Internal supersymmetry and unification

We construct a family of finite-dimensional representations of the superalgebra sl(n/m) that depend on an integer parameter for m > 1 and on a complex parameter, b, for m = 1. We describe some models of elementary particles for sl(2/1), sl(3/1), and sl(5/1). This involves the choice of the parameter b and the choice of the operators I(3) (the third component of the weak left-handed isospin) and U (the weak hypercharge). These must commute, and are related to the electric charge by the usual formula Q = I(3) + (1/2)U. In particular, taking I(3) to be in its standard form in su(2) subset sl(5) subset sl(5/1) and requiring that U commute with color su(3) subset sl(5) subset sl(5/1) leaves three free parameters, two for the choice of U and one for the choice of b. We show that there are just two possible choices of these parameters yielding exactly all 32 quark and lepton charges: the Georgi-Glashow U in su(5), corresponding to U(1,-(2/3)) and arbitrary b and U(0,(1/3)) negated set membership su(5), with b = 2. We provide a general construction of representations of sl(n/1) consisting exactly of sequences of generations of quarks and leptons.