An Introduction to Heisenberg Groups in Analysis and Geometry, Volume 50, Number 6

640 NOTICES OF THE AMS VOLUME 50, NUMBER 6 H eisenberg groups, in discrete and continuous versions, appear in many parts of mathematics, including Fourier analysis, several complex variables, geometry, and topology. In the present survey we shall not focus too much on any particular aspect, but try to give a kind of sampler. We begin with something which is, in effect, basic calculus. Commutators of Multiplication and Differentiation Operators Fix a positive integer n, and let v ·w denote the usual dot product on Rn, so that v ·w =∑nj=1 vj wj , where vj and wj denote the jth components of v and w , respectively. If v and w are elements of Rn and f is a function on Rn, let us write Mw (f ) for the function on Rn defined by Mw (f )(x) = (w · x) f (x), and let us write Dv (f ) for the function on R which is the directional derivative of f associated to v, i.e., Dv (f )(x) = v · ∇f (x). We shall not dwell on differentiability issues here, and so the reader is invited to assume that the functions are sufficiently smooth as might be convenient. The usual Leibniz rule for differentiating products implies that