Lie superalgebras in characteristic 2 and mixed characteristic
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908462276935680 |
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| author | Etingof, Pavel Hu, Serina |
| author_facet | Etingof, Pavel Hu, Serina |
| contents | We define the notion of a Lie superalgebra over a field $k$ of characteristic $2$ which unifies the two pre-existing ones - $\mathbb{Z}/2$-graded Lie algebras with a squaring map and Lie algebras in the Verlinde category ${\rm Ver}_4^+(k)$, and prove the PBW theorem for this notion. We also do the same for the restricted version. Finally, discuss mixed characteristic deformation theory of such Lie superalgebras (for perfect $k$), introducing and studying a natural lift of our notion of Lie superalgebra to characteristic zero - the notion of a mixed Lie superalgebra over a ramified quadratic extension $R$ of the ring of Witt vectors $W(k)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_17457 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lie superalgebras in characteristic 2 and mixed characteristic Etingof, Pavel Hu, Serina Representation Theory Category Theory Quantum Algebra Rings and Algebras We define the notion of a Lie superalgebra over a field $k$ of characteristic $2$ which unifies the two pre-existing ones - $\mathbb{Z}/2$-graded Lie algebras with a squaring map and Lie algebras in the Verlinde category ${\rm Ver}_4^+(k)$, and prove the PBW theorem for this notion. We also do the same for the restricted version. Finally, discuss mixed characteristic deformation theory of such Lie superalgebras (for perfect $k$), introducing and studying a natural lift of our notion of Lie superalgebra to characteristic zero - the notion of a mixed Lie superalgebra over a ramified quadratic extension $R$ of the ring of Witt vectors $W(k)$. |
| title | Lie superalgebras in characteristic 2 and mixed characteristic |
| topic | Representation Theory Category Theory Quantum Algebra Rings and Algebras |
| url | https://arxiv.org/abs/2507.17457 |