Pre-Lie algebras with divided powers and the Deligne groupoid in positive characteristic
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911333826428928 |
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| author | Verstraete, Marvin |
| author_facet | Verstraete, Marvin |
| contents | The purpose of this paper is to develop a deformation theory controlled by pre-Lie algebras with divided powers over a ring of positive characteristic. We show that every differential graded pre-Lie algebra with divided powers comes with operations, called weighted braces, which we use to generalize the classical deformation theory controlled by Lie algebras over a field of characteristic $0$. Explicitly, we define the Maurer-Cartan set, as well as the gauge group, and prove that there is an action of the gauge group on the Maurer-Cartan set. This new deformation theory moreover admits a Goldman-Millson theorem which remains valid on the integers. As an application, we give the computation of the $π_0$ of a mapping space $\text{Map}(B^c(\mathcal{C}),\mathcal{P})$ with $\mathcal{C}$ and $\mathcal{P}$ suitable cooperad and operad respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_20300 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Pre-Lie algebras with divided powers and the Deligne groupoid in positive characteristic Verstraete, Marvin Algebraic Topology 17B55 18M70 17A30 20L05 The purpose of this paper is to develop a deformation theory controlled by pre-Lie algebras with divided powers over a ring of positive characteristic. We show that every differential graded pre-Lie algebra with divided powers comes with operations, called weighted braces, which we use to generalize the classical deformation theory controlled by Lie algebras over a field of characteristic $0$. Explicitly, we define the Maurer-Cartan set, as well as the gauge group, and prove that there is an action of the gauge group on the Maurer-Cartan set. This new deformation theory moreover admits a Goldman-Millson theorem which remains valid on the integers. As an application, we give the computation of the $π_0$ of a mapping space $\text{Map}(B^c(\mathcal{C}),\mathcal{P})$ with $\mathcal{C}$ and $\mathcal{P}$ suitable cooperad and operad respectively. |
| title | Pre-Lie algebras with divided powers and the Deligne groupoid in positive characteristic |
| topic | Algebraic Topology 17B55 18M70 17A30 20L05 |
| url | https://arxiv.org/abs/2310.20300 |