Embedding of non-simple Lie groups,coupling constant relations and non-uniqueness of models of unification

A general derivation of the coupling constant relations which result on embedding a non-simple group like SU _L (2) ⇔ U(1) in a larger simple group (or graded Lie group) is given. It is shown that such relations depend only on the requirement (i) that the multiplet of vector fields form an irreducible representation of the unifying algebra and (ii) the transformation properties of the fermions under SU _L (2). This point is illustrated in two ways, one by constructing two different unification groups containing the same fermions and therefore have same Weinberg angle; the other by putting different SU _L (2) structures on the same fermions and consequently have different Weinberg angles. In particular the value sin²ϑ=3/8 is characteristic of the sequential doublet models or models which invoke a large number of additional leptons likeE ₆, while addition of extra charged fermion singlets can reduce the value of sin² ϑ to 1/4. We point out that at the present time the models of grand unification are far from unique.