Grassmannian perspectives of classical Lie groups and Cartan involutions
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866910007400857600 |
|---|---|
| author | Chen, Yunxia Leung, Naichung Conan |
| author_facet | Chen, Yunxia Leung, Naichung Conan |
| contents | Classical noncompact reductive Lie group $G$ admits a compactification $\overline{G}$ as a Riemannian symmetric space by He. First, we provide a unified construction of these compactifications via Grassmannian geometry and realize the group structures in terms of the geometry of configurations of linear subspaces.
Second, we show that the Cartan involution $ρ$ on $G$ extends uniquely to an isometric involution $\barρ$ on $\overline{G}$ and $\overline{G}^{\barρ} = G^ρ = K$, the maximal compact subgroup of $G$.
Third, we show that $η(g) = ρ(g)^{-1}$ extends uniquely to an isometric involution $\barη$ on $\overline{G}$ and $\overline{G}^{\barη} = G_c/K$, the compact symmetric space dual to $(G^η)_0 = G/K$. This provides a natural generalization of the classical Borel embeddings $G/K \hookrightarrow G_c/K$. Furthermore, $K$ and $G_c/K$ form a complementary pair of reflective submanifolds in $\overline{G}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_00602 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Grassmannian perspectives of classical Lie groups and Cartan involutions Chen, Yunxia Leung, Naichung Conan Differential Geometry Classical noncompact reductive Lie group $G$ admits a compactification $\overline{G}$ as a Riemannian symmetric space by He. First, we provide a unified construction of these compactifications via Grassmannian geometry and realize the group structures in terms of the geometry of configurations of linear subspaces. Second, we show that the Cartan involution $ρ$ on $G$ extends uniquely to an isometric involution $\barρ$ on $\overline{G}$ and $\overline{G}^{\barρ} = G^ρ = K$, the maximal compact subgroup of $G$. Third, we show that $η(g) = ρ(g)^{-1}$ extends uniquely to an isometric involution $\barη$ on $\overline{G}$ and $\overline{G}^{\barη} = G_c/K$, the compact symmetric space dual to $(G^η)_0 = G/K$. This provides a natural generalization of the classical Borel embeddings $G/K \hookrightarrow G_c/K$. Furthermore, $K$ and $G_c/K$ form a complementary pair of reflective submanifolds in $\overline{G}$. |
| title | Grassmannian perspectives of classical Lie groups and Cartan involutions |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2602.00602 |