Jordan–Chevalley Decomposition in Lie Algebras. Issue 2 (28th February 2019)

• Record Type: Journal Article
• Authors: Cagliero, Leandro; Szechtman, Fernando
• Is Part Of: Canadian mathematical bulletin =. Volume 62:Issue 2(2019)
• Page Start: 349
• Page End: 354
• Publication Date: 2019-02-28
• Subjects: 17-08, -- 17B05, -- 20C40, -- 15A21; solvable Lie algebra, -- Jordan–Chevalley decomposition, -- representation; Mathematics -- Periodicals; Mathematics; Periodicals; 510.5
• DOI: 10.4153/CMB-2018-023-7
• Languages: English
• ISSNs: 0008-4395

Abstract

Abstract: We prove that if $\mathfrak{s}$ is a solvable Lie algebra of matrices over a field of characteristic 0 and $A\in \mathfrak{s}$, then the semisimple and nilpotent summands of the Jordan–Chevalley decomposition of $A$ belong to $\mathfrak{s}$ if and only if there exist $S, N\in \mathfrak{s}$, $S$ is semisimple, $N$ is nilpotent (not necessarily $[S, N]=0$ ) such that $A=S+N$ .