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9 - Lattices of Compact Semisimple Lie Groups

Published online by Cambridge University Press:  05 July 2011

Jiří Patera
Affiliation:
Université de Montréal
Decio Levi
Affiliation:
Università degli Studi Roma Tre
Peter Olver
Affiliation:
University of Minnesota
Zora Thomova
Affiliation:
SUNY Institute of Technology
Pavel Winternitz
Affiliation:
Université de Montréal
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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 FMLM. 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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Chapter
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Publisher: Cambridge University Press
Print publication year: 2011

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References

[1] Hrivnák, J., and Patera, J. 2009. On discretization of tori of compact simple Lie groups. J. Phys. A, 42(38), 385208.CrossRefGoogle Scholar
[2] Klimyk, A., and Patera, J. 2006. Orbit functions. SIGMA Symmetry Integrability Geom. Methods Appl., 2, Paper 006.Google Scholar
[3] Klimyk, A., and Patera, J. 2007. Antisymmetric orbit functions. SIGMA Symmetry Integrability Geom. Methods Appl., 3, Paper 023.Google Scholar
[4] Moody, R. V., and Patera, J. 1984. Characters of elements of finite order in Lie groups. SIAM J. Algebraic Discrete Methods, 5(3), 359–383.CrossRefGoogle Scholar
[5] Moody, R. V., and Patera, J. 1987. Computation of character decompositions of class functions on compact semisimple Lie groups. Math. Comp., 48(178), 799–827.CrossRefGoogle Scholar
[6] Moody, R. V., and Patera, J. 2006. Orthogonality within the families of C-, S-, and E-functions of any compact semisimple Lie group. SIGMA Symmetry Integrability Geom. Methods Appl., 2, Paper 076.Google Scholar
[7] Nesterenko, M., Patera, J., and Tereszkiewicz, A. 2010. Orthogonal polynomials of compact simple Lie groups. arXiv:1001.3683.
[8] Nesterenko, M., Patera, J., Szjewska, M., and Tereszkiewicz, A. 2010. Orthogonal polynomials of compact simple Lie groups. The branching rules for polynomials, J. Phys. A: Math. Theor. 43, 495207 (20pp); arXiv:1007.4431v1CrossRefGoogle Scholar
[9] Patera, J. 2005. Compact simple Lie groups and their C-, S-, and E-transforms. SIGMA Symmetry Integrability Geom. Methods Appl., 1, Paper 025.Google Scholar
[10] Patera, J., and Zaratsyan, A. 2005a. Discrete and continuous cosine transform generalized to Lie groups SU(2)×SU(2) and O(5). J. Math. Phys., 46(5), 053514.Google Scholar
[11] Patera, J., and Zaratsyan, A. 2005b. Discrete and continuous cosine transform generalized to Lie groups SU(3) and G(2). J. Math. Phys., 46(11), 113506.Google Scholar

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