Group action analysis developed and applied mainly by Louis Michel to the study of N-dimensional periodic lattices is the main subject of the book. Different basic mathematical tools currently used for the description of lattice geometry are introduced and illustrated through applications to crystal structures in two- and three-dimensional space, to abstract multi-dimensional lattices and to lattices associated with integrable dynamical systems. Starting from general Delone sets authors turn to different symmetry and topological classifications including explicit construction of orbifolds for two- and three-dimensional point and space groups. Voronoï and Delone cells together with positive quadratic forms and lattice description by root systems are introduced to demonstrate alternative approaches to lattice geometry study. Zonotopes and zonohedral families of 2-, 3-, 4-, 5-dimensional lattices are explicitly visualized using graph theory approach. Along with crystallographic applications, qualitative features of lattices of quantum states appearing for quantum problems associated with classical Hamiltonian integrable dynamical systems are shortly discussed. The presentation of the material is done through a number of concrete examples with an extensive use of graphical visualization. The book is addressed to graduated and post-graduate students and young researches in theoretical physics, dynamical systems, applied mathematics, solid state physics, crystallography, molecular physics, theoretical chemistry, ...
Author(s): Boris Zhilinskii
Series: Current Natural Sciences
Publisher: EDP Sciences
Year: 2015
Language: French
Pages: 272
City: Paris
Introduction to Louis Michel’s lattice geometry through group action
Contents
Preface
1 - Introduction
2 - Group action. Basic definitions and examples
2.1 The action of a group on itself
2.2 Group action on vector space
3 - Delone sets and periodic lattices
3.1 Delone sets
3.2 Lattices
3.3 Sublattices of L
3.4 Dual lattices
4 - Lattice symmetry
4.1 Introduction
4.2 Point symmetry of lattices
4.3 Bravais classes
4.4 Correspondence between Bravais classes and lattice point symmetry groups
4.5 Symmetry, stratification, and fundamental domains
4.6 Point symmetry of higher dimensional lattices
5 - Lattices and their Voronoïand Delone cells
5.1 Tilings by polytopes: some basic concepts
5.2 Voronoï cells and Delone polytopes
5.3 Duality
5.4 Voronoï and Delone cells of point lattices
5.5 Classification of corona vectors
6 - Lattices and positive quadratic forms
6.1 Introduction
6.2 Two dimensional quadratic forms and lattices
6.3 Three dimensional quadratic forms and 3D-lattices
6.4 Parallelohedra and cells for N-dimensional lattices
6.5 Partition of the cone of positive-definite quadratic forms
6.6 Zonotopes and zonohedral families of parallelohedra
6.7 Graphical visualization of members of the zonohedral family
6.8 Graphical visualization of non-zonohedral lattices
6.9 On Voronoï conjecture
7 - Root systems and root lattices
7.1 Root systems of lattices and root lattices
7.2 Lattices of the root systems
7.3 Low dimensional root lattices
8 - Comparison of lattice classifications
8.1 Geometric and arithmetic classes
8.2 Crystallographic classes
8.3 Enantiomorphism
8.4 Time reversal invariance
8.5 Combining combinatorial and symmetry classification
9 - Applications
9.1 Sphere packing, covering, and tiling
9.2 Regular phases of matter
9.3 Quasicrystals
9.4 Lattice defects
9.5 Lattices in phase space. Dynamical models. Defects
9.6 Modular group
9.7 Lattices and Morse theory
A - Basic notions of group theory with illustrative examples
B - Graphs, posets, and topological invariants
C - Notations for point and crystallographic groups
C.1 Two-dimensional point groups
C.2 Crystallographic plane and space groups
C.3 Notation for four-dimensional parallelohedra
D - Orbit spaces for planecrystallographic groups
E - Orbit spaces for 3D-irreducible Bravais groups
Bibliography
Index
