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arXiv · 2212.13298

Invariants of a semi-direct sum of Lie algebras

J C Ndogmo
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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.

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J C Ndogmo. 2022-12-26. Invariants of a semi-direct sum of Lie algebras. https://doi.org/10.1088/0305-4470%2F37%2F21%2F009

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