I am trying to solve the following. Given the special unitary group $SU(n)$ and its adjoint action $Ad_{U}: \mathfrak{su}(n) \rightarrow \mathfrak{su}(n)$, what is the largest subset of $SU(n)$ such that this action has an invariant subspace?
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$\begingroup$ I'm not sure I understand the question. The whole group has an invariant subspace: it just happens to be zero-dimensional. Are you asking about an invariant subspace of positive dimension? $\endgroup$– José Figueroa-O'FarrillJun 28, 2015 at 2:30
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$\begingroup$ Yes, only non trivial subspaces are important. $\endgroup$– BenjaminJun 28, 2015 at 3:13
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1$\begingroup$ Clearly a sub lie group has its own lie algebra as an example. Are there any examples larger than the largest subgroup of $SU(n)$? $\endgroup$– BenjaminJun 28, 2015 at 3:16
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$\begingroup$ I suggest you rewrite the question, making it clear that you are after proper subspaces and that by "invariant" you don't mean pointwise. $\endgroup$– José Figueroa-O'FarrillJun 28, 2015 at 3:33
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3$\begingroup$ The question is not "what is the largest" but "what are the largest", because there's nonuniqueness. And it's clear that any such subset has to be a closed subgroup. Conversely since we precisely have the adjoint action, any closed subgroup (infinite and distinct from the whole group) has an invariant subspace (its own Lie algebra), so the question boils down to the standard question of classifying (infinite) maximal closed subgroups of $SU(n)$, whose main step is the classification of maximal Lie subalgebras of $\mathfrak{su}(n)$, which is due to Dynkin or before. $\endgroup$– YCorJun 28, 2015 at 5:42
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