Harmonic morphisms from 5-dimensional Lie groups

Abstract

We consider 5-dimensional Lie groups equipped with a left-invariant Riemannian metric. On such groups we construct left-invariant conformal foliations with minimal leaves of codimension 2. These foliations produce complex-valued harmonic morphisms locally defined on the Lie group.

Appendix 1: The 3-dimensional Lie algebras

At the end of the nineteenth century, L. Bianchi classified the 3-dimensional real Lie algebras. They fall into nine disjoint types I–IX. Each contains a single isomorphy class except types VI and VII which are continuous families of different classes.

Example 10.8

(I) By $\mathbb R^3$ we denote both the 3-dimensional abelian Lie algebra and the corresponding simply connected Lie group.

Example 10.9

(II) The nilpotent Heisenberg algebra ${\mathfrak {nil}^3}$ is defined by

$$\begin{aligned} {[}Z,Y{]}=X. \end{aligned}$$

The corresponding simply connected Lie group is $\text {Nil}^3$.

Example 10.10

(III) The solvable Lie algebra $\mathfrak {h}^2\oplus \mathbb R$ is given by

$$\begin{aligned} {[}Z,X{]}=X. \end{aligned}$$

The corresponding simply connected Lie group is the product $H^2\times \mathbb R$ of the real line and the group $H^2$ classically modelling the standard hyperbolic plane.

Example 10.11

(IV) By $\mathfrak {g}_4$ we denote the solvable Lie algebra given by

$$\begin{aligned} {[}Z,X{]}=X,\quad [Z,Y]=X+Y. \end{aligned}$$

The corresponding simply connected Lie group is $G_4$.

Example 10.12

(V) The solvable Lie algebra $\mathfrak {h}^3$ is defined by

$$\begin{aligned}{}[Z,X]=X,\quad [Z,Y]=Y. \end{aligned}$$

The corresponding simply connected group $H^3$ classically models the standard hyperbolic 3-space.

Example 10.13

(VI) For $\alpha \in \mathbb R^+$, the solvable Lie algebra ${\mathfrak {sol}}_\alpha ^3$ is given by

$$\begin{aligned} {[}Z,X{]}=\alpha X,\quad [Z,Y]=-Y. \end{aligned}$$

The corresponding simply connected Lie group is denoted by $\text {Sol}_\alpha ^3$.

Example 10.14

(VII) For $\alpha \in \mathbb R$, the solvable Lie algebra $\mathfrak {g}_7(\alpha )$ is defined by

$$\begin{aligned} {[}Z,X{]}=\alpha X-Y,\quad [Z,Y]=X+\alpha Y. \end{aligned}$$

We denote the corresponding simply connected Lie group by $G_7(\alpha )$.

Example 10.15

(VIII) The simple Lie algebra $\mathfrak {sl}_{2}(\mathbb R)$ satisfies

$$\begin{aligned} {[}X,Y{]}=2Z,\quad [Z,X]=2Y,\quad [Y,Z]=-2X. \end{aligned}$$

The corresponding simply connected Lie group is denoted by $\widetilde{\mathbf{SL}_{2}(\mathbb R)}$ as it is the universal cover of the special linear group $\mathbf{SL}_{2}(\mathbb R)$.

Example 10.16

(IX) The simple Lie algebra $\mathfrak {su}(2)$ satisfies

$$\begin{aligned} {[}X,Y{]}=2Z,\quad [Z,X]=2Y,\quad [Y,Z]=2X. \end{aligned}$$

The corresponding simply connected Lie group is $\mathbf{SU}(2)$ diffeomorphic to the standard 3-dimensional sphere.