# Non-split groups

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

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You should edit your question and click on the "Community Wiki" flag. This is what we do for questions that ask for a list of examples rather than questions that have a focused answer. – Ryan Budney Oct 16 '11 at 20:38
Ok, I have done that now. Thanks. I wasn't sure since I was just looking for a reference. – Thomas Oct 16 '11 at 20:53
Kevin's link has changed to encyclopediaofmath.org/index.php?title=Split_group – George Lowther Dec 2 '11 at 1:39

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.