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Discrete Heisenberg-Weyl group and modular group. (English) Zbl 0836.47012

Summary: It is shown that the generators of two discrete Heisenberg-Weyl groups with irrational rotation numbers \(\theta\) and \(-1/ \theta\) generate the whole algebra \({\mathcal B}\) of operators on \(L_2 (\mathbb{R})\). The natural action of the modular group in \({\mathcal B}\) is implied. Applications to dynamical algebras appearing in lattice regularization and some duality principles are discussed.

MSC:

47A60 Functional calculus for linear operators
47A67 Representation theory of linear operators
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
81S05 Commutation relations and statistics as related to quantum mechanics (general)

References:

[1] Schwinger, J.:Proc. Nat. Acad. Sci. USA 46 (1960), 1401. · Zbl 0125.21603 · doi:10.1073/pnas.46.10.1401
[2] Faddeev, L. and Volkov, A.:Phys. Lett. B 315 (1993), 311. · Zbl 0864.17042 · doi:10.1016/0370-2693(93)91618-W
[3] Faddeev, L.: Lectures delivered at the International School of Physics ’Enrico Fermi’ Varenna, Italy, 1994, hep-th/9408041.
[4] Connes, A.:Non-commutative Geometry: Academic Press, New York, 1994. · Zbl 0818.46076
[5] Polyakov, A.:Gauge Fields and Strings, Ellis Harwood, Chichester, 1991. · Zbl 1021.81879
[6] Faddeev, L.:Lett. ZhETP 21 (1975), 141.
[7] Rieffel, M. A.,Pacific J. Math. 93 (1981), 715.
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