Book contents
- Frontmatter
- Contents
- List of figures
- List of contributors
- Preface
- Introduction
- 1 Lagrangian and Hamiltonian Formalism for Discrete Equations: Symmetries and First Integrals
- 2 Painlevé Equations: Continuous, Discrete and Ultradiscrete
- 3 Definitions and Predictions of Integrability for Difference Equations
- 4 Orthogonal Polynomials, their Recursions, and Functional Equations
- 5 Discrete Painlevé Equations and Orthogonal Polynomials
- 6 Generalized Lie Symmetries for Difference Equations
- 7 Four Lectures on Discrete Systems
- 8 Lectures on Moving Frames
- 9 Lattices of Compact Semisimple Lie Groups
- 10 Lectures on Discrete Differential Geometry
- 11 Symmetry Preserving Discretization of Differential Equations and Lie Point Symmetries of Differential-Difference Equations
- References
9 - Lattices of Compact Semisimple Lie Groups
Published online by Cambridge University Press: 05 July 2011
- Frontmatter
- Contents
- List of figures
- List of contributors
- Preface
- Introduction
- 1 Lagrangian and Hamiltonian Formalism for Discrete Equations: Symmetries and First Integrals
- 2 Painlevé Equations: Continuous, Discrete and Ultradiscrete
- 3 Definitions and Predictions of Integrability for Difference Equations
- 4 Orthogonal Polynomials, their Recursions, and Functional Equations
- 5 Discrete Painlevé Equations and Orthogonal Polynomials
- 6 Generalized Lie Symmetries for Difference Equations
- 7 Four Lectures on Discrete Systems
- 8 Lectures on Moving Frames
- 9 Lattices of Compact Semisimple Lie Groups
- 10 Lectures on Discrete Differential Geometry
- 11 Symmetry Preserving Discretization of Differential Equations and Lie Point Symmetries of Differential-Difference Equations
- References
Summary
Abstract
An efficient construction is to be described of lattice points FM of any density and any admissible symmetry in a finite region F of a real n-dimensional Euclidean space. The shape of F and the lattice symmetry of FM is determined by a compact semisimple Lie group of rank n. The density of FM is fixed by our choice of a positive integer M, where 1 ≤ M < ∞. The Lie group allows one to introduce systems of special functions discretely orthogonal on FM.
Introduction
The goal of this chapter is to provide all of the details necessary for construction of an n-dimensional lattice LM of any symmetry and density in the real Euclidean space ℝn. The motivation for such a construction might be the need to process digital data on LM. This typically requires a system of orthogonal functions on a finite fragment FM ⊂ LM. Such functions are available although their description is outside of the scope of this chapter. Some of the functions are shown here. But for their properties one needs to go to the references provided.
The starting point is a compact simple Lie group G of rank n, or equivalently, the corresponding simple Lie algebra g. Symmetry of its weight lattice P(g) is the symmetry of the lattice LM we construct. Density of LM is determined by our choice of natural number M, that is LM = P(g)/M.
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- Information
- Symmetries and Integrability of Difference Equations , pp. 247 - 258Publisher: Cambridge University PressPrint publication year: 2011