Communications in mathematical physics vol:253 issue:2 pages:385-422
A class of Z(2)-graded Lie algebra and Lie superalgebra extensions of the pseudo-orthogonal algebra of a spacetime of arbitrary dimension and signature is investigated. They have the form g = g(0)+ g(1), with g(0) = s(o)(V) + W-0 and g(1) = W-1, where the algebra of generalized translations W = W-0 + W-1 is the maximal solvable ideal of g, W-0 is generated by W-1 and commutes with W. Choosing W-1 to be a spinorial s(o)(V)-module ( a sum of an arbitrary number of spinors and semispinors), we prove that W-0 consists of polyvectors, i.e. all the irreducible so(V)-submodules of W-0 are submodules of boolean AND V. We provide a classification of such Lie (super) algebras for all dimensions and signatures. The problem reduces to the classification of so(V)-invariant boolean AND(k)V-valued bilinear forms on the spinor module S.