The Structure of Primitive Leibniz Algebras via Maximal Subalgebras
Mathematics, vol.14, no.9, 2026 (SCI-Expanded, Scopus)
- Publication Type: Article / Article
- Volume: 14 Issue: 9
- Publication Date: 2026
- Doi Number: 10.3390/math14091531
- Journal Name: Mathematics
- Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, zbMATH, Directory of Open Access Journals
- Keywords: centraliser, Leibniz algebra, maximal subalgebra, minimal ideal, primitive algebra
- Süleyman Demirel University Affiliated: Yes
Abstract
In this paper, we investigate the structure of primitive Leibniz algebras via their maximal subalgebras and minimal ideals. Using a two-sided definition of the centraliser, we show that the centraliser of a minimal ideal is again an ideal. Unlike the Lie algebra case, the use of this two-sided centraliser is essential in the Leibniz setting and accommodates genuinely new structural phenomena. In particular, we prove that a primitive Leibniz algebra has at most two minimal ideals and classify such algebras into three distinct types according to the structure of the socle, extending the classical Lie-theoretic classification. In the solvable case, we obtain an alternative characterisation of primitive Leibniz algebras of type 1 in terms of split extensions by self-centralising minimal ideals.