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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 $S= {\germ {sl}}(2,\bbfK)$, 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:
17B10Representations of Lie algebras, algebraic theory
22E70Applications of Lie groups to physics; explicit representations
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