Non-split groups - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T06:29:46Z http://mathoverflow.net/feeds/question/78284 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/78284/non-split-groups Non-split groups Tom 2011-10-16T20:25:14Z 2011-10-17T04:32:37Z <p>I am looking for a reference with definitions on what it means for an algebraic group to be split, quasi-split, and non-split. I would like to see some examples of the different "types".</p> <p>Thanks, Tom</p> http://mathoverflow.net/questions/78284/non-split-groups/78306#78306 Answer by S. Carnahan for Non-split groups S. Carnahan 2011-10-17T03:25:14Z 2011-10-17T03:25:14Z <p>As far as references go, you can look in pretty much any book on algebraic groups, like Borel or Humphreys or Platonov-Rapinchuk.</p> <p>Here is a standard example: The norm torus of the extension $\mathbb{C}/\mathbb{R}$ is the circle group, whose analytification is the compact Lie group $U(1)$. It is not split, since it is not isomorphic to the multiplicative group (whose analytification is $\mathbb{R}^\times$). It is quasi-split because all tori are quasi-split.</p> <p>One example of a non-quasi-split group is $U_{2,\mathbb{C}/\mathbb{R}}$. The compactness of the analytification obstructs the existence of a real Borel subgroup.</p> http://mathoverflow.net/questions/78284/non-split-groups/78309#78309 Answer by Amritanshu Prasad for Non-split groups Amritanshu Prasad 2011-10-17T04:32:37Z 2011-10-17T04:32:37Z <p>Besides the basic definitions and examples, you will find a concise description of the vocabulary needed to talk about linear algebraic groups over fields that are not algebraically closed in T. A. Springer's article titled <em>Reductive Groups</em>, which appears in Part I of <em>Automorphic Forms, Representaions, and L-Functions</em> (Proceedings of a conference in Corvallis), A.Borel and W.Casselman (editors), AMS, 1979.</p>