Classification of N-(Super)-Extended Poincaré Algebras and Bilinear Invariants of the Spinor Representation of Spin (p,q)

Abstract:

We classify extended Poincaré Lie super algebras and Lie algebras of any signature (p, q), that is Lie super algebras (resp. Z₂-graded Lie algebras) , where is the (generalized) Poincaré Lie algebra of the pseudo-Euclidean vector space of signature (p,q) and is the spinor -module extended to a -module with kernel V. The remaining super commutators (respectively, commutators ) are defined by an -equivariant linear mapping

Denote by (respectively, ) the vector space of all such Lie super algebras (respectively, Lie algebras), where and is the classical signature. The description of reduces to the construction of all -invariant bilinear forms on S and to the calculation of three -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 of arbitrary signature (p,q). As a result of the classification, we obtain the numbers of independent Lie super algebras and algebras, which take values 0,1,2,3,4 or 6. Due to Bott periodicity, 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 with respect to the axis n= 0 interchanges L ⁺ and L ^-: