unitary group

Definition

The subgroup $\text{U}(n)=\{A\in {\text{Mat}}_{n\times n}(\mathbb{C})|A^{\ast }A=I\}$ is called the **unitary group.

The set $\text{SU}(n)\subset \text{U}(n)$ of unitary matrices with determinant 1 are called the special unitary group.

Importance

Unitary matrices preserve the Hermetian inner product.

Additionally, $U(1)$ is isomorphic to the torus $S^1$.

Lie group structure

We can write each entry of the matrix multiplication

$(A^{\ast }A{)}_{ij}={\sum }_{k=1}^n\overline{A_{ki}}{A}_{kj}$

Thus, this is a continuous function, and we can consider the inverse image of the closed set where

$\{\begin{array}{ll}(A^{\ast }A{)}_{ij}=1& i=j\\ (A^{\ast }A{)}_{ij}=0& i\ne j\end{array}.$

Since the set above is closed (it consists of one “point” in ${\mathbb{R}}^{n^2}$ or one matrix depending on how you choose to look at it), so thus the inverse image, which is $\text{U}(n)$ is also closed.

Note that $\text{SU}(n)=\text{U}(n)\cap \text{SO}(n)$ so it is clearly closed as well.

Therefore, by the closed subgroup theorem, $\text{U}(n)$ and $\text{SU}(n)$ is a Lie group.

Lie algebra $\mathfrak{u}(n)$

Let $\mathfrak{u}(n)$ and $\mathfrak{su}(n)$ be the Lie algebra for $\text{U}(n)$ and $\text{SU}(n)$, respectively.

$\mathfrak{u}(n)=\{A\in {\text{Mat}}_{n\times n}(\mathbb{C})|A^{\ast }=-A\}=\{\text{skew-Hermitian }n\times n\text{matrices}\}$

and

$\mathfrak{su}(n)=\{\text{skew-Hermitian }n\times n\text{matrices with trace}=0\}$