Invariants of a semi-direct sum of Lie algebras

TL;DR

This paper investigates the invariants of semi-direct sum Lie algebras, establishing conditions under which all invariants are Casimir operators and providing explicit invariant counts for certain cases.

Contribution

It introduces a new approach to find invariants using differential equations and characterizes invariants based on subalgebra elements, extending previous methods.

Findings

All invariants are Casimir operators when the representation lacks trivial components.

Explicit invariants are computed for semi-direct sums with Levi factor sl(2) up to dimension five.

A theorem relating invariants to elements of specific subalgebras is established.

Abstract

We show that any semi-direct sum $L$ of Lie algebras with Levi factor $S$ must be perfect if the representation associated with it does not possess a copy of the trivial representation. As a consequence, all invariant functions of $L$ must be Casimir operators. When $S= \frak{sl}(2,\mathbb{K}),$ the number of invariants is given for all possible dimensions of $L$. Replacing the traditional method of solving the system of determining PDEs by the equivalent problem of solving a system of total differential equations, the invariants are found for all dimensions of the radical up to five. An analysis of the results obtained is made, and this lead to a theorem on invariants of Lie algebras depending only on the elements of certain subalgebras.