2024/03/22 by Nikola Kovačević, Kovačević, Nikola
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2403.14959
openalex publication_date 2024/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we prove that the variety Cm(L) of commuting m-tuples of elements of simple Lie algebra L is often reducible. Explicitely, we prove it is reducible for all simple Lie algebra L not isomorphic to \mathfraksl2 and \mathfraksl)3, and all m ≥ 4. We also prove it is reducible for C3(L) for L of types Bk,Ck,E7,E8,F4,G2, k ≥ 2, as well as for Dl for l ≥ 10. We do this by proving Theorem on Adding Diagonals, that says that if we can find a simple Lie subalgebra L' whose Dynkin diagram is a subdiagram of the Dynkin diagram of L, then under mild conditions, from the fact that Cm(L') is reducible, it follows that Cm(L) is also reducible.