Is there an explicit(!) classification of all(!) connected real simple Lie groups up to isomorphism? Not just simply connected or adjoint, but all of them?
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- $\begingroup$ Wouldn't be hard to accomplish. You have the simple connected ones and in each case you determine the center. $\endgroup$user473423– user4734232023-12-26 07:27:15 +00:00Commented Dec 26, 2023 at 7:27
- $\begingroup$ "up to isomorphism"!!! Here we need to take into account external automorphisms and its actions on the center! $\endgroup$Vladimir47– Vladimir472023-12-26 09:23:19 +00:00Commented Dec 26, 2023 at 9:23
- 2$\begingroup$ @Vladimir47 Yep. It means determining the orbits of the Out on the set of subgroups of the center of the simply connected representative. In most cases, e.g., when the center is cyclic, this action is the trivial action. And it is easy in general anyway. For instance for $\mathrm{Spin}(2n,2n)$, $4n\ge 8$ the center is a Klein group, the action is of order 2 (2 orbits of subgroups of order 2, the nontrivial orbit defining the so-called half-spin group and the fixed point defining $\mathrm{SO}(2n,2n)$), except when $2n=8$ in which case there is a single orbit. $\endgroup$YCor– YCor2023-12-26 09:27:56 +00:00Commented Dec 26, 2023 at 9:27
- 2$\begingroup$ For a list including both Out and the center, see " The Component Group of the Automorphism Group of a Simple Lie Algebra and the Splitting of the Corresponding Short Exact Sequence" by Gündogan. Link at journal website. In each case the action of Out on the center should be easy to determine. $\endgroup$YCor– YCor2023-12-26 09:30:26 +00:00Commented Dec 26, 2023 at 9:30
- $\begingroup$ Thank you very much for your answers. But such a classification of Lie algebras has not been published anywhere in its final form? Why? $\endgroup$Vladimir47– Vladimir472023-12-26 10:56:19 +00:00Commented Dec 26, 2023 at 10:56
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There is an old paper by Kobayashi and Goto [On The subgroups of the centers of simply connected simple Lie groups - classification of simple Lie groups in the large; Osaka J. Math. 6 (1969), 251-281] which is devoted exactly to the problem you ask about. But be careful - this article cites (and applies) the results of a paper by Sirota and Solodovnikov which, according to H. Freudenthal's (!) review (in the Mathematical Reviews) contains a number of errors.
