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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Online-Zugang: | https://arxiv.org/abs/2403.14005 |
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| _version_ | 1866914722618540032 |
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| author | Grigorian, Sergey |
| author_facet | Grigorian, Sergey |
| contents | In this paper we explore algebraic and geometric structures that arise on parallelizable manifolds. Given a parallelizable manifold $\mathbb{L}$, there exists a global trivialization of the tangent bundle, which defines a map $ρ_p:\mathfrak{l} \longrightarrow T_p \mathbb{L}$ for each point $p \in \mathbb{L}$, where $\mathfrak{l}$ is some vector space. This allows us to define a particular class of vector fields, known as fundamental vector fields, that correspond to each element of $\mathfrak{l}$. Furthermore, flows of these vector fields give rise to a product between elements of $% \mathfrak{l}$ and $\mathbb{L}$, which in turn induces a local loop structure (i.e. a non-associative analog of a group). Furthermore, we also define a generalization of a Lie algebra structure on $\mathfrak{l}$. We will describe the properties and examples of these constructions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_14005 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Algebraic structures on parallelizable manifolds Grigorian, Sergey Rings and Algebras Differential Geometry 17D99 (Primary), 53B05 (Secondary) In this paper we explore algebraic and geometric structures that arise on parallelizable manifolds. Given a parallelizable manifold $\mathbb{L}$, there exists a global trivialization of the tangent bundle, which defines a map $ρ_p:\mathfrak{l} \longrightarrow T_p \mathbb{L}$ for each point $p \in \mathbb{L}$, where $\mathfrak{l}$ is some vector space. This allows us to define a particular class of vector fields, known as fundamental vector fields, that correspond to each element of $\mathfrak{l}$. Furthermore, flows of these vector fields give rise to a product between elements of $% \mathfrak{l}$ and $\mathbb{L}$, which in turn induces a local loop structure (i.e. a non-associative analog of a group). Furthermore, we also define a generalization of a Lie algebra structure on $\mathfrak{l}$. We will describe the properties and examples of these constructions. |
| title | Algebraic structures on parallelizable manifolds |
| topic | Rings and Algebras Differential Geometry 17D99 (Primary), 53B05 (Secondary) |
| url | https://arxiv.org/abs/2403.14005 |