Knowlpedia

Heisenberg group

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

For n≥1n\ge 1, the (real) Heisenberg group HnH_n can be realized as R2n×R\mathbb R^{2n}\times \mathbb R with elements written (x,y,z)(x,y,z) where x,y∈Rnx,y\in\mathbb R^n and z∈Rz\in\mathbb R, and group law

(x,y,z)⋅(x′,y′,z′)=(x+x′, y+y′, z+z′+12(x⋅y′−y⋅x′)).(x,y,z)\cdot(x',y',z')=\Big(x+x',\,y+y',\,z+z' + \tfrac12(x\cdot y' - y\cdot x')\Big).

This makes HnH_n a connected of dimension 2n+12n+1. Its Lie algebra is the Heisenberg Lie algebra (compare ), which is .

Remarks

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

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

one computes the commutator

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

In particular, the commutator always lies in the “zz-axis,” which is the :

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

Thus Hn/Z(Hn)≅R2nH_n/Z(H_n)\cong \mathbb R^{2n} is abelian, and HnH_n is 2-step nilpotent (compare for the analogous Lie-algebra notion).

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