Classi cation of 5−dimensional complex nilpotent Leibniz algebras
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- Tez No: 403082
- Danışmanlar: DR. KAILASH MISRA, DR. ERNEST STITZINGER
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- Üniversite: North Carolına State Unıversıty
- Enstitü: Yurtdışı Enstitü
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Özet
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Özet (Çeviri)
Leibniz algebras are certain generalization of Lie algebras. They were introduced by Bloh (1965) who called them D-algebras. Then it was popularized by Loday (1993) and the subject has been studied since then. Lie algebras are known to have many applications in mathematical areas including algebraic geometry, di erential geometry, di erential equations, number theory and also in physical areas such as general relativity, quantum mechanics, quantum eld theory, string theory, particle physics and nuclear physics. The classi cation problem is one of the fundamental and important problems in Lie algebras. The famous Levi-Malcev (1905, 1950) theorem reduce the problem of classifying Lie algebras to classifying semisimple and solvable Lie algebras over a eld of characteristic 0. The semisimple Lie algebras was classi ed by Cartan (1894) and later re ned by Dynkin (1947). Malcev (1950) showed that the problem of classifying solvable Lie algebras can be reduced to classifying nilpotent Lie algebras. So far the complete classi cation of complex nilpotent Lie algebras of dimension n B 7 is known and the classi cation problem of complex nilpotent Lie algebras is wild in higher dimensions. The lack of antisymmetry property in Leibniz algebras makes the classi cation problem more di cult for Leibniz algebras. In this work, we give the classi cation of complex nilpotent non-split non-Lie Leibniz algebras of dimension n B 5. A Leibniz algebras is called non-split if it doesn't have nontrivial ideals as a summand. We introduce the technique involving bilinear forms to obtain the classi cation of complex nilpotent non-split non-Lie Leibniz algebras with one dimensional derived algebra. The remaining cases are done by using some algebraic invariants.
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