Is every locally compact abelian group which admits a Heisenberg central extension isomorphic to the product of a locally compact abelian group and its Pontryagin dual? An affirmative answer is obtained for all the commonly occurring types of abelian groups having Heisenberg central extensions, including Lie groups and certain finite Cartesian products of local fields and adèles. Furthermore, f...

Heisenberg, in constructing quantum mechanics, explicitly followed certain principles exemplified, as he believed, in Einstein's construction of the special theory of relativity which for him was the paradigm for radical theoretic change in physics. These were the principles of (i) scientific realism, (ii) stability of background knowledge, (iii) E-observability, (iv) contextual re-interpretati...

We compute the braided groups and braided matrices B(R) for the solution R of the Yang-Baxter equation associated to the quantum Heisenberg group. We also show that a particular extension of the quantum Heisenberg group is dual to the Heisenberg universal enveloping algebra U q (h), and use this result to derive an action of U q (h) on the braided groups. We then demonstrate the various covaria...

The study of step 2 subRiemannian manifolds has the Heisenberg group as a prototype. This is a noncommutative Lie group with the base manifold R and endowed with a nonintegrable distribution spanned by two of the noncommutative left invariant vector fields. This structure enjoys also the property of being a contact structure or a CR-manifold. The study of the subRiemannian geodesics on the Heis...

Given k ∈ N, the k’th discrete Heisenberg group, denoted H Z , is the group generated by the elements a1, b1, . . . , ak, bk, c, subject to the commutator relations [a1, b1] = . . . = [ak, bk] = c, while all the other pairs of elements from this generating set are required to commute, i.e., for every distinct i, j ∈ {1, . . . , k} we have [ai, aj ] = [bi, bj ] = [ai, bj ] = [ai, c] = [bi, c] = ...

The aim of this paper is to establish higher order Poincaré-Sobolev, Hardy-Sobolev-Maz'ya, Adams and Hardy-Adams inequalities on Siegel domains complex hyperbolic spaces using the method Helgason-Fourier analysis spaces. Firstly, we give a factorization theorem for operators space which closely related Geller's operator, as well CR invariant differential Heisenberg group sphere. (See Theorem 1....