On primitive Lie algebras (Q1947660)

The authors give sufficient conditions for Lie algebra primitiveness and examples of primitive Lie algebras and nonprimitive Lie algebras. They prove the following theorems. Theorem 1. Let \(L\) be an Artinian Lie algebra over a field. Let \(L\) have a unique minimal ideal. Then the Lie algebra \(L\) is primitive. Theorem 2. Let \(L_1\) and \(L_2\) be primitive algebras having faithful irreducible representations \(\varphi_i : L_i\to M_i\) such that the centroids \(\Delta_i\) of the modules \(M_i\) coincide with the basic field and \(\varphi_i(L_i) \cap \Delta_i = 0\), where \(i = 1, 2\). Then their direct sum \(L_1 \oplus L_2\) is also primitive.