unitary group

Definition

The subgroup is called the **unitary group.

The set of unitary matrices with determinant 1 are called the special unitary group.

Importance

Unitary matrices preserve the Hermetian inner product.

Additionally, is isomorphic to the torus .

Lie group structure

We can write each entry of the matrix multiplication

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

Since the set above is closed (it consists of one “point” in or one matrix depending on how you choose to look at it), so thus the inverse image, which is is also closed.

Note that so it is clearly closed as well.

Therefore, by the closed subgroup theorem, and is a Lie group.

Lie algebra

Let and be the Lie algebra for and , respectively.

and