📂Abstract Algebra

Special Unitary Group

Definition

The set of unitary matrices whose determinant is $1$ is denoted by $\mathrm{SU}(n)$ and called the special unitary group of degree $n$.

$$\mathrm{SU}(n) := \left\{ n \times n \text{ unitary matrix} \right\} = {\left\{ A \in M_{n \times n}(\mathbb{C}) : A A^{\ast} = I \right\}}$$

Here, $A^{\ast}$ is the conjugate transpose matrix.

Explanation

Since it consists only of unitary matrices, it forms a group with respect to matrix multiplication. It is a subgroup of the general linear group $\mathrm{GL}(n, \mathbb{C})$.

$$\mathrm{SU}(n) \subset \mathrm{U}(n) \subset \mathrm{GL}(n, \mathbb{C})$$

It has a differentiable structure, thus it is a Lie group.