Topic
Split Lie algebra
About: Split Lie algebra is a research topic. Over the lifetime, 8 publications have been published within this topic receiving 108 citations.
Papers
Journal Article•
TL;DR: In this article, the notion of compatible actions was introduced in the context of split ex- tensions of Lie algebras over a finite number of pages, and the cohomology of semi-direct products was computed using compatible actions.
Abstract: We introduce the notion of compatible actions in the context of split ex- tensions of Lie algebras over a eld k. Using compatible actions, we construct new resolutions to compute the cohomology of semi-direct products of Lie algebras and give an alternative way to construct the Hochschild-Serre spectral sequence associated to a split extension. Finally, we describe several instances in which this spectral sequence collapses at the second page and obtain a sharper bound for its length in the nite dimensional case.
10 citations
TL;DR: In this article, the notion of integrable root in the framework of split Lie triple systems T with a coherent 0-root space was introduced and a local finiteness theorem for T is given.
Abstract: We focus on the notion of an integrable root in the framework of split Lie triple systems T with a coherent 0-root space. As a main result, it is shown that if T has all its nonzero roots integrable, then its standard embedding is a split Lie algebra having all its nonzero roots integrable. As a consequence, a local finiteness theorem for split Lie triple systems, saying that whenever all nonzero roots of T are integrable then T is locally finite, is stated. Finally, a classification theorem for split simple Lie triple systems having all its nonzero roots integrable is given.
8 citations
29 May 2012
TL;DR: In this paper, the structure of split Malcev algebras of arbitrary dimension over an algebraically closed field of characteristic zero was studied, and it was shown that M is the direct sum of a semisimple split Lie algebra and a direct sum for simple non-Lie Malceve algesbras.
Abstract: We study the structure of split Malcev algebras of arbitrary dimension over an algebraically closed field of characteristic zero. We show that any such algebras M is of the form \(M={\mathcal U} +\sum_{j}I_{j}\) with \({\mathcal U}\) a subspace of the abelian Malcev subalgebra H and any Ij a well described ideal of M satisfying [Ij,Ik] = 0 if j ≠ k. Under certain conditions, the simplicity of M is characterized and it is shown that M is the direct sum of a semisimple split Lie algebra and a direct sum of simple non-Lie Malcev algebras.
2 citations
TL;DR: In this paper, the authors introduced strongly split Lie algebras, a class of strongly graded Lie algebra, and showed that if a strongly split lie algebra is centerless, then it is the direct sum of split ideals, each of which is a split-simple strongly split algebra.
Abstract: Split Lie algebras are possibly the most known examples of graded Lie algebras. Since an important category in the class of graded algebras is the category of strongly graded algebras, we introduce, in a natural way, the category of strongly split Lie algebras 𝔏 and show that if 𝔏 is centerless, then 𝔏 is the direct sum of split ideals each of which is a split-simple strongly split Lie algebra.
TL;DR: In this article, it was proved that the property of being a semisimple algebra is preserved under projections (lattice isomorphisms) for locally finite-dimensional Lie algebras over a perfect field of characteristic not equal to 2 and 3, except for the projection of a three-dimensional simple nonsplit algebra.
Abstract: It is proved that the property of being a semisimple algebra is preserved under projections (lattice isomorphisms) for locally finite-dimensional Lie algebras over a perfect field of characteristic not equal to 2 and 3, except for the projection of a three-dimensional simple nonsplit algebra. Over fields with the same restrictions, we give a lattice characterization of a three-dimensional simple split Lie algebra and a direct product of a one-dimensional algebra and a three-dimensional simple nonsplit one.
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2019 | 1 |
| 2018 | 1 |
| 2016 | 1 |
| 2012 | 2 |
| 2009 | 2 |
| 2008 | 1 |