Narrow and wide regular subalgebras of semisimple Lie algebras

A subalgebra of a semisimple Lie algebra is wide if every simple module of the semisimple Lie algebra remains indecomposable when restricted to the subalgebra. A subalgebra is narrow if the restrictions of all non-trivial simple modules to the subalgebra have proper decompositions. A semisimple Lie algebra is regular extreme if any regular subalgebra of the semisimple Lie algebra is either narrow or wide. We determine necessary and sufficient conditions for a simple module of a semisimple Lie algebra to remain indecomposable when restricted to a regular subalgebra. As a natural consequence, we establish necessary and sufficient conditions for regular subalgebras to be wide, a result which has already been established by Panyushev for essentially all regular solvable subalgebras [10]. Next, we show that establishing whether or not a regular subalgebra of a simple Lie algebra is wide does not require consideration of all simple modules. It is necessary and sufficient to only consider the adjoint representation. Then, we show that all simple Lie algebras are regular extreme. Finally, we show that no non-simple, semisimple Lie algebra is regular extreme.