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

Abstract

We classify extended Poincare 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) Poincare 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.