Affinization of algebraic structures: Leibniz algebras
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909666640920576 |
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| author | Brzeziński, Tomasz Radziszewski, Krzysztof Pérez, Brais Ramos |
| author_facet | Brzeziński, Tomasz Radziszewski, Krzysztof Pérez, Brais Ramos |
| contents | A general procedure of affinization of linear algebra structures is illustrated by the case of Leibniz algebras. Specifically, the definition of an affine Leibniz bracket, that is, a bi-affine operation on an affine space that at each tangent vector space becomes a (bi-linear) Leibniz bracket in terms of a tri-affine operation called a Leibnizian, is given. An affine space together with such an operation is called a Leibniz affgebra. It is shown that any Leibniz algebra can be extended to a family of Leibniz affgebras. Depending on the choice of a Leibnizian different types of Leibniz affgebras are introduced. These include: derivative-type, which captures the derivation property of linear Leibniz bracket; homogeneous-type, which is based on the simplest and least restrictive choice of the Leibnizian; Lie-type which includes all Lie affgebras introduced in [R.R. Andruszkiewicz, T. Brzeziński & K. Radziszewski, Lie affgebras vis-à-vis Lie algebras, Res. Math. 80 (2025), art. 61.]. Each type is illustrated by examples with prescribed Leibniz algebra fibres. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_23398 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Affinization of algebraic structures: Leibniz algebras Brzeziński, Tomasz Radziszewski, Krzysztof Pérez, Brais Ramos Rings and Algebras Quantum Algebra 14R10, 17A32, 20N10 A general procedure of affinization of linear algebra structures is illustrated by the case of Leibniz algebras. Specifically, the definition of an affine Leibniz bracket, that is, a bi-affine operation on an affine space that at each tangent vector space becomes a (bi-linear) Leibniz bracket in terms of a tri-affine operation called a Leibnizian, is given. An affine space together with such an operation is called a Leibniz affgebra. It is shown that any Leibniz algebra can be extended to a family of Leibniz affgebras. Depending on the choice of a Leibnizian different types of Leibniz affgebras are introduced. These include: derivative-type, which captures the derivation property of linear Leibniz bracket; homogeneous-type, which is based on the simplest and least restrictive choice of the Leibnizian; Lie-type which includes all Lie affgebras introduced in [R.R. Andruszkiewicz, T. Brzeziński & K. Radziszewski, Lie affgebras vis-à-vis Lie algebras, Res. Math. 80 (2025), art. 61.]. Each type is illustrated by examples with prescribed Leibniz algebra fibres. |
| title | Affinization of algebraic structures: Leibniz algebras |
| topic | Rings and Algebras Quantum Algebra 14R10, 17A32, 20N10 |
| url | https://arxiv.org/abs/2506.23398 |