LIE ALGEBRAS CONSTRUCTED WITH LIE MODULES AND THEIR POSITIVELY AND NEGATIVELY GRADED MODULES

In this paper, we shall give a way to construct a graded Lie algebra L(g, rho, V, nu, B-0) from a standard pentad (g, rho, V, nu, B-0) which consists of a Lie algebra g which has a non-degenerate invariant bilinear form B-0 and g-modules (rho, V) and nu subset of Hom(V, F) all defined over a field F with characteristic 0. In general, we do not assume that these objects are finite-dimensional. We can embed the objects g, rho, V, nu into L(g, rho, V, nu, B-0). Moreover, we construct specific positively and negatively graded modules of L(g, rho, V, nu, B-0). Finally, we give a chain rule on the embedding rules of standard pentads.