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Generation and Coherent Control of Pulsed Quantum Frequency Combs

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

Coincidence lattices in the hyperbolic plane.

M A Rodríguez-Andrade¹, G Aragón-González, J L Aragón

¹Departamento de Matemáticas, Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional, Unidad Profesional Adolfo López Mateos, Edificio 9. 07300 México DF, México.

Acta Crystallographica. Section A, Foundations of Crystallography

|December 22, 2010

Summary

This study analyzes lattice coincidences in R(p,q) spaces using Clifford algebra. It demonstrates that coincidence isometries decompose into lattice vector reflections, offering a metric-independent method applicable to the hyperbolic plane.

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Area of Science:

Mathematics
Geometry
Clifford Algebra

Background:

Lattice theory and isometry analysis are fundamental in geometry.
Understanding coincidences in R(p,q) spaces requires advanced algebraic tools.

Purpose of the Study:

To analyze lattice coincidences in R(p,q) spaces using Clifford algebra.
To demonstrate the decomposition of coincidence isometries into reflections.
To provide a metric-independent procedure applicable to spaces like the hyperbolic plane.

Main Methods:

Application of Clifford algebra for analyzing lattice structures.
Decomposition of coincidence isometries into products of reflections.
Explicit construction of bases and coincidence indices for specific lattices.

Main Results:

Any coincidence isometry in R(p,q) (p+q=2) can be expressed as at most two reflections by lattice vectors.
A metric-independent procedure for analyzing lattice coincidences is developed.
The hyperbolic plane (p=q=1) is shown to be a specific case within this framework.

Conclusions:

The Clifford algebra approach provides a unified method for studying lattice coincidences.
The findings offer a constructive proof and algorithm for the Cartan-Dieudonné theorem in R(p,q) spaces.
This work generalizes geometric isometry analysis to various metric spaces.