Semisimple graded Lie algebras

The concept of metric is introduced for graded Lie algebras. Semisimple graded Lie algebras are defined in terms of metric conditions of nonsingularity. It is shown that for this class of algebras the metric tensor generates a quadratic Casimir operator. Also for this class, the grading representation is irreducible and its weights are related to the roots of the Lie algebra (’’root-weight theorem’’). The problem is solved to find all semisimple graded Lie algebras. For SU(N), N≳2, for O(N), N≳5, and for all exceptional groups there are none. For all other semisimple Lie algebras there is one and only one. These are explicity constructed in terms of a convenient realization of Sp(2N) matrices. SU(2) is discussed in some detail and a new group [GSU(2)] is found which leaves a mixed c-number/q-number quadratic form invariant. We also define irreducible tensor operators for this group. SU(N), N≳2, provides examples of nonsemisimple gradings.