Introduction to Louis Michel's lattice geometry through group action / B. Zhilinskii.

Format:

Book

Author/Creator:

Zhilinskiĭ, B., author.

Le Bellac, Michel, Author.

Leduc, Michel, Author.

Series:

Current Natural Sciences

Language:

English

Subjects (All):

Lattice theory.

Physical Description:

1 online resource (271 pages) : illustrations.

Edition:

1st ed.

Place of Publication:

Paris, [France] : EDP Sciences, 2015.

Summary:

Group action analysis developed and applied mainly by Louis Michel to the study of N-dimensional periodic lattices is the central subject of the book. Di erent 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 the authors turn to di erent symmetry and topological classi- cations 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 presented through a number of concrete examples with an extensive use of graphical visualization. The book is aimed at graduated and post-graduate students and young researchers in theoretical physics, dynamical systems, applied mathematics, solid state physics, crystallography, molecular physics, theoretical chemistry, .

Contents:

Frontmatter

Contents

Preface

1 Introduction

2 Group action. Basic definitions and examples

3 Delone sets and periodic lattices

4 Lattice symmetry

5 Lattices and their Voronoï and Delone cells

6 Lattices and positive quadratic forms

7 Root systems and root lattices

8 Comparison of lattice classifications

9 Applications

A. Basic notions of group theory with illustrative examples

B. Graphs, posets, and topological invariants

C. Notations for point and crystallographic groups

D. Orbit spaces for plane crystallographic groups

E. Orbit spaces for 3D-irreducible Bravais groups

Bibliography

Index

Notes:

Includes bibliographical references and index.

This eBook is made available Open Access under a CC BY-NC-ND 4.0 license: https://creativecommons.org/licenses/by-nc-nd/4.0 https://www.degruyter.com/dg/page/open-access-policy

Description based on online resource; title from PDF title page (ebrary, viewed October 20, 2017).

ISBN:

9782759819522

2759819523

OCLC:

1004831978

Access Restriction:

Open Access Unrestricted online access