# Diagrams and exercises

## Exercises

1. 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.
2. Describe the geodesics between arbitrary pairs of points in $$\mathbb(\mathbb{R})$$
1. 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.

2. 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.

Created: 2017-08-21 Mon 13:50

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