Classification of Tori of GL2, up to conjugation

Question

Over an algebraically closed field $k$, every one-dimensional torus embedded (as a closed algebraic subgroup) into GL2 is diagonalisable, and the embedding is $t\mapsto (t^m,t^n)$ for some integers $m,n$ with $gcd(m,n)=1$.

What happens when the field $k$ is not algebraically closed? Can I find somewhere the classification of all subtori of GL2, up to conjugation?

By a torus I of course mean a closed algebraic subgroup of GL2, which is isomorphic to $K^{*}$ over the algebraical closure $K$ of $k$.

One example is, when $k=\mathbb{R}$, the group of rotations. This group is algebraically given by matrices of the form $[a,b,c,d]$ (sorry it seems that matrices do not appear correctly, but you should see what I mean) with $a=d$, $b=-c$ and $a^2+b^2=1$. The same example works over any field, and it is not diagonalisable if $-1$ is not a square in the field.

I would like to have a complete classification.

Answer 1

In general, a group that is isomorphic to a $d$-dimensional torus over the algebraic closure of the ground field can be identified by the action of the Galois group on its character group / group of one-parameter subgroups, which is $\mathbb Z^d$, so you get a homomorphism $Gal(\bar{k}/k) \to GL_d(\mathbb Z)$. In the one-dimensional case, this is just a quadratic character, so tori correspond to quadratic fields, in the manner discussed in the comments. In particular, all $1$-dimensional tori in any group embed into $GL_2$.

The reason you can recover it from the Galois action on the character group is that the points of the torus are exactly the homomorphisms from the character group to $GL_1(\bar{k})$ that are Galois-equivariant. This is a version of Pontryagin duality.

This is part of the general theory that twists of an object $X$ over a field $k$ are classified by the Galois cohomology group $H^1(k,X)$, or, for twists over an arbitrary base, an etale cohomology group.

EDIT: For a very explicit description, the Galois rep associated to the quadratic field extension $\mathbb Q(\sqrt{D})$ is the group of determinant $1$ matrices that preserve the quadratic form $x^2-Dy^2$, which means

$\left(\begin{array}{cc} a & Db \\ b & a \end{array}\right) $

for $a^2-D b^2 =1 $

for $D=-1$ this reduces to the classic $SO_2$, and for $D=1$ this is just a regular split torus.

Answer 2

The $n$-dimensional tori up to isomorphism can be described by Galois cohomology, see for instance, Serre's Galois cohomology, or my thesis (http://arxiv.org/abs/math/0409453) for more details. These are given by (equivalence classes of) Galois representations taking values in $GL_n(\mathbb{Z})$.

For instance, for real numbers, the Galois group is $\mathbb{Z}/2\mathbb{Z}$ and has three indecomposable integral representations; the trivial 1-dimensional representation, the sign representation and the 2 dimensional permutation representation (where the Galois group operates by permuting a basis). Thus we get three 2-dimensional tori over $\mathbb{R}$; the split one, the restriction of scalars from $\mathbb{G}_m$ from $\mathbb{C}$ to $\mathbb{R}$, and the $SO_2 \times \mathbb{G}_m$. Since the last torus is not embeddable in $GL_2$, we get only two types of tori in $GL_2$ over $\mathbb{R}$.

Over $\mathbb{Q}$, the Galois group is much more complicated and hence has many more two-dimensional representations (one corresponding to each quadratic extension). Thus we get many non-conjugate maximal tori in $GL_2$ over $\mathbb{Q}$.

Answer 3

In the special case $G= \mathrm{GL}_2(k)$, a more direct approach than those indicated in the answers and comments is certainly possible even though it doesn't do much to illuminate the general case.

To simplify a bit, note that the center (consisting of nonzero scalar matrices) is itself a 1-dimensional torus not conjugate to others; so it's safe to consider just the derived group $\mathrm{SL}_2(k)$. Here the 1-dimensional tori are just the maximal tori in the algebraic group setting, usually called Cartan subgroups in the more specialized Lie group version when $k= \mathbb{C}$ or $\mathbb{R}$.

In the Lie group case, you are asking for all conjugacy classes of Cartan subgroups in the real group. This is an old problem, solved in general by Kostant for semisimple Lie groups (and independently by Sugiura, available online here). It's equivalent to finding the conjugacy classes of Cartan subalgebras. In your special case there are two classes, represented by the diagonal torus and a compact version.

In the algebraic group case, you are similarly asking for all conjugacy classes of maximal $k$-tori in the case when $k$ fails to be algebraically closed. Here again there is a lot of general theory, organized by Borel-Tits, with the machinery of Galois cohomology then being invoked to discuss $k$-forms. But some short-cuts are likely in your rank 1 situation. What approach you take depends a lot on what your ultimate interest is and which fields are most important.

For higher rank groups, the whole problem has a different flavor. In the algebraic group setting, you'd be looking at the dual of the character group of a maximal torus consisting on co-characters (or 1-parameter subgroups) relative to a splitting field and its subfield.

Answer 4

A finite dimensional commutative $k$-algebra is étale if it is isomorphic to a product of separable field extensions of $k$. For any $n \ge 1$, the conjugacy classes of maximal $k$-tori in $\operatorname{GL}_n = \operatorname{GL}(V)$ are in 1-1 correspondence with isomorphism classes of étale $k$-algebras $E$ of dimension $n$ over $k$.

Here is how the correspondence goes: given a maximal torus $T$ of $\operatorname{GL}(V)$, the Lie algebra $\operatorname{Lie}(T)$ is an $n$-dimensional etale subalgebra of $\operatorname{End}_k(V)$.

Conversely, given $E$ an étale algebra of dimension $n$ over $k$, view $E$ as a left $E$-module. The module structure determines a $k$-algebra embedding $E \to \operatorname{End}_k(E)$ and hence an injective homomorphism of algebraic groups $E^\times \to \operatorname{GL}(E) = \operatorname{GL}_n$, where $E^\times$ is the "unit group of $E$ viewed as an algebraic group". The image of this homomorphism is the desired maximal torus of $\operatorname{GL}_n$.

It remains to see that étale subalgebras $E$ and $F$ of $\operatorname{End}_k(V)$ are conjugate by an element of $\operatorname{GL}_n(k)$ if and only if they are isomorphic $k$-algebras. This follows from the observation that if $E$ is an étale subalgebra of $\operatorname{End}_k(V)$ with $\dim_k E = \dim_k V$, then viewed as $E$-module, $V$ is isomorphic to the regular representation of $E$.

From the point of view of Galois cohomology, here is what is going on. Write $G = \operatorname{GL}_n$, let $T_0$ be a split maximal $k$-torus, and let $N = N_G(T_0)$. The $k$-variety $\mathcal{T}$ of all maximal tori of $G$ is $\mathcal{T} = G/N$, and there is an exact sequence of pointed sets $$1 \to N(k) \to G(k) \to \mathcal{T}(k) \to H^1(k,N) \to 1$$ since $H^1(k,G)$ is trivial (Hilbert 90).

Now, $N$ is the semidirect product of the split torus $T_0$ and the symmetric group $S_n$. Since $H^1(k,T_0)$ is trivial (Hilbert 90 again), and since the quotient homomorphism $N \to S_n$ has a section, $H^1(k,N)$ identifies with $H^1(k,S_n)$, which in turn identifies with the set $\operatorname{Et}_n(k)$ of iso. classes of étale $k$-algebras of dimension $n$.

Thus the mapping $\mathcal{T}(k) \to H^1(k,N) = \operatorname{Et}_n(k)$ is onto, and by "twisting" one identifies the fibers of this mapping with the $G(k)$-orbits on $\mathcal{T}(k)$; i.e. with the conjugacy classes of maximal $k$-tori in $\operatorname{GL}_n$.