The University of Southampton
University of Southampton Institutional Repository

Moebius transformations in noncommutative conformal geometry

Moebius transformations in noncommutative conformal geometry
Moebius transformations in noncommutative conformal geometry
We study the projective linear group PGL2(A) associated with an arbitrary algebra A and its subgroups from the point of view of their action on the space of involutions in A. This action formally resembles Möbius transformations known from complex geometry. By specifying A to be an algebra of bounded operators in a Hilbert space H, we rediscover the Möbius group wev(M) defined by Connes and study its action on the space of Fredholm modules over the algebra A. There is an induced action on the K-homology of A, which turns out to be trivial. Moreover, this action leads naturally to a simpler object, the polarized module underlying a given Fredholm module, and we discuss this relation in detail. Any polarized module can be lifted to a Fredholm module, and the set of different lifts forms a category, whose morphisms are given by generalized Möbius tranformations. We present an example of a polarized module canonically associated with the differentiable structure of a smooth manifold V. Using our lifting procedure we obtain a class of Fredholm modules characterizing the conformal structures on V. Fredholm modules obtained in this way are a special case of those constructed by Connes, Sullivan and Teleman.
0010-3616
35-60
Bongaarts, P.J.M.
c3d3ed33-a2aa-4dc7-b0fc-c519ad4aabdc
Brodzki, J.
b1fe25fd-5451-4fd0-b24b-c59b75710543
Bongaarts, P.J.M.
c3d3ed33-a2aa-4dc7-b0fc-c519ad4aabdc
Brodzki, J.
b1fe25fd-5451-4fd0-b24b-c59b75710543

Bongaarts, P.J.M. and Brodzki, J. (1999) Moebius transformations in noncommutative conformal geometry. Communications in Mathematical Physics, 201 (1), 35-60. (doi:10.1007/s002200050548).

Record type: Article

Abstract

We study the projective linear group PGL2(A) associated with an arbitrary algebra A and its subgroups from the point of view of their action on the space of involutions in A. This action formally resembles Möbius transformations known from complex geometry. By specifying A to be an algebra of bounded operators in a Hilbert space H, we rediscover the Möbius group wev(M) defined by Connes and study its action on the space of Fredholm modules over the algebra A. There is an induced action on the K-homology of A, which turns out to be trivial. Moreover, this action leads naturally to a simpler object, the polarized module underlying a given Fredholm module, and we discuss this relation in detail. Any polarized module can be lifted to a Fredholm module, and the set of different lifts forms a category, whose morphisms are given by generalized Möbius tranformations. We present an example of a polarized module canonically associated with the differentiable structure of a smooth manifold V. Using our lifting procedure we obtain a class of Fredholm modules characterizing the conformal structures on V. Fredholm modules obtained in this way are a special case of those constructed by Connes, Sullivan and Teleman.

Full text not available from this repository.

More information

Published date: 1999

Identifiers

Local EPrints ID: 29845
URI: http://eprints.soton.ac.uk/id/eprint/29845
ISSN: 0010-3616
PURE UUID: 4651adfa-eb85-4b63-bb63-acb3b823dd85

Catalogue record

Date deposited: 27 Jul 2006
Last modified: 15 Jul 2019 19:08

Export record

Altmetrics

Contributors

Author: P.J.M. Bongaarts
Author: J. Brodzki

University divisions

Download statistics

Downloads from ePrints over the past year. Other digital versions may also be available to download e.g. from the publisher's website.

View more statistics

Atom RSS 1.0 RSS 2.0

Contact ePrints Soton: eprints@soton.ac.uk

ePrints Soton supports OAI 2.0 with a base URL of http://eprints.soton.ac.uk/cgi/oai2

This repository has been built using EPrints software, developed at the University of Southampton, but available to everyone to use.

We use cookies to ensure that we give you the best experience on our website. If you continue without changing your settings, we will assume that you are happy to receive cookies on the University of Southampton website.

×