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".
Thanks, Tom
2 Answers
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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 Reductive Groups, which appears in Part I of Automorphic Forms, Representaions, and L-Functions (Proceedings of a conference in Corvallis), A.Borel and W.Casselman (editors), AMS, 1979.
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$\begingroup$ The entire thing used to be available online for free on the AMS website. This no longer seems to be the case. Does anyone know why? $\endgroup$ Commented Oct 17, 2011 at 4:33
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As far as references go, you can look in pretty much any book on algebraic groups, like Borel or Humphreys or Platonov-Rapinchuk.
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.
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.