Birational equivalence of the Zassenhaus varieties for basic classical Lie superalgebras and their purely-even reductive Lie subalgebras in odd characteristic
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866912066746449920 |
|---|---|
| author | Shu, Bin Zheng, Lisun Ren, Ye |
| author_facet | Shu, Bin Zheng, Lisun Ren, Ye |
| contents | Let $\mathfrak{g}=\mathfrak{g}_{\bar 0}\oplus\mathfrak{g}_{\bar 1}$ be a basic classical Lie superalgebra over an algebraically closed field $\textbf{k}$ of characteristic $p>2$. Denote by $\mathcal{Z}$ the center of the universal enveloping algebra $U(\mathfrak{g})$. Then $\mathcal{Z}$ turns out to be finitely-generated purely-even commutative algebra without nonzero divisors.
In this paper, we demonstrate that the fraction $\text{Frac}(\mathcal{Z})$ is isomorphic to $\text{Frac}(\mathfrak{Z})$ for the center $\mathfrak{Z}$ of $U(\mathfrak{g}_{\bar 0})$. Consequently, both Zassenhaus varieties for $\mathfrak{g}$ and $\mathfrak{g}_{\bar 0}$ are birationally equivalent via a subalgebra $\widetilde{mathcal{Z}}\subset\mathcal{Z}$, and $\text{Spec}(\mathcal{Z})$ is rational under the standard hypotheses. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_07292 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Birational equivalence of the Zassenhaus varieties for basic classical Lie superalgebras and their purely-even reductive Lie subalgebras in odd characteristic Shu, Bin Zheng, Lisun Ren, Ye Representation Theory Primary 17B50, Secondary 17B35, 17B45, 14E08, 14M20 Let $\mathfrak{g}=\mathfrak{g}_{\bar 0}\oplus\mathfrak{g}_{\bar 1}$ be a basic classical Lie superalgebra over an algebraically closed field $\textbf{k}$ of characteristic $p>2$. Denote by $\mathcal{Z}$ the center of the universal enveloping algebra $U(\mathfrak{g})$. Then $\mathcal{Z}$ turns out to be finitely-generated purely-even commutative algebra without nonzero divisors. In this paper, we demonstrate that the fraction $\text{Frac}(\mathcal{Z})$ is isomorphic to $\text{Frac}(\mathfrak{Z})$ for the center $\mathfrak{Z}$ of $U(\mathfrak{g}_{\bar 0})$. Consequently, both Zassenhaus varieties for $\mathfrak{g}$ and $\mathfrak{g}_{\bar 0}$ are birationally equivalent via a subalgebra $\widetilde{mathcal{Z}}\subset\mathcal{Z}$, and $\text{Spec}(\mathcal{Z})$ is rational under the standard hypotheses. |
| title | Birational equivalence of the Zassenhaus varieties for basic classical Lie superalgebras and their purely-even reductive Lie subalgebras in odd characteristic |
| topic | Representation Theory Primary 17B50, Secondary 17B35, 17B45, 14E08, 14M20 |
| url | https://arxiv.org/abs/2410.07292 |