Lifts of endomorphisms of Weyl algebras modulo $p^2$
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866911411476627456 |
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| author | Lauritzen, Niels Thomsen, Jesper Funch |
| author_facet | Lauritzen, Niels Thomsen, Jesper Funch |
| contents | Let $φ$ denote a $k$-algebra endomorphism of the $n$-th Weyl algebra $A_n(k)$ over a perfect field $k$ of positive characteristic $p$. We prove that $φ$ can be lifted to an endomorphism of the Weyl algebra $A_n(W_2(k))$ over the Witt vectors $W_2(k)$ of length two over $k$ if and only if $φ$ induces a Poisson morphism of the center of $A_n(k)$. Furthermore, we improve a result of Tsuchimoto, which enables us to conclude that these equivalent statements hold at least when ${\rm deg}(φ) < p$. In particular, we conclude that $φ$ is injective if ${\rm deg}(φ) < p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_23110 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Lifts of endomorphisms of Weyl algebras modulo $p^2$ Lauritzen, Niels Thomsen, Jesper Funch Rings and Algebras Algebraic Geometry Quantum Algebra Let $φ$ denote a $k$-algebra endomorphism of the $n$-th Weyl algebra $A_n(k)$ over a perfect field $k$ of positive characteristic $p$. We prove that $φ$ can be lifted to an endomorphism of the Weyl algebra $A_n(W_2(k))$ over the Witt vectors $W_2(k)$ of length two over $k$ if and only if $φ$ induces a Poisson morphism of the center of $A_n(k)$. Furthermore, we improve a result of Tsuchimoto, which enables us to conclude that these equivalent statements hold at least when ${\rm deg}(φ) < p$. In particular, we conclude that $φ$ is injective if ${\rm deg}(φ) < p$. |
| title | Lifts of endomorphisms of Weyl algebras modulo $p^2$ |
| topic | Rings and Algebras Algebraic Geometry Quantum Algebra |
| url | https://arxiv.org/abs/2601.23110 |