Rubenthaler independently showed that any finite-dimensional reductive Lie algebra and its finite-dimensional representation can be embedded into some (finite or infinite-dimensional) graded Lie algebra ([5, the author], [3,H.

In the sense of Theorem 1.2, we can obtain a graded Lie algebra such that a given representation of a reductive Lie algebra can be embedded into its local part.

In [3, Theorem 3.1.2], he constructed a local Lie algebra [GAMMA]([g.sub.0], [B.sub.0], [pi]) from a fundamental triplet ([g.sub.0], [B.sub.0], ([pi], U)), which consists of a quadratic Lie algebra ([g.sub.0], [B.sub.0]) and its finite-dimensional representation ([pi], U), and constructed a graded Lie algebra [g.sub.min] ([GAMMA]([g.sub.0], [B.sub.0], [pi])) using [1, Proposition 4] by V.

Thus, the theories of graded Lie algebras by the author and by H.

PVs and graded Lie algebras. In this section, we shall consider how to describe the regularity of PVs using the theory of graded Lie algebras.

Let Gr(G) denote the graded Lie algebra over z defined by

([a.sub.1], ..., [a.sub.q] [member of] [H.sub.1](X)), we obtain a graded Lie algebra homomorphism

of graded Lie algebras induce an isomorphism Gr(F)/I [equivalent] Gr(G)?

Among their topics are no-go theorems and

graded Lie algebras, representations of the super-Poincare algebra, superspace formalism and superfields, supersymmetric Lagrangians, and supersymmetric Gauge theories.

Their topics include supersymmetric (SUSY) field theory in four and more dimensions, highlights on SUSY phenomenology, and SUSY from a string point of view in particles and fields;

graded Lie algebras and applications and experimental tests of SUSY in atomic nuclei; and SUSY in quantum mechanics and random matrices.