LIE ALGEBRAS GENERATED BY LIE MODULES

It is known that a finite-dimensional reductive Lie algebra has a non-degenerate symmetric invariant bilinear form. In this paper, for a given reductive Lie algebra and its finite-dimensional completely reducible representation, we will construct a graded Lie algebra by using a non-degenerate symmetric invariant bilinear form on the reductive Lie algebra. This graded Lie algebra also has a non-degenerate symmetric invariant bilinear form and, moreover, the reductive Lie algebra, its representation and the bilinear form which are used to construct the graded Lie algebra can be embedded into it