Just wondering if there are any matrix groups out there waiting to be discovered, or if they are all known?
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I think the classification of finite-dimensional Lie algebras is close to an answer for your question. – Rob Arthan Aug 22 '19 at 22:25
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This is a cool question. How can you prove that you found all subgroups of a group? – Spencer Kraisler Aug 22 '19 at 22:27
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1To be clear, this link referenced by @RobArthan is for simple and semisimple finite dimensional Lie algebras. There is quite a beastly collection of nonsemisimple finite dimensional Lie algebras. – Lee Mosher Aug 22 '19 at 22:46
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@LeeMosher: agreed: that's why I said "close to an answer" as I didn't really know what the OP was looking for. – Rob Arthan Aug 22 '19 at 22:50
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There is no known classification of finite-dimensional real Lie algebras. Since any such Lie algebra is the Lie algebra of some matrix group, the answer to your question is negative.
José Carlos Santos
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Has anyone attempted to classify the lie groups directly? Or is it considered more manageable to approach it via the lie algebras? – vibe Aug 23 '19 at 14:44
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It is considered more manageable to approach it via Lie algebras, since the same Lie algebra can be the Lie algebra of several non-isomorphic Lie groups. And also because the classification of the Lie algebras is basically a problem concerning finite-dimensional vector spaces. – José Carlos Santos Aug 23 '19 at 14:49
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Actually, the classification of nilpotent Lie algebras, even over the complex numbers, is only known up to dimension $7$. In dimension $8$, there are only partial results. So it is clear that the classification is a wild problem.
References:
This MO-question,
Dietrich Burde
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