Semi-simple Leibniz algebras I
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866914641638064128 |
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| author | Feldvoss, Jörg |
| author_facet | Feldvoss, Jörg |
| contents | The goal of this paper is to describe the structure of finite-dimensional semi-simple Leibniz algebras in characteristic zero. Our main tool in this endeavor are hemi-semidirect products. One of the major results of this paper is a simplicity criterion for hemi-semidirect products. In addition, we characterize when a hemi-semidirect product is semi-simple or Lie-simple. Using these results we reduce the classification of finite-dimensional semi-simple Leibniz algebras over fields of characteristic zero to the well-known classification of finite-dimensional semi-simple Lie algebras and their finite-dimensional irreducible modules. As one consequence of our structure theorem, we determine the derivation algebra of a finite-dimensional semi-simple Leibniz algebra in characteristic zero as a vector space. This generalizes a recent result of Ayupov et al. from the complex numbers to arbitrary fields of characteristic zero. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_05588 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Semi-simple Leibniz algebras I Feldvoss, Jörg Rings and Algebras Representation Theory 17A32, 17A60, 17A36, 17A70 The goal of this paper is to describe the structure of finite-dimensional semi-simple Leibniz algebras in characteristic zero. Our main tool in this endeavor are hemi-semidirect products. One of the major results of this paper is a simplicity criterion for hemi-semidirect products. In addition, we characterize when a hemi-semidirect product is semi-simple or Lie-simple. Using these results we reduce the classification of finite-dimensional semi-simple Leibniz algebras over fields of characteristic zero to the well-known classification of finite-dimensional semi-simple Lie algebras and their finite-dimensional irreducible modules. As one consequence of our structure theorem, we determine the derivation algebra of a finite-dimensional semi-simple Leibniz algebra in characteristic zero as a vector space. This generalizes a recent result of Ayupov et al. from the complex numbers to arbitrary fields of characteristic zero. |
| title | Semi-simple Leibniz algebras I |
| topic | Rings and Algebras Representation Theory 17A32, 17A60, 17A36, 17A70 |
| url | https://arxiv.org/abs/2401.05588 |