Alfonso Di Bartolo, Gianmarco La Rosa, Manuel Mancini
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Non-Nilpotent Leibniz Algebras with One-Dimensional Derived Subalgebra
In this paper we study non-nilpotent non-Lie Leibniz \(\mathbb {F}\)-algebras with one-dimensional derived subalgebra, where \(\mathbb {F}\) is a field with \({\text {char}}(\mathbb {F}) \ne 2\). We prove that such an algebra is isomorphic to the direct sum of the two-dimensional non-nilpotent non-Lie Leibniz algebra and an abelian algebra. We denote it by \(L_n\), where \(n=\dim _\mathbb {F}L_n\). This generalizes the result found in Demir et al. (Algebras and Representation Theory 19:405-417, 2016), which is only valid when \(\mathbb {F}=\mathbb {C}\). Moreover, we find the Lie algebra of derivations, its Lie group of automorphisms and the Leibniz algebra of biderivations of \(L_n\). Eventually, we solve the coquecigrue problem for \(L_n\) by integrating it into a Lie rack.
期刊介绍:
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.