Special unitary group of degree two

Definition

Suppose $L$ is a separable quadratic extension of a field $K$ and $\sigma :L\to L$ is the unique automorphism of $L$ that fixes $K$ pointwise. The special unitary group of degree two $SU(2,L)$ corresponding to this field extension is the special unitary group of degree two: it is the subgroup of the special linear group of degree two over $L$ comprising those matrices for which the entry-wise image under $\sigma$ coincides with the effects of the transpose-inverse map. Explicitly, this works out to be:

$SU(2,L):=\left\{{\begin{pmatrix}a&b\\-\sigma (b)&\sigma (a)\\\end{pmatrix}}\mid a,b\in L,a\sigma (a)+b\sigma (b)=1\right\}$

For the real and complex numbers

The most typical usage of the term special unitary group is in the context where $K$ is the field of real numbers, $L$ is the field of complex numbers, and the automorphism $\sigma$ is complex conjugation. In this case, the group is described explicitly as follows:

$SU(2,\mathbb {C} ):=\left\{{\begin{pmatrix}a&b\\-{\overline {b}}&{\overline {a}}\\\end{pmatrix}}\mid a,b\in \mathbb {C} ,|a|^{2}+|b|^{2}=1\right\}$

Here, ${\overline {a}}$ and ${\overline {b}}$ denote the complex conjugates of $a$ and $b$ respectively.

For a finite field

If $K$ is the (unique up to isomorphism) finite field of size a prime power $q$, there is a unique quadratic extension $L$ of $K$, and this extension is separable. The extension field is the finite field (unique up to isomorphism) of order $q^{2}$. The automorphism $\sigma$ is the map $x\mapsto x^{q}$. The special unitary group for this extension may be denoted $SU(2,q)$ (the more standard choice) or $SU(2,q^{2})$ (a less standard choice). It is given explicitly as:

$SU(2,q):=\left\{{\begin{pmatrix}a&b\\-b^{q}&a^{q}\\\end{pmatrix}}\mid a,b\in L,a^{q+1}+b^{q+1}=1\right\}$

It turns out that this group is isomorphic to the special linear group of degree two $SL(2,q)$: special unitary group of degree two equals special linear group of degree two over a finite field.