Special unitary group

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 $n$ for this quadratic extension, denoted $SU(n,L)$ (if the extension being referred to is understood) is defined as the subgroup of the special linear group $SL(n,L)$ comprising those matrices on which the transpose-inverse map gives the same result as the entry-wise application of $\sigma$.

$SU(n,L)=\{A\in SL(n,L)\mid \sigma (A)=(A^{t})^{-1}\}$

Here, $\sigma (A)$ is the matrix obtained by applying $\sigma$ to each of the entries of $A$.

Alternatively, we can define it as the intersection of the unitary group and the special linear group, both viewed as subgroups of the general linear group:

$SU(n,L)=U(n,L)\cap SL(n,L)$

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 $SU(n,\mathbb {C} )$ is the subgroup of the special linear group $SL(n,\mathbb {C} )$ comprising those matrices whose complex conjugate equals the transpose-inverse. When it's understood that we are working over the complex numbers, this group is sometimes just denoted $SU(n)$.

This group may often be written as

$SU(n)=\{A\in GL_{n}(\mathbb {C} ):A{\overline {A^{T}}}=I,\det A=1\}$

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 $SU(n,q)$ (the more standard choice) or $SU(n,q^{2})$ (a less standard choice). Note that due to the ambiguity of notation, it is important to understand from context what exactly is meant.

Note that, if we denote this group by $SU(n,q)$, then, somewhat confusingly:

$SU(n,q)=U(n,q)\cap SL(n,q^{2})$