Semi-group compactifications of Algebraic Groups
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916206727921664 |
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| author | Leibtag, Elyasheev |
| author_facet | Leibtag, Elyasheev |
| contents | We show that for algebraic groups over local fields of characteristic zero, the following are equivalent: Every homomorphism has a closed image, every unitary representation decomposes into a direct sum of finite-dimensional and mixing representations, and that the matrix coefficients are dense within the algebra of weakly almost periodic functions over the group. In our proof, we employ methods from semi-group theory. We establish that algebraic groups are \emph{compactification-centric}, meaning $sG = Gs$ for any element $s$ in the weakly almost periodic compactification of the group $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_09878 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Semi-group compactifications of Algebraic Groups Leibtag, Elyasheev Group Theory 20G05, 22E50, 20M99, 20G25, 43A99, 54H11, 54H13, 54D25 We show that for algebraic groups over local fields of characteristic zero, the following are equivalent: Every homomorphism has a closed image, every unitary representation decomposes into a direct sum of finite-dimensional and mixing representations, and that the matrix coefficients are dense within the algebra of weakly almost periodic functions over the group. In our proof, we employ methods from semi-group theory. We establish that algebraic groups are \emph{compactification-centric}, meaning $sG = Gs$ for any element $s$ in the weakly almost periodic compactification of the group $G$. |
| title | Semi-group compactifications of Algebraic Groups |
| topic | Group Theory 20G05, 22E50, 20M99, 20G25, 43A99, 54H11, 54H13, 54D25 |
| url | https://arxiv.org/abs/2404.09878 |