Duflo-Moore Operator for The Square-Integrable Representation of 2-Dimensional Affine Lie Group
DOI:
https://doi.org/10.15642/mantik.2020.6.2.114-122Keywords:
Affine Lie group, Duflo-Moore operator, Square-integrable representation;Abstract
In this paper, we study the quasi-regular and the irreducible unitary representation of affine Lie group of dimension two. First, we prove a sharpening of Fuhr’s work of Fourier transform of quasi-regular representation of . The second, in such the representation of affine Lie group is square-integrable then we compute its Duflo-Moore operator instead of using Fourier transform as in F hr’s work.
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References
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[2] A. . Farashahi, “Square-integrability of metaplectic wave-packet representations of L^2(R),” J. Math. Anal. Appl., vol. 449, no. 1, 2017.
[3] A. Grossmann, J. Morlet, and T. Paul, “Transform associated to square-integrable group of representations I,” J.Math.Phys, vol. 26, pp. 2473--2479, 1985.
[4] A. Grossmann, J. Morlet, and T. Paul, “Trsansform associated to square-integrable group representations .II. Examples,” Ann.Inst.H.Poincare Phys.Theor, vol. 45, pp. 293--309, 1986.
[5] P. Stachura, “On the quantum ax+b group,” J. Geom. Phys., vol. 73, pp. 125--149, 2013.
[6] Zeitlin,A.M, “Unitary representations of a loop ax+b group, Wiener measure and \Gamma -function,” J. Funct. Anal., vol. 263, no. 3, pp. 529--548, 2012.
[7] M. . Dyer and G. . Lehres, “Parabolic subgroup orbits on finite root systems,” J. Pure Appl. Algebr., vol. 222, no. 12, pp. 3849--3857, 2018.
[8] M. Calvez and et al, “Conjugacy stability of parabolic subgroups of Artin-Tits groups of spherical type,” J. Algebr., vol. 556, pp. 621--633, 2020.
[9] E. Kurniadi and H. Ishi, “Harmonic analysis for 4-dimensional real Frobenius Lie algebras,” in Springer Proceeding in Mathematics & Statistics, 2019.
[10] W. Rump, “Affine stucture of decomposable solvable groups,” J. Algebr., vol. 556, pp. 725--749, 2020.
[11] H. Li and Q. Wang, “Trigonometric Lie algebras, affine Lie algebras, and vertex algebras,” J. Adv. Math., vol. 363, 2020.
[12] J. . Souza, “Sufficient conditions for dispersiveness of invariant control affine system on the Heisenberg group,” Syst. &Control Lett., vol. 124, pp. 68--74, 2019.
[13] E. Marberg, “On some actions the 0-zero hecke monoids of affine symmetric groups,” J. Comb. Theory, vol. 161, pp. 178--219, 2019.
[14] Ayala,V, A. Da Silva, and M. Ferreira, “Affine and bilinear systems on Lie groups,” Syst. &Control Lett., vol. 117, pp. 23--29, 2018.
[15] F. Catino, I. Colazzo, and P. Stefanella, “Regular subgroups of the affine group and asymmetric product of radical braces,” J. Algebr., vol. 455, pp. 164--182, 2016.
[16] D. Burde and et al, “Affine actions on Lie groups and post-Lie algebra structures,” J. Linear Algebr. Its Appl., vol. 437, no. 5, pp. 1250--1263, 2012.
[17] H. Kato, “Low dimensional Lie groups admitting left-invariant flat projective or affine structures,” J. Differ. Geom. Its Appl., vol. 30, no. 2, pp. 153--163, 2012.
[18] H. Fuhr, Abstrac harmonic analysis of continuous wavelet transforms, Lecture notes in mathematics. Berlin: Springer-Verlag, 2005.
[19] E. Kurniadi, “On Square-Integrable Representations of A Lie Group of 4-Dimensional Standard Filiform Lie Algebra,” CauchyJurnal Mat. Murni dan Apl., 2020.
[20] A. A. Kirillov, “Lectures on the Orbit Method, Graduate Studies in Mathematics,” Am. Math. Soc., vol. 64, 2004.
[21] R. Berndt, Representation of linear groups. An introduction based on examples from physics and number theory. Wiesbaden: Vieweg, 2007.
[22] P. Aniello, G. Cassinelli, E. de Vito, and Levrero,A., “Square-integrability of induced representations of semidirect products,” Rev.Math.Phys, vol. 10, pp. 301--313, 1998.
[23] M. Duflo and C. C. Moore, “On the Regular Representation of a nonunimodular Locally Compact,” J. Funct. Anal., vol. 21, pp. 209–243, 1976.
[24] E. Kurniadi, “Harmonic analysis for finite dimensional real Frobenius Lie algebras,Ph.D thesis, ” Nagoya University, 2019.
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Published
2020-10-31
How to Cite
Kurniadi, E., Gusriani, N., & Subartini, B. (2020). Duflo-Moore Operator for The Square-Integrable Representation of 2-Dimensional Affine Lie Group . Jurnal Matematika MANTIK, 6(2), 114–122. https://doi.org/10.15642/mantik.2020.6.2.114-122
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