Connections and Finsler geometry of the structure group of a JB-algebra
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866914932557086720 |
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| author | Larotonda, Gabriel Luna, José |
| author_facet | Larotonda, Gabriel Luna, José |
| contents | We endow the Banach-Lie structure group $Str(V)$ of an infinite dimensional JB-algebra $V$ with a left-invariant connection and Finsler metric, and we compute all the quantities of its connection. We show how this connection reduces to $G(Ω)$, the group of transformations that preserve the positive cone $Ω$ of the algebra $V$, and to $Aut(V)$, the group of Jordan automorphisms of the algebra. We present the cone $Ω$ as an homogeneous space for the action of $G(Ω)$, therefore inducing a quotient Finsler metric and distance. With the techniques introduced, we prove the minimality of the one-parameter groups in $Ω$ for any symmetric gauge norm in $V$. We establish that the two presentations of the Finsler metric in $Ω$ give the same distance there, which helps us prove the minimality of certain paths in $G(Ω)$ for its left-invariant Finsler metric. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_09208 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Connections and Finsler geometry of the structure group of a JB-algebra Larotonda, Gabriel Luna, José Differential Geometry Functional Analysis Group Theory Metric Geometry 22E65, 58B20 (Primary) 53C22 (Secondary) We endow the Banach-Lie structure group $Str(V)$ of an infinite dimensional JB-algebra $V$ with a left-invariant connection and Finsler metric, and we compute all the quantities of its connection. We show how this connection reduces to $G(Ω)$, the group of transformations that preserve the positive cone $Ω$ of the algebra $V$, and to $Aut(V)$, the group of Jordan automorphisms of the algebra. We present the cone $Ω$ as an homogeneous space for the action of $G(Ω)$, therefore inducing a quotient Finsler metric and distance. With the techniques introduced, we prove the minimality of the one-parameter groups in $Ω$ for any symmetric gauge norm in $V$. We establish that the two presentations of the Finsler metric in $Ω$ give the same distance there, which helps us prove the minimality of certain paths in $G(Ω)$ for its left-invariant Finsler metric. |
| title | Connections and Finsler geometry of the structure group of a JB-algebra |
| topic | Differential Geometry Functional Analysis Group Theory Metric Geometry 22E65, 58B20 (Primary) 53C22 (Secondary) |
| url | https://arxiv.org/abs/2206.09208 |