Heisenberg group

Example

The (real, three-dimensional) Heisenberg group $H$ is the group of real upper unitriangular $3 \times 3$ matrices $$[a, b, c] = \begin{pmatrix} 1 & a & c \\ 0 & 1 & b \\ 0 & 0 & 1 \end{pmatrix}, \qquad a, b, c \in \R,$$ with matrix multiplication. Multiplying out gives $$[a, b, c]\,[a', b', c'] = [a + a', \ b + b', \ c + c' + a b'],$$ and $[a, b, c]^{-1} = [-a, -b, -c + ab]$. So $H$ is $\R^3$ with a polynomial group law, a Lie group of dimension $3$.

Check that $H$ is a nilpotent Lie group. The commutator of $g = [a, b, 0]$ and $g' = [a', b', 0]$ is $g g' g^{-1} g'^{-1} = [0, 0, ab' - a'b]$. The centre of $H$ is $Z = \set{[0, 0, c] : c \in \R}$, and all commutators lie in $Z$. So the lower central series is $H \supset Z \supset \set{1}$, and $H$ is a nilpotent group of class $2$; it is not abelian.