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

SIGMA 18 (2022), 065, 21 pages      arXiv:2105.13811      https://doi.org/10.3842/SIGMA.2022.065

Tuning Co- and Contra-Variant Transforms: the Heisenberg Group Illustration

Amerah A. Al Ameer a and Vladimir V. Kisil b
a) School of Science, Mathematics Department, University of Hafr Al Batin, Hafr Al Batin 31991 P.O Box 1803, Saudi Arabia
b) School of Mathematics, University of Leeds, Leeds LS29JT, UK

Received December 26, 2021, in final form August 26, 2022; Published online September 01, 2022

Abstract
We discuss a fine tuning of the co- and contra-variant transforms through construction of specific fiducial and reconstructing vectors. The technique is illustrated on three different forms of induced representations of the Heisenberg group. The covariant transform provides intertwining operators between pairs of representations. In particular, we obtain the Zak transform as an induced covariant transform intertwining the Schrödinger representation on $\mathsf{L}_2(\mathbb{R})$ and the lattice (nilmanifold) representation on $\mathsf{L}_2\big(\mathbb{T}^2\big)$. Induced covariant transforms in other pairs are Fock-Segal-Bargmann and theta transforms. Furthermore, we describe peelings which map the group-theoretical induced representations to convenient representation spaces of analytic functions. Finally, we provide a condition which can be imposed on the reconstructing vector in order to obtain an intertwining operator from the induced contravariant transform.

Key words: Heisenberg group; covariant transform; coherent states; Zak transform; Fock-Segal-Bargmann space.

pdf (539 kb)   tex (32 kb)  

References

  1. Al Ameer A.A., Singularities of analytic functions and group representations, Ph.D. Thesis, University of Leeds, 2019, available at https://etheses.whiterose.ac.uk/24776/.
  2. Albargi A.H.A., Covariant transforms on locally convex spaces, Ph.D. Thesis, University of Leeds, 2015, available at https://etheses.whiterose.ac.uk/12308/.
  3. Ali S.T., Antoine J.P., Gazeau J.P., Coherent states, wavelets, and their generalizations, 2nd ed., Theoretical and Mathematical Physics, Springer, New York, 201,.
  4. Almalki F., Kisil V.V., Geometric dynamics of a harmonic oscillator, arbitrary minimal uncertainty states and the smallest step 3 nilpotent Lie group, J. Phys. A: Math. Theor. 52 (2019), 025301, 25 pages, arXiv:1805.01399.
  5. Almalki F., Kisil V.V., Solving the Schrödinger equation by reduction to a first-order differential operator through a coherent states transform, Phys. Lett. A 384 (2020), 126330, 7 pages, arXiv:1903.03554.
  6. Arefijamaal A.A., Ghaani Farashahi A., Zak transform for semidirect product of locally compact groups, Anal. Math. Phys. 3 (2013), 263-276, arXiv:1203.1509.
  7. Barbieri D., Hernández E., Mayeli A., Bracket map for the Heisenberg group and the characterization of cyclic subspaces, Appl. Comput. Harmon. Anal. 37 (2014), 218-234, arXiv:1303.2350.
  8. Barbieri D., Hernández E., Paternostro V., The Zak transform and the structure of spaces invariant by the action of an LCA group, J. Funct. Anal. 269 (2015), 1327-1358, arXiv:1410.7250.
  9. Berezin F.A., The method of second quantization, 2nd ed., Nauka, Moscow, 1986.
  10. Berndt R., Representations of linear groups: an introduction based on examples from physics and number theory, Vieweg, Wiesbaden, 2007.
  11. Bohm A., Quantum mechanics: foundations and applications, 3rd ed., Texts and Monographs in Physics, Springer-Verlag, New York, 1993.
  12. Cartier P., Quantum mechanical commutation relations and theta functions, in Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., Providence, R.I., 1966, 361-383.
  13. Feichtinger H.G., Gröchenig K.H., Banach spaces related to integrable group representations and their atomic decompositions. I, J. Funct. Anal. 86 (1989), 307-340.
  14. Feichtinger H.G., Gröchenig K.H., Banach spaces related to integrable group representations and their atomic decompositions. II, Monatsh. Math. 108 (1989), 129-148.
  15. Folland G.B., Harmonic analysis in phase space, Annals of Mathematics Studies, Vol. 122, Princeton University Press, Princeton, NJ, 1989.
  16. Folland G.B., A course in abstract harmonic analysis, 2nd ed., Textbooks in Mathematics, CRC Press, Boca Raton, FL, 2016.
  17. Gröchenig K., Foundations of time-frequency analysis, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2001.
  18. Grossmann A., Morlet J., Paul T., Transforms associated to square integrable group representations. I. General results, J. Math. Phys. 26 (1985), 2473-2479.
  19. Grossmann A., Morlet J., Paul T., Transforms associated to square integrable group representations. II. Examples, Ann. Inst. H. Poincaré Phys. Théor. 45 (1986), 293-309.
  20. Hernández E., Luthy P.M., Šikić H., Soria F., Wilson E.N., Spaces generated by orbits of unitary representations: a tribute to Guido Weiss, J. Geom. Anal. 31 (2021), 8735-8761.
  21. Hernández E., Šikić H., Weiss G., Wilson E., Cyclic subspaces for unitary representations of LCA groups; generalized Zak transform, Colloq. Math. 118 (2010), 313-332.
  22. Hernández E., Šikić H., Weiss G.L., Wilson E.N., The Zak transform(s), in Wavelets and Multiscale Analysis, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, New York, 2011, 151-157.
  23. Howe R., On the role of the Heisenberg group in harmonic analysis, Bull. Amer. Math. Soc. (N.S.) 3 (1980), 821-843.
  24. Iverson J.W., Subspaces of $L^2(G)$ invariant under translation by an abelian subgroup, J. Funct. Anal. 269 (2015), 865-913, arXiv:1411.1014.
  25. Iverson J.W., Frames generated by compact group actions, Trans. Amer. Math. Soc. 370 (2018), 509-551, arXiv:1509.06802.
  26. Iverson J.W., The Zak transform and representations induced from characters of an abelian subgroup, in 13th International conference on Sampling Theory and Applications (SampTA) (July 08-12, 2019, Bordeaux, France), IEEE, 2020, 19451251, 4 pages, arXiv:1904.03527.
  27. Kaniuth E., Taylor K.F., Induced representations of locally compact groups, Cambridge Tracts in Mathematics, Vol. 197, Cambridge University Press, Cambridge, 2013.
  28. Kirillov A.A., Elements of the theory of representations, Grundlehren der Mathematischen Wissenschaften, Vol. 220, Springer-Verlag, Berlin - New York, 1976.
  29. Kirillov A.A., Introduction to the theory of representations and noncommutative harmonic analysis, in Representation Theory and Noncommutative Harmonic Analysis, I, Encyclopaedia Math. Sci., Vol. 22, Springer, Berlin, 1994, 1-156, 227-234.
  30. Kirillov A.A., Lectures on the orbit method, Graduate Studies in Mathematics, Vol. 64, Amer. Math. Soc., Providence, RI, 2004.
  31. Kisil V.V., Wavelets in Banach spaces, Acta Appl. Math. 59 (1999), 79-109, arXiv:math.FA/9807141.
  32. Kisil V.V., $p$-mechanics as a physical theory: an introduction, J. Phys. A: Math. Gen. 37 (2004), 183-204, arXiv:quant-ph/0212101.
  33. Kisil V.V., Erlangen program at large: an overview, in Advances in applied analysis, Trends Math., Birkhäuser/Springer Basel AG, Basel, 2012, 1-94, arXiv:1106.1686.
  34. Kisil V.V., Operator covariant transform and local principle, J. Phys. A: Math. Theor. 45 (2012), 244022, 10 pages, arXiv:1201.1749.
  35. Kisil V.V., Calculus of operators: covariant transform and relative convolutions, Banach J. Math. Anal. 8 (2014), 156-184, arXiv:1304.2792.
  36. Kisil V.V., The real and complex techniques in harmonic analysis from the point of view of covariant transform, Eurasian Math. J. 5 (2014), 95-121, arXiv:1209.5072.
  37. Kisil V.V., Symmetry, geometry and quantization with hypercomplex numbers, in Geometry, Integrability and Quantization XVIII, Bulgar. Acad. Sci., Sofia, 2017, 11-76, arXiv:1611.05650.
  38. Massopust P.R., Fractal functions, fractal surfaces, and wavelets, 2nd ed., Elsevier/Academic Press, London, 2016.
  39. Neretin Yu.A., Lectures on Gaussian integral operators and classical groups, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich, 2011.
  40. Perelomov A., Generalized coherent states and their applications, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1986.
  41. Rudin W., Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991.
  42. Vasilevski N.L., On the structure of Bergman and poly-Bergman spaces, Integral Equations Operator Theory 33 (1999), 471-488.
  43. Weil A., Sur certains groupes d'opérateurs unitaires, Acta Math. 111 (1964), 143-211.
  44. Zak J., Finite translations in solid-state physics, Phys. Rev. Lett. 19 (1967), 1385-1387.

Previous article  Next article  Contents of Volume 18 (2022)