Analogues of the exponential map associated with complex structures on noncommutative two-tori

Alexander Polishchuk

Vol. 226 (2006), No. 1, 153–178

DOI: 10.2140/pjm.2006.226.153

Abstract

We define and study analogues of exponentials for functions on noncommutative two-tori that depend on the choice of a complex structure. The major difference with the commutative case is that our exponentials can be defined only for sufficiently small functions. We show that this phenomenon is related to the existence of certain discriminant hypersurfaces in an irrational rotation algebra. As an application of our methods we give a very explicit characterization of connected components in the group of invertible elements of this algebra.

Keywords

exponential map, noncommutative tori

Mathematical Subject Classification 2000

Primary: 46L87

Milestones

Received: 2 September 2004

Accepted: 25 January 2005

Published: 1 July 2006