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.

Then there exists a (finite or infinite dimensional) graded Lie algebra L(g, [pi], U, U, B) = [[direct sum].

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.

0]) and its finite-dimensional representation ([pi], U), and constructed a graded Lie algebra [g.

2] by the author, we can find the structure of a graded Lie algebra [g.

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

1](X)), we obtain a graded Lie algebra homomorphism

q+1)] be the graded Lie algebra associated to the lower central series [{[F.

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

Graded Lie algebras and regular prehomogeneous vector spaces with one- dimensional scalar multiplication Nagatoshi SASANO

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.