Diagrams and exercises

Exercises

The integer Heisenberg group and the Lie algebra of the real Heisenberg group are related by a change-of-fonts isomorphism, i.e., \[\mathbb{H}=\langle x,y,z\mid [x,y]=z, [x,z]=[y,z]=1\rangle\] vs. \[\mathfrak{h}=\langle X,Y,Z\mid [X,Y]=Z, [X,Z]=[Y,Z]=0\rangle.\] Conjecture and prove a generalization and/or counterexample.
Describe the geodesics between arbitrary pairs of points in \(\mathbb(\mathbb{R})\)

The five-dimensional integer Heisenberg group is the group
\begin{align*} \mathbb{H}^5(\mathbb{Z})&=\left\{\begin{pmatrix} 1 & x & u & z\\ 0 & 1 & 0 & y\\ 0 & 0 & 1 & v\\ 0 & 0 & 0 & 1 \end{pmatrix}\middle| x,y,u,v\in \mathbb{Z}\right\} \\ &=\langle x,y,u,v,z\mid [x,y]=[u,v]=z, [x,z]=[y,z]=[u,z]=[v,z]=1, [x,u]=[x,v]=[y,u]=[y,v]=1\rangle. \end{align*}
Show that \(\mathbb{H}^5(\mathbb{Z})\) has a presentation \[\mathbb{H}^5(\mathbb{Z})=\langle x,y,u,v\mid [xv,yu]=1, [x,u]=[x,v]=[y,u]=[y,v]=1\rangle,\] in which all relations are commutators.
Show that the Lie algebra \(\mathfrak{h}^5\) of \(\mathbb{H}^5(\mathbb{R})\) can be presented \[\mathfrak{h}^5=\langle X,Y,U,V\mid [X+V,Y+U]=1, [X,U]=[X,V]=[Y,U]=[Y,V]=1\rangle.\] A Lie algebra presentation like this (where the relators are all linear combinations of brackets of the generators) is called a quadratic presentation. Find other examples of finite-dimensional Lie algebras with quadratic presentations.