Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 1 (2005), 025, 6 pages      math-ph/0512029      https://doi.org/10.3842/SIGMA.2005.025

Compact Simple Lie Groups and Their C-, S-, and E-Transforms

Jiri Patera
Centre de Recherches Mathématiques, Université de Montréal, C.P.6128-Centre ville, Montréal, H3C 3J7, Québec, Canada

Received December 01, 2005; Published online December 09, 2005

Abstract
New continuous group transforms, together with their discretization over a lattice of any density and admissible symmetry, are defined for a general compact simple Lie groups of rank 2 ≤ n < ∞. Rank 1 transforms are known. Rank 2 exposition of C- and S-transforms is in the literature. The E-transforms appear here for the first time.

Key words: compact simple Lie groups; C-, S-, and E-transforms; discretization; fundamental region; Weyl group; weight lattice.

pdf (250 kb)   ps (391 kb)   tex (332 kb)

References

  1. Patera J., Invited talk at the Sixth International Conference "Symmetry in Nonlinear Mathematical Physics" (June 20-26, 2005, Kyiv).
  2. Atoyan A., Patera J., Properties of continuous Fourier extension of the discrete cosine transform and its multidimensional generalization, J. Math. Phys., 2004, V.45, 2468-2491.
  3. Wang Zhongde, Interpolation using type I discrete cosine transform, Electron. Lett., 1990, V.26, 1170-1171.
  4. Agbinya J.I., Two dimensional interpolation of real sequences using the DCT, Electron. Lett., 1993, V.29, 204-205.
  5. Moody R.V., Patera J., Computation of character decompositions of class functions on compact semisimple Lie groups, Math. Comp., 1987, V.48, 799-827.
  6. Moody R.V., Patera J., Elements of finite order in Lie groups and their applications, XIII Int. Colloq. on Group Theoretical Methods in Physics, Editor W. Zachary, Singapore, World Scientific Publishers, 1984, 308-318.
  7. McKay W.G., Moody R.V., Patera J., Decomposition of tensor products of E8 representations, Algebras Groups Geom., 1986, V.3, 286-328.
  8. McKay W.G., Moody R.V., Patera J., Tables of E8 characters and decomposition of plethysms, in Lie Algebras and Related Topics, Editors D.J. Britten, F.W. Lemire and R.V. Moody, Providence R.I., Amer. Math. Society, 1985, 227-264.
  9. Grimm S., Patera J., Decomposition of tensor products of the fundamental representations of E8, in Advances in Mathematical Sciences: CRM's 25 Years, Editor L. Vinet, CRM Proc. Lecture Notes, Providence RI, Amer. Math. Soc., 1997, V.11, 329-355.
  10. Patera J., Zaratsyan A., Cosine transform generalized to Lie groups SU(2)×SU(2) and O(5), J. Math. Phys., 2005, V.46, 053514, 25 pages.
  11. Patera J., Zaratsyan A., Cosine transform generalized to Lie groups SU(3) and G(2), J. Math. Phys., 2005, V.46, 113506, 17 pages.
  12. Atoyan A., Patera J., Sahakian V., Akhperjanian A., Fourier transform method for imaging atmospheric Cherenkov telescopes, Astroparticle Phys., 2005, V.23, 79-95.
  13. Patera J., Orbit functions of compact semisimple Lie groups as special functions, in Proceedinds of Fifth International Conference "Symmetry in Nonlinear Mathematical Physics" (June 23-29, 2003, Kyiv), Editors A.G. Nikitin, V.M. Boyko, R.O. Popovych and I.A. Yehorchenko, Proceedings of Institute of Mathematics, Kyiv, 2004, V.50, Part 3, 1152-1160.
  14. Patera J., Algebraic solutions of the Neumann boundary value problems on fundamental region of a compact semisimple Lie group, talk given at the Workshop on Group Theory and Numerical Methods (May 26-31, 2003, Montreal).
  15. Atoyan A., Patera J., Continuous extension of the discrete cosine transform, and its applications to data processing, Proceedings of the Workshop on Group Theory and Numerical Methods (May 26-31, 2003, Montreal).
  16. Klimyk A., Patera J., Orbit functions, in preparation.
  17. Patera J., Zaratsyan A., The sine transform generalized to semisimple Lie groups rank 2, Preprint, 2005.
  18. Kashuba I., Patera J., Zaratsyan A., The E-functions of compact semisimple Lie groups and their discretization, in preparation.
  19. Bremner M.R., Moody R.V., Patera J., Tables of dominant weight multiplicities for representations of simple Lie algebras, New York, Marcel Dekker, 1985, 340 pages.
  20. Moody R.V., Patera J., Discrete and continuous orthogonality of C-, S-, and E-functions of a compact semisimple Lie group, in preparation.


Previous article   Next article   Contents of Volume 1 (2005)