In this paper, we shall give a way to construct a graded Lie algebra $L(\mathfrak{ g},\rho,V,{ \cal V},B_0)$ from a standard pentad $(\mathfrak{ g},\rho,V,{ \cal V},B_0)$ which consists of a Lie algebra $\mathfrak{ g}$ which has a non-degenerate invariant bilinear form $B_0$ and $\mathfrak{ g}$-modules $(\rho, V)$ and ${ \cal V}\subset \mathrm { Hom }(V,\mathfrak{ k})$ all defined over a field $\mathfrak{ k}$. In general, we do not assume that these objects are finite-dimensional. We can embed the objects $\mathfrak{ g},\rho,V,{ \cal V}$ into $L(\mathfrak{ g},\rho,V,{ \cal V},B_0)$. Moreover, we construct specific positively and negatively graded modules of $L(\mathfrak{ g},\rho,V,{ \cal V},B_0)$. Finally, we give a chain rule on the embedding rules of standard pentads.
Categories: math.RT