Simple supersymmetries

Abstract

Two infinite families of simple graded Lie algebras (GLA’s) over the complex numbers are described: the special linear algebras SL(m‖n) [whose Bose sector is the direct sum of a one-dimensional algebra with the ordinary Lie algebra SL(m) ×SL(n)], and the orthosymplectic algebras OSp(2r‖s) [with Bose sector Sp(2r) ×O(s)]. The GLA’s of physics fit into these two families either directly or via Inönü–Wigner contraction. These algebras along with further exceptional GLA’s constitute all the classical GLA’s, i.e., all GLA’s with a nondegenerate metric (not necessarilly Killing) form. The existence of infinite families of hyperexceptional GLA’s, i.e., of GLA’s that are simple but not classical is pointed out.