On automorphisms of affine superspaces
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914968510660608 |
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| author | Shu, Bin |
| author_facet | Shu, Bin |
| contents | In this note, we propose a super version of Jacobian conjecture on the automorphisms of affine superspaces over an algebraically closed field $\mathbb{F}$ of characteristic $0$, which predicts that for a homomorphism $φ$ of the polynomial superalgebra $\mathcal{R}:=\mathbb{F}[x_1,\ldots,x_m; ξ_1,\ldots,ξ_m]$ over $\mathbb{F}$, if $φ$ satisfies the super version of Jacobian condition (SJ for short), then $φ$ gives rise to an automorphism of the affine superspace $\mathbb{A}_{\mathbb{F}}^{m|n}$. We verify the conjecture if additionally, the set $\mathscr{M}$ of maximal $\mathbb{Z}_2$-homogeneous ideals of $\mathcal{R}$ is assumed to be preserved under $φ$. The statement is actually proved in any characteristic, i.e. a homomorphism $φ$ gives rise to an automorphism of $\mathbb{A}_{\mathbb{F}}^{m|n}$ if SJ is satisfied with $φ$ and the set $\mathscr{M}$ is preserved under $φ$ for an algebraically closed field $\mathbb{F}$ of any characteristic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_07008 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On automorphisms of affine superspaces Shu, Bin Algebraic Geometry Group Theory Representation Theory 14A10, 14R15, 14A22, 16S38 In this note, we propose a super version of Jacobian conjecture on the automorphisms of affine superspaces over an algebraically closed field $\mathbb{F}$ of characteristic $0$, which predicts that for a homomorphism $φ$ of the polynomial superalgebra $\mathcal{R}:=\mathbb{F}[x_1,\ldots,x_m; ξ_1,\ldots,ξ_m]$ over $\mathbb{F}$, if $φ$ satisfies the super version of Jacobian condition (SJ for short), then $φ$ gives rise to an automorphism of the affine superspace $\mathbb{A}_{\mathbb{F}}^{m|n}$. We verify the conjecture if additionally, the set $\mathscr{M}$ of maximal $\mathbb{Z}_2$-homogeneous ideals of $\mathcal{R}$ is assumed to be preserved under $φ$. The statement is actually proved in any characteristic, i.e. a homomorphism $φ$ gives rise to an automorphism of $\mathbb{A}_{\mathbb{F}}^{m|n}$ if SJ is satisfied with $φ$ and the set $\mathscr{M}$ is preserved under $φ$ for an algebraically closed field $\mathbb{F}$ of any characteristic. |
| title | On automorphisms of affine superspaces |
| topic | Algebraic Geometry Group Theory Representation Theory 14A10, 14R15, 14A22, 16S38 |
| url | https://arxiv.org/abs/2410.07008 |