Lie algebras constructed with Lie modules and their positively and negatively graded modules

Abstract

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

Cited by in corpus (4)

• Contragredient Lie algebras and Lie algebras associated with a standard pentad
• Reduced contragredient Lie algebras and PC Lie algebras
• A class of Lie algebras who contains a class of Kac-Moody algebras
• Graded Lie algebras and regular prehomogeneous vector spaces with one-dimensional scalar multiplication