Lie subalgebras inside simple Lie algebras (of type ABCDEFG) have been classified up to equivalence, and linear equivalence (by Dynkin et al). How does one classify embeddings of a Lie algebra h inside a Lie algebra g, where the equivalence relation is given by saying that embeddings are ('adjoint') equivalent when they are conjugate in the adjoint representation of g ? (Thus, equivalence, and linear equivalence imply (adjoint) equivalence, but not the other way around.) More generally, how does one classify embeddings of subalgebras up to equivalence in a (=one) given irreducible representation ?
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3$\begingroup$ Lie subalgebras of Lie algebras have not been classified for Lie algebras (only maximal subalgebras of complex semisimple ones of finite dimension). $\endgroup$– Dietrich BurdeCommented Apr 1, 2015 at 14:25
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$\begingroup$ Question modified to reflect that I wished to ask the question within the context of simple Lie algebras of type ABCDEFG. $\endgroup$– tomatosoupCommented Apr 1, 2015 at 15:07
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$\begingroup$ Dietrich's comment is well-taken and points to a fundamental difficulty with this kind of broad question: every finite dimensional Lie algebra embeds in some general linear algebra by Ado/Iwasawa theorems, so a lot of Lie algebras embed in the trace zero subalgebra (which is simple in most characteristics and acts irreducibly on the underlying vector space). $\endgroup$– Jim HumphreysCommented Apr 1, 2015 at 16:01
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As peoples' comments suggest, it is perhaps somewhat ambitious to hope for a classification of all Lie subalgebras of a given complex simple Lie algebra $\mathfrak{g}$. However, Jacobson and Morozov have an answer concerning embeddings of $\mathfrak{sl}_2(\mathbb{C})$ into $\mathfrak{g}$.
Given a nilpotent orbit $\mathcal{O}\subseteq\mathfrak{g}$, one can find an $\mathfrak{sl}_2(\mathbb{C})$-triple $(e,h,f)$ for which the nilpositive element $e$ lies in $\mathcal{O}$. This establishes a correspondence between the nilpotent orbits in $\mathfrak{g}$ and the adjoint-conjugacy classes of $\mathfrak{sl}_2(\mathbb{C})$-triples in $\mathfrak{g}$. The conjugacy classes of embeddings of $\mathfrak{sl}_2(\mathbb{C})$ into $\mathfrak{g}$ then correspond to the non-zero nilpotent orbits. There are finitely many of the latter.
answered Apr 1, 2015 at 15:16
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$\begingroup$ Thanks, this does answer my question for the adjoint representation. $\endgroup$ Commented Apr 1, 2015 at 15:20