A Wiener Lemma for the discrete Heisenberg group

Poincaré’s Lemma on the Heisenberg Group

It is well known that the system ∂xf = a, ∂yf = b on R 2 has a solution if and only if the closure condition ∂xb = ∂ya holds. In this case the solution f is the work done by the force U = (a, b) from the origin to the point (x, y). This paper deals with a similar problem, where the vector fields ∂x, ∂y are replaced by the Heisenberg vector fields X1, X2. In this case the subRiemannian system X1...

Discrete Heisenberg-Weyl Group and Modular Group

It is shown that the generators of two discrete Heisenberg-Weyl groups with irrational rotation numbers θ and −1/θ generate the whole algebra B of bounded operators on L2(R). The natural action of the modular group in B is implied. Applications to dynamical algebras appearing in lattice regularization and some duality principles are discussed. Writing a contribution to a memorial volume one alw...

Automorphisms of the Discrete Heisenberg Group

This note describes some computational results that were obtained while studying the associated group of automorphisms Aut(H), [2]. Specifically, we give an explicit description of a splitting of this group as an extension of GL(2,Z) by Z. That such a splitting exists could, in principle, be shown by computing a 2-cocycle for the extension and showing that it represents 0 in the cohomology grou...

The Noncommutative Geometry of the Discrete Heisenberg Group

Motivated by the search for new examples of “noncommutative manifolds”, we study the noncommutative geometry of the group C*-algebra of the three dimensional discrete Heisenberg group. We present a unified treatment of the K-homology, cyclic cohomology and derivations of this algebra. 1. The discrete Heisenberg group Recently, there has been considerable interest in the notion of a “noncommutat...

The Wiener algebra and Wiener’s lemma