Last update: 12 May 2014
This is an excerpt of the lecture notes Discrete complex reflection groups by V.L. Popov. Lectures delivered at the Mathematical Institute, Rijksuniversiteit Utrecht, October 1980.
1. | Notation and formulation of the problem. |
1.1. | Notation. |
1.2. | Motions and reflections. |
1.3. | Main problem. |
1.4. | Irreducibility. |
1.5. | What is already known: $k=\mathbb{R}\text{.}$ |
1.6. | What is already known: $k=\u2102\text{.}$ |
2. | Formulation of the results. |
2.1. | Complexifications and real forms. |
2.2. | Classification of infinite irreducible complex noncrystallographic $r\text{-groups:}$ |
2.3. | Ingredients of the description. |
2.4. | Cohomology. |
2.5. | Description of the group of linear parts: the result. |
2.6. | The list of irreducible infinite crystallographic complex groups. |
2.7. | Equivalence. |
2.8. | The structure of an extension of $\text{Tran}\hspace{0.17em}W$ by $\text{Lin}\hspace{0.17em}W\text{.}$ |
2.9. | The rings and fields of definition of $\text{Lin}\hspace{0.17em}W\text{.}$ |
2.10. | Further remarks. |
3. | Several auxiliary results and the classification of irreducible infinite noncrystallographic complex $r\text{-groups.}$ |
3.1. | The subgroups of translations. |
3.2. | Some auxiliary results. |
3.3. | Semidirect products. |
3.4. | Classification of the irreducible infinite complex noncrystallographic $r\text{-groups.}$ |
4. | Invariant lattices |
4.1. | Root lattices. |
4.2. | The lattices with a fixed root lattice. |
4.3. | Further remarks on lattices. |
4.4. | Properties of the operator $S\text{.}$ |
4.5. | Root lattices and infinite $r\text{-groups.}$ |
4.6. | Description of the groups of linear parts: proof. |
4.7. | Description of root lattices. |
4.8. | Invariant lattices in the case $s=n+1\text{.}$ |
5. | The structure of $r\text{-groups}$ in the case $s=n+1\text{.}$ |
5.1. | The cocycle $c\text{.}$ |
5.2. | Example. |
References. |