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Invariants of a semi-direct sum of Lie algebras. (English) Zbl 1058.17002

Summary: 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 \(S= {\mathfrak {sl}}(2,\mathbb{K})\), 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 5. An analysis of the results obtained is made, and this leads to a theorem on invariants of Lie algebras depending only on the elements of certain subalgebras.

MSC:

17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
22E70 Applications of Lie groups to the sciences; explicit representations

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