An introduction to Heisenberg groups in analysis and geometry. (English) Zbl 1050.22012

Semmes, Stephen

Notices Am. Math. Soc. 50, No. 6, 640-646 (2003).

Various aspects of the general Heisenberg group theory, in discrete and continuous versions, and “the ways in which their algebraic structure is reflected in Fourier analysis, complex analysis and geometry” are discussed. In the paper, the following topics are considered: commutators of multiplication and differentiation operators, some groups of linear operators, definition of the \(n\)-th Heisenberg group, discrete versions of the Heisenberg groups, connections with several complex variables, tube domains and spaces of holomorphic functions on them, some geometric aspects.

Reviewer: A. A. Bogush (Minsk)

MSC:

22E25	Nilpotent and solvable Lie groups
42B10	Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
51P05	Classical or axiomatic geometry and physics (should also be assigned at least one other classification number from Sections 70-XX–86-XX)