Finite-dimensional faithful representations of compact groups - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T22:37:31Z http://mathoverflow.net/feeds/question/63982 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/63982/finite-dimensional-faithful-representations-of-compact-groups Finite-dimensional faithful representations of compact groups Sabri 2011-05-05T10:50:04Z 2011-05-05T17:25:53Z <p>Is it true that a compact group always has a faithful, finite-dimensional unitary representation? If not, are there any reasonably simple counter-examples?</p> <p>I've done some research and know that every group has <em>some</em> faithful representation, all irreducible reps of a compact group are finite, and that the irreducible reps separate the points of the group. However, that doesn't quite answer the question!</p> http://mathoverflow.net/questions/63982/finite-dimensional-faithful-representations-of-compact-groups/63983#63983 Answer by Bugs Bunny for Finite-dimensional faithful representations of compact groups Bugs Bunny 2011-05-05T11:08:54Z 2011-05-05T11:08:54Z <p>No, it is false! Take a product of alef-2011 copies of \$C_2\$. It is a bit too big to fit into \$GL_n (C)\$...</p> http://mathoverflow.net/questions/63982/finite-dimensional-faithful-representations-of-compact-groups/63985#63985 Answer by Mark Schwarzmann for Finite-dimensional faithful representations of compact groups Mark Schwarzmann 2011-05-05T11:44:10Z 2011-05-05T11:44:10Z <p>A famous theorem is that this is true if and only if \$G\$ is a Lie group.</p> http://mathoverflow.net/questions/63982/finite-dimensional-faithful-representations-of-compact-groups/63991#63991 Answer by Alain Valette for Finite-dimensional faithful representations of compact groups Alain Valette 2011-05-05T12:14:58Z 2011-05-05T12:25:30Z <p>See <a href="http://mathoverflow.net/questions/61921/on-closed-totally-disconnected-subgroups-of-connected-real-lie-groups/63030#63030" rel="nofollow">http://mathoverflow.net/questions/61921/on-closed-totally-disconnected-subgroups-of-connected-real-lie-groups/63030#63030</a>: a non-discrete totally diisconnected group has no faithful representation in <code>\$GL_n(\mathbb{C})\$</code>.</p> http://mathoverflow.net/questions/63982/finite-dimensional-faithful-representations-of-compact-groups/64028#64028 Answer by Dick Palais for Finite-dimensional faithful representations of compact groups Dick Palais 2011-05-05T17:25:53Z 2011-05-05T17:25:53Z <p>Another famous and interesting NASC for a compact (or even locally compact) group to have a faithful finite dimensional representation is that it "not have arbitrarily small subgroups", i.e., that there exist a neighborhood of the identity with no non-trivial subgroup. This was the way that von Neumann solved the Hilbert 5th Problem in the compact case, and is explained (starting on page 1243) in:</p> <p><a href="http://www.ams.org/notices/200910/rtx091001236p.pdf" rel="nofollow">http://www.ams.org/notices/200910/rtx091001236p.pdf</a></p>