# Compact Lie Groups, Generalised Euler Angles, and Applications

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## Abstract

**:**

## 1. Introduction

#### 1.1. Euler Parametrisation of $\mathit{Su}\left(\mathbf{2}\right)$

#### 1.2. Split Euler Parametrisation of $\mathit{Su}\left(\mathbf{3}\right)$

#### 1.3. Non-Split Euler Parametrisation of $\mathit{Su}\left(\mathbf{3}\right)$

#### 1.4. Euler Parametrisation and Dyson Integrals

#### 1.4.1. The Roots Structure

#### 1.4.2. Invariant Measure

#### 1.4.3. The Dyson Integral

## 2. Macdonald’s Conjecture

**Theorem**

**1**

## 3. Compact Connected Lie Groups

#### 3.1. Lie Groups and Lie Algebras

#### 3.1.1. Lie Groups

- There is an associative product$$\begin{array}{c}\hfill \circ :\mathit{G}\times \mathit{G}\to \mathit{G},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}({g}_{1},{g}_{2})\u21dd\circ ({g}_{1},{g}_{2})\equiv {g}_{1}{g}_{2},\end{array}$$
- There is a privileged point $e\in \mathit{G}$ such that $eg=ge=g,$$\forall g\in \mathit{G}$, called the unit element;
- There is an inverse map$$\begin{array}{c}\hfill \nu :\mathit{G}\to \mathit{G},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}g\u21dd\nu \left(g\right),\end{array}$$

#### 3.1.2. Lie Algebras

#### 3.1.3. The Exponential Map

**Theorem**

**2.**

- 1.
- ${\mathrm{Exp}}_{\mathit{G}}\left(0\right)=e$;
- 2.
- ${\mathrm{Exp}}_{\mathit{G}}\left(tX\right)={\gamma}_{X}\left(t\right)$;
- 3.
- ${\mathrm{Exp}}_{\mathit{G}}\left((t+s)X\right)={\mathrm{Exp}}_{\mathit{G}}\left(tX\right){\mathrm{Exp}}_{\mathit{G}}\left(sX\right)$;
- 4.
- ${\mathrm{Exp}}_{\mathit{G}}(-X)={\mathrm{Exp}}_{\mathit{G}}{\left(X\right)}^{-1}$;
- 5.
- ${\mathrm{Exp}}_{\mathit{G}}(X+Y)={\mathrm{Exp}}_{\mathit{G}}\left(X\right){\mathrm{Exp}}_{\mathit{G}}\left(Y\right)$ if $[X,Y]=0$;
- 6.
- ${\mathrm{Exp}}_{\mathit{G}}$ defines a local diffeomorphism between an open neighbourhood of 0 in $\mathfrak{g}$ and an open neighbourhood of e in $\mathit{G}$;
- 7.
- ${T}_{e}{\mathrm{Exp}}_{\mathit{G}}:{T}_{e}\mathit{G}\to \mathfrak{g}$ is an isomorphism of vector spaces;
- 8.
- ${\mathrm{ev}}_{e}\circ {T}_{e}{\mathrm{Exp}}_{\mathit{G}}:{T}_{e}\mathit{G}\to {T}_{e}\mathit{G}$ is the identity map over ${T}_{e}\mathit{G}$.

#### 3.2. Semisimple Lie Groups and Algebras

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

#### 3.3. Abelian Compact Lie Groups

#### 3.4. All Compact Lie Groups

**Theorem**

**3.**

**Proof.**

#### 3.5. Cohomology of Compact Lie Groups

**Theorem**

**4.**

#### 3.6. Fundamental Group of Compact Lie Groups

#### 3.7. Representations

**Definition**

**5.**

**Definition**

**6.**

**Proposition**

**1.**

**Proposition**

**2.**

**Proposition**

**3.**

**Definition**

**7.**

**Definition**

**8.**

#### 3.7.1. The Adjoint Representations

#### 3.7.2. Simple Algebras and the Cartan Criterion

**Theorem**

**5.**

#### 3.8. Roots and Classifications

#### 3.8.1. Classification of Complex Simple Lie Algebras

- For any non-vanishing root $\lambda \in {H}^{*}$, also $k\lambda $ is a root if and only if $k=0,\pm 1$;
- The number of non-vanishing roots is at least $2r$;
- If $\lambda $ is a non-vanishing root, then its corresponding eigenspace has dimension one;
- If $\alpha $ and $\beta $ are two non-vanishing roots and a, b are in the corresponding eigenspaces, then either $[a,b]=0$ and $\alpha +\beta $ is not a root or $\alpha +\beta $ is a root and $[a,b]\ne 0$ belongs to the corresponding eigenspace;
- The eigenspaces ${\mathfrak{g}}_{\alpha}$ and ${\mathfrak{g}}_{\beta}$ of the roots $\alpha $ and $\beta $ are mutually orthogonal with regard to the Killing form unless $\alpha +\beta =0$.

**Theorem**

**6**

- The diagonal elements are al ltwo, whereas the non-diagonal elements are non-positive integers;
- ${p}_{\alpha \beta}$ is zero if and only if ${p}_{\beta \alpha}$ is zero;
- The root can be ordered so that the non-vanishing elements below the diagonal are $-1$.

- $\mathit{sl}(n,\mathbb{C})$ of rank $n-1$, $n\ge 2$ of traceless $n\times n$ complex matrices;
- $\mathit{so}(2n+1)$ of rank n, $n\ge 2$ of $(2n+1)\times (2n+1)$ antisymmetric complex matrices;
- $\mathit{sp}\left(2n\right)$ of rank n, $n\ge 2$ of $\left(2n\right)\times \left(2n\right)$ symplectic complex matrices;
- $\mathit{so}\left(2n\right)$ of rank n, $n\ge 3$ of $\left(2n\right)\times \left(2n\right)$ antisymmetric complex matrices;

- ${\mathfrak{g}}_{2}$, the linear algebra generated by the derivations acting on the octonionic algebra $\mathbb{O}$;
- ${\mathfrak{f}}_{4}$, the Lie algebra associated with the isometry group of the octonionic projective plane $\mathbb{O}{\mathbb{P}}^{2}$;
- ${\mathfrak{e}}_{6}$, the Lie algebra associated with the isometry group of the complex octonionic projective plane $(\mathbb{C}\otimes \mathbb{O}){\mathbb{P}}^{2}$;
- ${\mathfrak{e}}_{7}$, the Lie algebra associated with the isometry group of the quaternionic octonionic projective plane $(\mathbb{H}\otimes \mathbb{O}){\mathbb{P}}^{2}$;
- ${\mathfrak{e}}_{8}$, the Lie algebra associated with the isometry group of the bi-octonionic projective plane $(\mathbb{O}\otimes \mathbb{O}){\mathbb{P}}^{2}$.

#### 3.8.2. Classification of Real Simple Lie Algebras

**Proposition**

**4.**

#### 3.9. Root Systems

#### 3.10. Compact Forms

**Proposition**

**5.**

#### 3.11. Realizations

## 4. Compact Symmetric Spaces

#### 4.1. Globally Symmetric Spaces

#### 4.2. A Little Bit of Differential Geometry

#### 4.3. Real Forms, Subgroups, and Lattices

- The numbers$$\begin{array}{c}\hfill {p}_{\alpha ,\beta}=2\frac{\left(\alpha \right|\beta )}{\left(\beta \right|\beta )}\end{array}$$
- $\alpha -{p}_{\alpha ,\beta}\beta \in \mathcal{R}$.

#### Non-Compact Real Forms

**AI.**The diagrams coincide with the Dynkin diagrams and correspond to the split real forms. The lattice is the same as the one of the algebra. There are only shorter roots with multiplicity one. They correspond to the algebras $\mathfrak{sl}(r+\mathbf{1},\mathbb{R})$, consisting of the $(r+1)\times (r+1)$ traceless matrices with real entries and have $r(r+1)$ compact directions corresponding to the subalgebra of $(r+1)\times (r+1)$ antisymmetric matrices. They generate the symmetrically embedded maximal compact subalgebra (MCS) $\mathfrak{so}(r+\mathbf{1})$ of $r+1$ dimensional rotations.

**AII.**These diagrams exist for rank $r=2n-1$, $n>1$.

**AIIIa.**These are defined for rank $r=n-1$, $n=p+q$ with $1<p<q$.

**AIIIb.**These are defined for rank $r=2p-1$, $p>1$.

**AIV.**Exist for rank $n>1$.

**BI.**Exist for rank $n>1$, $2\le p\le n$.

**BII.**Exist for rank $n>1$.

**CI.**Exist for rank $n>2$.

**CIIa.**Exist for rank $n>2$, $0<p<n/2$.

**CIIb.**Exist for rank $2n$, $n>1$.

**DIa.**They exist for rank $n\ge 4$.

**DIb.**Exist for rank $n>2$.

**DIc.**Exist for rank $n>2$, $1<p<n-1$.

**DII.**Exist for rank $n>2$.

**DIIIa.**Exist for rank $2n+1$, $n>1$.

**DIIIb.**Exist for rank $2n$, $n>1$.

**EI.**It is of type ${E}_{6}$.

**EII.**It is of type ${E}_{6}$.

**EIII.**It is of type ${E}_{6}$.

**EIV.**It is of type ${E}_{6}$.

**EV.**It is of type ${E}_{7}$.

**EVI.**It is of type ${E}_{7}$.

**EVII.**It is of type ${E}_{7}$.

**EVIII.**It is of type ${E}_{8}$.

**EIX.**It is of type ${E}_{8}$.

**FI.**It is of type ${F}_{4}$.

**FII.**It is of type ${F}_{4}$.

**G.**It is of type ${G}_{2}$.

## 5. Generalised Euler Angles

#### 5.1. The General Strategy

#### 5.2. MCS Euler Parametrisation

#### 5.3. The Non-Split Case

## 6. Euler versus Dyson

#### 6.1. The Split Integrals

**Conjecture**

**1.**

#### 6.2. The Non-Split Integrals

## 7. Open Questions and Further Applications

#### 7.1. Non-Symmetric Embeddings

#### 7.1.1. The Group ${\mathit{G}}_{2}$ and Its Lie Algebra

**Definition**

**9.**

**Proposition**

**6.**

#### 7.1.2. A Euler Parametrisation of ${\mathit{G}}_{2}$

#### 7.2. Open Questions

#### 7.3. Applications

#### The Problem of Measure Concentration

#### 7.4. Applications to Nuclear Physics

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Simple Lie Algebras

## Appendix B. Matrices of Lie(**G** 2)

## Notes

1 | That is, it is the sum of simple roots with largest possible non-negative coefficients. |

2 | recall that the surface of a sphere ${S}^{2d-1}$ of radius 1 is $2\frac{{\pi}^{d}}{(d-1)!}$ |

3 | The vector space $\mathbb{K}\simeq {\mathbb{C}}^{n}$ is the space of all complex valued Weyl invariant functions on R, m equals the numbers of conjugacy classes of roots in R and elements of $\mathbb{K}$ are called multiplicity functions on R. The notation ${k}_{\alpha}$ denotes the evaluation of $k\in \mathbb{K}$ on $\alpha \in R$. |

4 | in general no requirements are done on the dimensions. |

5 | We are grateful to S. Pigola for explaining us these points. |

6 | Indeed, on ${\mathit{T}}^{r}$ there is the adjoint action of the normalizer $\mathit{N}$: ${\mathit{T}}^{r}\to {\left({\mathit{T}}^{r}\right)}^{\mathit{N}}\subseteq {\mathit{T}}^{r}$. Moreover, $\mathit{N}/{\mathit{T}}^{r}=\mathit{W}$ is the Weyl group. Since the invariant measure restricted to the torus is just $d{\mu}_{{\mathit{T}}^{r}}={\prod}_{i=1}^{r}d{s}_{i}$, we see that the action of the Weyl group sends ${\mathit{T}}^{r}$ isometrically onto itself. Thus, the cube is divided in equivalent sectors by the Weyl group action. The maximal number of such sectors is thus $\left|\mathit{W}\right|$, the cardinality of the Weyl group. More precisely, the adjoint action $\sigma :{\mathit{T}}^{r}\mapsto {\left({\mathit{T}}^{r}\right)}^{\mathit{W}}$ is a surjective homomorphism over ${\mathit{T}}^{r}$, with a non-trivial kernel given by $\mathrm{Ker}\sigma \simeq {\mathsf{\Lambda}}_{\mathit{W}}/{\mathsf{\Lambda}}_{\mathit{R}}$, the quotient between the weight lattice ${\mathsf{\Lambda}}_{\mathit{W}}$ with regard to the root lattice ${\mathsf{\Lambda}}_{\mathit{R}}$. This lattice is isomorphic to the center $\mathit{Z}$ of the (covering) group. Then, we find that the number of cells in the cube is $\nu =\frac{\left|\mathit{W}\right|}{|{\mathsf{\Lambda}}_{\mathit{W}}/{\mathsf{\Lambda}}_{\mathit{R}}|}=\frac{\left|\mathit{W}\right|}{\left|\mathit{Z}\right|}$. We will see another way to compute the number of cells. |

7 | Notice that we cannot use associativity in general, but the reader can check that $\mathit{a}(\mathit{b}\mathit{c})$ satisfies associativity if two among $\mathit{a},\mathit{b},\mathit{c}$ are equal. For example if associativity would be true we would have $[{\mathit{e}}_{4}{\mathit{e}}_{1},{\mathit{e}}_{2}]=\left({\mathit{e}}_{4}{\mathit{e}}_{1}\right){\mathit{e}}_{2}-{\mathit{e}}_{2}\left({\mathit{e}}_{4}{\mathit{e}}_{1}\right)=\left({\mathit{e}}_{4}{\mathit{e}}_{1}\right){\mathit{e}}_{2}-\left({\mathit{e}}_{2}{\mathit{e}}_{4}\right){\mathit{e}}_{1}=\left({\mathit{e}}_{4}{\mathit{e}}_{1}\right){\mathit{e}}_{2}+\left({\mathit{e}}_{4}{\mathit{e}}_{2}\right){\mathit{e}}_{1}={\mathit{e}}_{4}({\mathit{e}}_{1}{\mathit{e}}_{2}+{\mathit{e}}_{2}{\mathit{e}}_{1})=0$, which is wrong since ${\mathit{e}}_{4}{\mathit{e}}_{1}={\mathit{e}}_{7}$ and $[{\mathit{e}}_{7},{\mathit{e}}_{2}]=-2{\mathit{e}}_{5}$. |

8 | with respect to the action of $SU(n+1)$. |

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**Figure 2.**The parallelogram is the range determined by the simple coroots direction only. The coloured half is the fundamental region.

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