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

TL;DR

This paper introduces a method to construct graded Lie algebras from standard pentads involving a Lie algebra, modules, and bilinear forms, and explores their graded modules and embedding rules.

Contribution

It provides a new construction framework for graded Lie algebras from standard pentads, including embedding and module construction without finite-dimensional restrictions.

Findings

Constructed graded Lie algebras from standard pentads.

Embedded original objects into the constructed Lie algebra.

Developed a chain rule for embedding rules of standard pentads.

Abstract

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.