📂Representation Theory

Heisenberg Group

Definition¹

The set of $3 \times 3$ matrices below is called the Heisenberg group.

$$ H := \left\{ \begin{bmatrix} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{bmatrix} : a, b, c \in \mathbb{R} \right\} $$

Explanation

The set of upper triangular matrices whose diagonal entries are $1$.

Group

With matrix multiplication as the binary operation, the set $H$ is a group.

• Closedness

  $$ \begin{bmatrix} 1 & a & c \\ 0 & 1 & b \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & \alpha & \gamma \\ 0 & 1 & \beta \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & a + \alpha & c + a\beta + \gamma \\ 0 & 1 & b + \beta \\ 0 & 0 & 1 \end{bmatrix} $$

  Due to the $a \beta$ of the entry in row $1$ and column $3$ on the right-hand side, one can see that it is not an abelian group.

• Associativity

  Holds because matrix multiplication is associative.

• Identity

  Since the operation is matrix multiplication, the identity matrix is the identity element.

• Inverse

  The inverses are as follows.

$$ \begin{bmatrix} 1 & a & c \\ 0 & 1 & b \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & -a & ab-c \\ 0 & 1 & -b \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} $$

Subgroup

Because every element has an inverse, $H$ is a subset of $\operatorname{GL}(3, \mathbb{R})$, and since it is itself a group under matrix multiplication, $H$ is a subgroup of $\operatorname{GL}(3, \mathbb{R})$.

Matrix Lie group

$H$ is a closed subgroup of $\operatorname{GL}(3, \mathbb{R})$, hence it is a matrix Lie group. Define the function $f : \operatorname{GL}(3, \mathbb{R}) \to M_{3 \times 3}(\mathbb{R})$ as follows. It is the function that extracts only the diagonal entries and the entries below them.

$$ f(A) = \begin{bmatrix} A_{11} & A_{22} & A_{33} & A_{21} & A_{31} & A_{32} \end{bmatrix}, \qquad A \in M_{3 \times 3}(\mathbb{R}) $$

Then $f$ is a continuous function. Recall that the preimage of a closed set under a continuous function is closed. Since the set $\left\{ \begin{bmatrix} 1 & 1 & 1 & 0 & 0 & 0 \end{bmatrix} \right\}$ is a closed subset of $M_{3 \times 3}$, its preimage $H$ is also closed.

$$ f^{-1}(\left\{ \begin{bmatrix} 1 & 1 & 1 & 0 & 0 & 0 \end{bmatrix} \right\}) = \left\{ \begin{bmatrix} 1 & a & c \\ 0 & 1 & b \\ 0 & 0 & 1 \end{bmatrix} : a, b, c \in \mathbb{R} \right\} = H $$

Therefore $H$ is a closed subgroup, and thus a matrix Lie group.