Give two simple Lie Algebras $\mathfrak{g_1}$ and $\mathfrak{g_2}$, can we say anything about the conjugacy of $\mathfrak{g_1}$ and $\mathfrak{g_2}$ based on the properties of the corresponding Dynkin diagrams $\mathfrak{D_1}$ and $\mathfrak{D_2}$ ? (Say isomorphism or something ?)
Simple Lie Algebra Conjugacy and Dykin Diagrams
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lie-algebras
dynkin-diagrams
1 Answers
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Semisimple Lie algebra conjugacy (as matrix algebra conjugacy) is indeed equivalent to graph isomorphism. The article Lie algebra conjugacy by Joshua A. Grochow discusses this in detail, also for the general case of an arbitrary Lie algebra.
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0I am familiar with the paper by grochow. Could you let me know if simple lie algebra as opposed to semisimple algebra be any easier to test for conjugacy since the dynkin diagram has only one connected component ? – 2017-02-17
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0Yes, it is easier. For the complexity however, the $O(\log(n))$ may not be improved; but big Oh is a class of functions anyway. – 2017-02-17