On Inner Derivations of Leibniz Algebras

Sutida Patlertsin, Suchada Pongprasert () and Thitarie Rungratgasame
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Sutida Patlertsin: Department of Mathematics, Faculty of Science, Srinakharinwirot University, 114 Sukhumvit 23, Bangkok 10110, Thailand
Suchada Pongprasert: Department of Mathematics, Faculty of Science, Srinakharinwirot University, 114 Sukhumvit 23, Bangkok 10110, Thailand
Thitarie Rungratgasame: Department of Mathematics, Faculty of Science, Srinakharinwirot University, 114 Sukhumvit 23, Bangkok 10110, Thailand

Mathematics, 2024, vol. 12, issue 8, 1-9

Abstract: Leibniz algebras are generalizations of Lie algebras. Similar to Lie algebras, inner derivations play a crucial role in characterizing complete Leibniz algebras. In this work, we demonstrate that the algebra of inner derivations of a Leibniz algebra can be decomposed into the sum of the algebra of left multiplications and a certain ideal. Furthermore, we show that the quotient of the algebra of derivations of the Leibniz algebra by this ideal yields a complete Lie algebra. Our results independently establish that any derivation of a semisimple Leibniz algebra can be expressed as a combination of three derivations. Additionally, we compare the properties of the algebra of inner derivations of Leibniz algebras with the algebra of central derivations.

Keywords: Leibniz algebra; Lie algebra; derivation; inner derivation; central derivation; completeness; semisimple (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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