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Székelyhidi, L.

A generalization of the Heisenberg group. (English) Zbl 07780488

J. Iran. Math. Soc. 4, No. 2, 105-119 (2023).
Summary: In our former paper we studied spectral synthesis on the Heisenberg group. This problem is closely connected with the finite dimensional representations of the Heisenberg group on the space of continuous complex valued functions. In this paper we make an attempt to generalise the Heisenberg group over any commutative topological group. Starting with a basic commutative topological group we define a non-commutative topological group whose elements are triplets consisting of an element of the basic group, an exponential on the basic group, and a nonzero complex number which serves as a scaling factor. The group operation is a combination of the addition on the basic group, the multiplication of the exponentials and the multiplication of complex nonzero numbers. Although there is no differentiability, our generalised Heisenberg group shares some basic properties with the classical one. In particular, we describe finite dimensional representations of this group on the space of continuous functions, and we show that finite dimensional translation invariant function spaces over this group consist of exponential polynomials.

MSC:

22E30 Analysis on real and complex Lie groups
22E25 Nilpotent and solvable Lie groups

Keywords:

Heisenberg group; Lie ring; representation; spectral synthesis
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\textit{L. Székelyhidi}, J. Iran. Math. Soc. 4, No. 2, 105--119 (2023; Zbl 07780488)
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References:

[1] [1] J. F. Cornwell, Group Theory in Physics, An introduction, Academic Press, Inc., San Diego, CA, 1997. · Zbl 0878.20001
[2] [2] G. B. Folland, Harmonic analysis in phase space, Princeton University Press, Princeton, NJ, 1989. · Zbl 0682.43001
[3] [3] H. Freudenthal and H. de Vries, Linear Lie Groups, Pure and Applied Mathematics 35 Academic Press, New York-London, 1969. · Zbl 0377.22001
[4] [4] J. E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, 9 Academic Press, Inc., Springer-Verlag, New York-Berlin, 1972. · Zbl 0254.17004
[5] [5] G. W. Mackey, A theorem of Stone and von Neumann, Duke Math. J. 16 (1949) 313-326. · Zbl 0036.07703
[6] [6] M. A. McKiernan, Functions of binomial type mapping groupoids into rings, Aequationes Math. 16 (1977), no. 2, 115-124. · Zbl 0335.39007
[7] [7] W. Rossmann, Lie Groups, Oxford Graduate Texts in Mathematics, 5 Oxford University Press, Oxford, 2002. · Zbl 0989.22001
[8] [8] L. Székelyhidi, Harmonic and Spectral Analysis, World Scienti_c Publishing Co. Pte. Ltd., Hackensack, NJ, 2014. · Zbl 1297.43001
[9] [9] L. Székelyhidi, Finite dimensional varieties over the Heisenberg group, Aequat. Math. 97 (2023), no. 2, 377-390. · Zbl 1519.43003
[10] [10] A. Weil, Sur certains groupes d’op_erateurs unitaires, Acta Math. 111 (1964) 143-211. · Zbl 0203.03305
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