2024/03/09 by Andrew Douglas, Joe Repka, Douglas, Andrew +1
Mathematics · #17B05 #17B10 #17B20 #17B22 #17B30 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2403.18847
openalex publication_date 2024/03/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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. From a finer viewpoint, a subalgebra is λ-wide if the simple module of a semisimple Lie algebra of highest weight λ remains indecomposable when restricted to the subalgebra. A subalgebra is narrow if the restriction of all non-trivial simple modules to the subalgebra have proper decompositions. We determine necessary and sufficient conditions for regular subalgebras of semisimple Lie algebras to be λ-wide. 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. 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. Finally, we show that a regular subalgebra of the special linear algebra \mathfraksln+1 is either narrow or wide; this property does not hold for non-regular subalgebras of \mathfraksln+1.