Heisenberg group

The basic nonabelian nilpotent Lie group, central extension of an abelian group.

Heisenberg group

For $n\ge 1$ , the (real) Heisenberg group $H_n$ can be realized as $\mathbb R^{2n}\times \mathbb R$ with elements written $(x,y,z)$ where $x,y\in\mathbb R^n$ and $z\in\mathbb R$ , and group law

(x,y,z)\cdot(x',y',z')=\Big(x+x',\,y+y',\,z+z' + \tfrac12(x\cdot y' - y\cdot x')\Big).

This makes $H_n$ a connected Lie group of dimension $2n+1$ . Its Lie algebra is the Heisenberg Lie algebra (compare the Heisenberg algebra example ), which is nilpotent .

Concrete calculation: commutator and center.
Let $p=(x,y,z)$ and $q=(x',y',z')$ . Using the group law and the inverse

(x,y,z)^{-1}=(-x,-y,-z),

one computes the commutator

[p,q]=pqp^{-1}q^{-1}=(0,0,\,x\cdot y' - y\cdot x').

In particular, the commutator always lies in the “ $z$ -axis,” which is the center :

Z(H_n)=\{(0,0,z): z\in\mathbb R\}.

Thus $H_n/Z(H_n)\cong \mathbb R^{2n}$ is abelian, and $H_n$ is 2-step nilpotent (compare lower central series for the analogous Lie-algebra notion).

Exponential/BCH viewpoint.
Because $H_n$ is nilpotent, the exponential map is a global diffeomorphism and the multiplication law is governed by a truncated BCH formula , reflecting that iterated brackets vanish after length 2.