Algebraic aspects of connections: from torsion, curvature, and post-Lie algebras to Gavrilov's double exponential and special polynomials
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866908284286402560 |
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| author | Al-Kaabi, M. J. H. Ebrahimi-Fard, K. Manchon, D. Munthe-Kaas, H. Z. |
| author_facet | Al-Kaabi, M. J. H. Ebrahimi-Fard, K. Manchon, D. Munthe-Kaas, H. Z. |
| contents | Understanding the algebraic structure underlying a manifold with a general affine connection is a natural problem. In this context, A. V. Gavrilov introduced the notion of framed Lie algebra, consisting of a Lie bracket (the usual Jacobi bracket of vector fields) and a magmatic product without any compatibility relations between them. In this work we will show that an affine connection with curvature and torsion always gives rise to a post-Lie algebra as well as a $D$-algebra. The notions of torsion and curvature together with Gavrilov's special polynomials and double exponential are revisited in this post-Lie algebraic framework. We unfold the relations between the post-Lie Magnus expansion, the Grossman-Larson product and the $K$-map, $α$-map and $β$-map, three particular functions introduced by Gavrilov with the aim of understanding the geometric and algebraic properties of the double-exponential, which can be understood as a geometric variant of the Baker-Campbell-Hausdorff formula. We propose a partial answer to a conjecture by Gavrilov, by showing that a particular class of geometrically special polynomials is generated by torsion and curvature. This approach unlocks many possibilities for further research such as numerical integrators and rough paths on Riemannian manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_04381 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Algebraic aspects of connections: from torsion, curvature, and post-Lie algebras to Gavrilov's double exponential and special polynomials Al-Kaabi, M. J. H. Ebrahimi-Fard, K. Manchon, D. Munthe-Kaas, H. Z. Differential Geometry Rings and Algebras 53C05, 17D25 Understanding the algebraic structure underlying a manifold with a general affine connection is a natural problem. In this context, A. V. Gavrilov introduced the notion of framed Lie algebra, consisting of a Lie bracket (the usual Jacobi bracket of vector fields) and a magmatic product without any compatibility relations between them. In this work we will show that an affine connection with curvature and torsion always gives rise to a post-Lie algebra as well as a $D$-algebra. The notions of torsion and curvature together with Gavrilov's special polynomials and double exponential are revisited in this post-Lie algebraic framework. We unfold the relations between the post-Lie Magnus expansion, the Grossman-Larson product and the $K$-map, $α$-map and $β$-map, three particular functions introduced by Gavrilov with the aim of understanding the geometric and algebraic properties of the double-exponential, which can be understood as a geometric variant of the Baker-Campbell-Hausdorff formula. We propose a partial answer to a conjecture by Gavrilov, by showing that a particular class of geometrically special polynomials is generated by torsion and curvature. This approach unlocks many possibilities for further research such as numerical integrators and rough paths on Riemannian manifolds. |
| title | Algebraic aspects of connections: from torsion, curvature, and post-Lie algebras to Gavrilov's double exponential and special polynomials |
| topic | Differential Geometry Rings and Algebras 53C05, 17D25 |
| url | https://arxiv.org/abs/2205.04381 |