Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.20639 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911127687921664 |
|---|---|
| author | An, Huihui Yan, Zaili Zhang, Shaoxiang |
| author_facet | An, Huihui Yan, Zaili Zhang, Shaoxiang |
| contents | This paper presents a systematic study of invariant Einstein metrics on basic classical Lie supergroups, whose Lie superalgebras belong to the Kac's classification of finite dimensional classical simple Lie superalgebras over $\mathbb{R}$. We consider a natural family of left invariant metrics parameterized by scaling factors on the simple and Abelian components of the reductive even part, using the canonical bi-invariant bilinear form. Explicit expressions for the Levi-Civita connection and Ricci tensor are derived, and the Einstein condition is reduced to a solvable algebraic system. Our main result shows that, except for the cases of $\mathbf{A}(m,n)$ with $m\neq n$, $\mathbf{F}(4)$, and their real forms, every real basic classical Lie superalgebra admits at least two distinct Einstein metrics. Notably, for $\mathbf{D}(n+1,n)$ and $\mathbf{D}(2,1;α)$, we obtain both Ricci flat and non Ricci flat Einstein metrics, a phenomenon not observed in the non-super setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_20639 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Invariant Einstein metrics on basic classical Lie supergroups An, Huihui Yan, Zaili Zhang, Shaoxiang Differential Geometry Rings and Algebras This paper presents a systematic study of invariant Einstein metrics on basic classical Lie supergroups, whose Lie superalgebras belong to the Kac's classification of finite dimensional classical simple Lie superalgebras over $\mathbb{R}$. We consider a natural family of left invariant metrics parameterized by scaling factors on the simple and Abelian components of the reductive even part, using the canonical bi-invariant bilinear form. Explicit expressions for the Levi-Civita connection and Ricci tensor are derived, and the Einstein condition is reduced to a solvable algebraic system. Our main result shows that, except for the cases of $\mathbf{A}(m,n)$ with $m\neq n$, $\mathbf{F}(4)$, and their real forms, every real basic classical Lie superalgebra admits at least two distinct Einstein metrics. Notably, for $\mathbf{D}(n+1,n)$ and $\mathbf{D}(2,1;α)$, we obtain both Ricci flat and non Ricci flat Einstein metrics, a phenomenon not observed in the non-super setting. |
| title | Invariant Einstein metrics on basic classical Lie supergroups |
| topic | Differential Geometry Rings and Algebras |
| url | https://arxiv.org/abs/2508.20639 |