINEAR INVARIANTS OF THE SPINOR REPRESENTATION OF SPIN (P, Q)

We classify extended Poincare Lie super algebras and Lie algebras of a ny signature (p,q), that is Lie super algebras (resp. Z(2)-graded Lie algebras) g = g(0) +g(1), where g(0) = so(V)+V is the (generalized) Po incare 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 B-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)-equiva riant linear mapping. V-2 g(1) --> V (respectively, boolean AND(2)g(1) --> V). Denote by P+(n, s) (respectively, P-(n,s)) the vector space o f all such Lie super algebras (respectively, Lie algebras), where n = p + g = dimV and s = p - q is the classical signature. The description of pi(n, s) reduces to the construction of all so(V)-invariant biline ar forms on S and to the calculation of three Z(2)-valued invariants f or some of them. This calculation is based on a simple explicit model of an irreducible Clifford module S for the Clifford algebra Cl-p,Cl-q of arbitrary signature (p,q). As a result of the classification, we o btain the numbers L(+/-)(n, s) = dim P+/-(n, s) of independent Lie sup er algebras and algebras, which take values 0,1,2,3,4 or 6. Due to Bot t periodicity, L(+/-)(n, s) may be considered as periodic functions wi th 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 = 3. Moreover, the reflection (n, s) --> (-n , s) with respect to the axis n = 0 interchanges L(+) and L(-) : L(+)( -n, s) = L(-)(n, s).