Are certain simple Lie groups linear algebraic groups? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T02:02:37Z http://mathoverflow.net/feeds/question/107577 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/107577/are-certain-simple-lie-groups-linear-algebraic-groups Are certain simple Lie groups linear algebraic groups? Petra Schwer 2012-09-19T15:01:08Z 2012-09-19T16:18:49Z <p>Assume you have an almost connected simple Lie group G with trivial center. (In particular excluding non-algebraic examples such as the universal cover of SL_2(R).)</p> <p>Such a group should automatically be an algebraic group over the reals resp. the complex numbers. </p> <p>Is this true and why? </p> <p>Can we in addition conclude (EDIT: under a good choice of the field and possibly additional assumptions?) that G is absolutely almost simple as an algebraic group? </p> <p>EDIT: Asking this I do not want to regard a complex Lie group as a real algebraic group. </p> http://mathoverflow.net/questions/107577/are-certain-simple-lie-groups-linear-algebraic-groups/107583#107583 Answer by Vesselin Dimitrov for Are certain simple Lie groups linear algebraic groups? Vesselin Dimitrov 2012-09-19T15:56:13Z 2012-09-19T16:18:49Z <p>The answer is yes for complex Lie groups, and follows from the classification. (Root data are in fact defined over \$\mathbb{Z}\$: a complex semisimple group has not only an underlying algebraic, but even an arithmetic structure). For a more direct explanation, see Theorem 6.3 in the book "Lie Groups and Lie Algebras III" by Onischik-Vinberg: any connected complex Lie group satisfying \$G = [G,G]\$ and admitting a faithful linear representation (which for semisimple groups is automatic), has a unique underlying complex algebraic structure.</p> <p>For real Lie groups this is not <em>quite</em> true, as noted above.</p>