Associative Structures in Pseudo-Riemannian Lie Algebras
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866913157888344064 |
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| author | Montoya, Santiago Castañeda Fernández-Culma, Edison Alberto |
| author_facet | Montoya, Santiago Castañeda Fernández-Culma, Edison Alberto |
| contents | This paper investigates the algebraic and geometric consequences of the associativity of the symmetric part $U$ of the Levi-Civita connection on a pseudo-Riemannian Lie algebra $(\mathfrak{g}, \langle \cdot, \cdot \rangle)$. We demonstrate that in the Riemannian (positive-definite) setting, the associativity of $U$ is an extremely restrictive condition that forces the tensor to vanish identically, thereby recovering the class of bi-invariant metrics. In contrast, in the pseudo-Riemannian setting, we focus on the subclass where $(\mathfrak{g}, U)$ is associative and unimodular. As a primary result, we establish that every connected Lie group endowed with a left-invariant pseudo-Riemannian metric whose $U$-tensor is associative and unimodular is geodesically complete. Finally, we explore the families of 2-step nilpotent and almost-abelian Lie algebras. For the latter, we obtain some rigid structural classifications, showing that the paradigmatic models are the $3$-dimensional Heisenberg algebra with certain (anti)-Lorentzian metrics or a semi-direct extension involving a nondegenerate infinitesimal $β$-transformation on the canonical neutral space $W \oplus W^*$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_24204 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Associative Structures in Pseudo-Riemannian Lie Algebras Montoya, Santiago Castañeda Fernández-Culma, Edison Alberto Differential Geometry Mathematical Physics 53C50 This paper investigates the algebraic and geometric consequences of the associativity of the symmetric part $U$ of the Levi-Civita connection on a pseudo-Riemannian Lie algebra $(\mathfrak{g}, \langle \cdot, \cdot \rangle)$. We demonstrate that in the Riemannian (positive-definite) setting, the associativity of $U$ is an extremely restrictive condition that forces the tensor to vanish identically, thereby recovering the class of bi-invariant metrics. In contrast, in the pseudo-Riemannian setting, we focus on the subclass where $(\mathfrak{g}, U)$ is associative and unimodular. As a primary result, we establish that every connected Lie group endowed with a left-invariant pseudo-Riemannian metric whose $U$-tensor is associative and unimodular is geodesically complete. Finally, we explore the families of 2-step nilpotent and almost-abelian Lie algebras. For the latter, we obtain some rigid structural classifications, showing that the paradigmatic models are the $3$-dimensional Heisenberg algebra with certain (anti)-Lorentzian metrics or a semi-direct extension involving a nondegenerate infinitesimal $β$-transformation on the canonical neutral space $W \oplus W^*$. |
| title | Associative Structures in Pseudo-Riemannian Lie Algebras |
| topic | Differential Geometry Mathematical Physics 53C50 |
| url | https://arxiv.org/abs/2605.24204 |