Cohomology and deformations of restricted Lie algebras and their morphisms in positive characteristic
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915233484767232 |
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| author | Ehret, Quentin Makhlouf, Abdenacer |
| author_facet | Ehret, Quentin Makhlouf, Abdenacer |
| contents | The main purpose of this paper is to study cohomology and develop a deformation theory of restricted Lie algebras in positive characteristic $p>0$. In the case $p\geq3$, it is shown that the deformations of restricted Lie algebras are controlled by the restricted cohomology introduced by Evans and Fuchs. Moreover, we introduce a new cohomology that controls the deformations of restricted morphisms of restricted Lie algebras. In the case $p=2$, we provide a full restricted cohomology complex with values in a restricted module and investigate its connections with formal deformations. Furthermore, we introduce a full deformation cohomology that controls deformations of restricted morphisms of restricted Lie algebras in characteristic $2$. As example, we discuss restricted cohomology with adjoint coefficients of restricted Heisenberg Lie algebras in characteristic $p\geq 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_05929 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cohomology and deformations of restricted Lie algebras and their morphisms in positive characteristic Ehret, Quentin Makhlouf, Abdenacer Representation Theory 17B50, 17B56, 17B60 The main purpose of this paper is to study cohomology and develop a deformation theory of restricted Lie algebras in positive characteristic $p>0$. In the case $p\geq3$, it is shown that the deformations of restricted Lie algebras are controlled by the restricted cohomology introduced by Evans and Fuchs. Moreover, we introduce a new cohomology that controls the deformations of restricted morphisms of restricted Lie algebras. In the case $p=2$, we provide a full restricted cohomology complex with values in a restricted module and investigate its connections with formal deformations. Furthermore, we introduce a full deformation cohomology that controls deformations of restricted morphisms of restricted Lie algebras in characteristic $2$. As example, we discuss restricted cohomology with adjoint coefficients of restricted Heisenberg Lie algebras in characteristic $p\geq 2$. |
| title | Cohomology and deformations of restricted Lie algebras and their morphisms in positive characteristic |
| topic | Representation Theory 17B50, 17B56, 17B60 |
| url | https://arxiv.org/abs/2504.05929 |