Examples of non-split algebraic groups

Question

I am interested in knowing various examples of non-split (added hypothesis reductive) reductive linear algebraic groups. In particular, I would like to collect the following examples in my counter-example toolbox.

1. Given an integer $n>2$, an extension $E/F$ of degree $n$ and a connected reductive $F$-group $G$ that splits over $E$ (added following Prof. Humphreys' comment:) and $E$ is the smallest such extension minimally;

2. Given a finite set $S$ of primes, a connected reductive group $G$ defined over $\mathbb Q$ which is quasi-split precisely over $\mathbb Q_v$ for $v \not\in S$ and not otherwise.

The motivation for this question was that I was reading Tits' article on Buildings in Corvallis and wanted to explicitly compute stuff about non-split groups. This prompted me to look for nonsplit groups, hence this question.

[I would like to make this question community-wiki since I am really asking 2 questions here and there is no unique answer, but for some reasons I can't find the CW box. Perhaps I need more reputation. ]

Answer 1

Standard examples are given by central simple algebras, as mentioned in the comments. Namely, let $A$ be a central simple algebra over a field $F$ of dimension $n^2$. Then $G = A^\times$ is an inner form of $GL(n)$, and it will be ramified (non-split) at a finite set of places. Further $A$ splits over an extension $E/F$ of degree $n$, and over global or local fields, one can take $E/F$ to be cyclic. If $n=2$, you get the case of quaternion algebras, which you are probably familiar with.

Just like the case of quaternion algebras over global fields, which must be ramified at an even number of places, one cannot make the ramification of CSAs arbitrary (for quaternion algebras, local there are only 2 choices, but for higher degree CSAs, there are more.) In particular one cannot let $S$ have cardinality 1--you need to be ramified at at least 2 places (and you can get by with 2 in higher degree as well).

There are many references one could give. From a number theory perspective, there's Weil's Basic Number Theory, Platonov-Rapinchuk, and I believe Milne's notes on Class Field Theory have some stuff. Standard references, from a less number theoretic point of view, are

• Reiner, Maximal Orders
• Pierce, Associative Algebras
• Gille-Szamuely, Central Simple Algebras and Galois Cohomology

Personally, I like Reiner and Pierce, and I think Platonov-Rapinchuk gives a nice (relatively short) overview in the beginning. The latter 3 references should all give explanations of how to explicitly construct such CSAs as cyclic algebras.

Similarly, one can come up with other examples by looking at non-split forms of classical groups over global fields.

Answer 2

If you allow yourself to work over fields of positive characteristic then finite reductive groups give you a raft of examples for (1). Indeed, let $G$ be a connected reductive algebraic group over $\mathbb{K} = \overline{\mathbb{F}_p}$ an algebraic closure of prime order $p>0$. Fix $T \leqslant B$ a maximal torus and Borel subgroup of $G$ and let $\{x_{\alpha} \mid \alpha \in \Delta\}$ be a pinning of $G$ with $\Delta$ the simple roots determined by $T \leqslant B$. For any $q$ a power of $p$ we can define a split Frobenius endomorphism $F_q : G \to G$ by $F_q(t) = t^q$ for all $t \in T$ and $F_q(x_{\alpha}(c)) = x_{\alpha}(c^q)$. Now let $\sigma : G \to G$ be any graph automorphism of $G$, i.e., $\sigma(x_{\alpha}(c)) = x_{\rho(\alpha)}(c)$ for some permutation $\rho$ of the roots. Then $F = \sigma \circ F_q = F_q\circ \sigma$ is also a Frobenius endomorphism of $G$ which makes $G$ into an $\mathbb{F}_q$-group. Indeed, we have the affine algebra $\mathbb{A}[G]$ is of the form $\mathbb{K}\otimes_{\mathbb{F}_q} A_0$ where $A_0 = \{a \in \mathbb{A}[G] \mid F^*(a) = a^q\}$ and $F^*$ is the comorphism. The minimal field over which $G$ splits will then be $\mathbb{F}_{q^n}$ where $n$ is the order of the automorphism $\sigma$.

One can get an example for any $n$ by taking $G = G_1 \times \cdots \times G_n$ a product of isomorphic factors and $\sigma$ acting by cyclic permutation. An example with $n>2$ and $G$ simple comes from the case of $\mathrm{D}_4$ and $\sigma$ the triality automorphism. But maybe these examples are not what you had in mind.