Kernels of Linear Representations of Lie Groups, Locally Compact Groups, and Pro-Lie Groups
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2010
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| _version_ | 1866910740662714368 |
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| author | Stroppel, Markus |
| author_facet | Stroppel, Markus |
| contents | For a topological group G the intersection KO(G) of all kernels of ordinary representations is studied. We show that KO(G) is contained in the center of G if G is a connected pro-Lie group. The class KO(C) is determined explicitly if C is the class ConnLie of connected Lie groups or the class almConnLie of almost connected Lie groups: in both cases, it consists of all compactly generated abelian Lie groups. Every compact abelian group and every connected abelian pro-Lie group occurs as KO(G) for some connected pro-Lie group G. However, the dimension of KO(G) is bounded by the cardinality of the continuum if G is locally compact and connected. Examples are given to show that KO(C) becomes complicated if C contains groups with infinitely many connected components. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1012_0540 |
| institution | arXiv |
| publishDate | 2010 |
| record_format | arxiv |
| spellingShingle | Kernels of Linear Representations of Lie Groups, Locally Compact Groups, and Pro-Lie Groups Stroppel, Markus Representation Theory Group Theory 22D05, 20G05, 22E65, 22E15 For a topological group G the intersection KO(G) of all kernels of ordinary representations is studied. We show that KO(G) is contained in the center of G if G is a connected pro-Lie group. The class KO(C) is determined explicitly if C is the class ConnLie of connected Lie groups or the class almConnLie of almost connected Lie groups: in both cases, it consists of all compactly generated abelian Lie groups. Every compact abelian group and every connected abelian pro-Lie group occurs as KO(G) for some connected pro-Lie group G. However, the dimension of KO(G) is bounded by the cardinality of the continuum if G is locally compact and connected. Examples are given to show that KO(C) becomes complicated if C contains groups with infinitely many connected components. |
| title | Kernels of Linear Representations of Lie Groups, Locally Compact Groups, and Pro-Lie Groups |
| topic | Representation Theory Group Theory 22D05, 20G05, 22E65, 22E15 |
| url | https://arxiv.org/abs/1012.0540 |