dorsal/arxiv
View SchemaMoebius Transformations in Noncommutative Conformal Geometry
| Authors | Peter Bongaarts, Jacek Brodzki |
|---|---|
| Categories | |
| ArXiv ID | physics/9710005 |
| URL | https://arxiv.org/abs/physics/9710005 |
| DOI | 10.1007/s002200050548 |
| Journal | Commun.Math.Phys. 201 (1999) 35-60 |
Abstract
We study the projective linear group PGL_2(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 Moebius transformations known from complex geometry. By specifying A to be an algebra of bounded operators in a Hilbert space H, we rediscover the Moebius group 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 Moebius 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.
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"abstract": "We study the projective linear group PGL_2(A), associated with an arbitrary\nalgebra A, and its subgroups from the point of view of their action on the\nspace of involutions in A. This action formally resembles Moebius\ntransformations known from complex geometry. By specifying A to be an algebra\nof bounded operators in a Hilbert space H, we rediscover the Moebius group\ndefined by Connes and study its action on the space of Fredholm modules over\nthe algebra A. There is an induced action on the K-homology of A, which turns\nout to be trivial. Moreover, this action leads naturally to a simpler object,\nthe polarized module underlying a given Fredholm module, and we discuss this\nrelation in detail. Any polarized module can be lifted to a Fredholm module,\nand the set of different lifts forms a category, whose morphisms are given by\ngeneralized Moebius tranformations. We present an example of a polarized module\ncanonically associated with the differentiable structure of a smooth manifold\nV. Using our lifting procedure we obtain a class of Fredholm modules\ncharacterizing the conformal structures on V. Fredholm modules obtained in this\nway are a special case of those constructed by Connes, Sullivan and Teleman.",
"arxiv_id": "physics/9710005",
"authors": [
"Peter Bongaarts",
"Jacek Brodzki"
],
"categories": [
"math-ph",
"hep-th",
"math.MP",
"math.QA",
"q-alg"
],
"doi": "10.1007/s002200050548",
"journal_ref": "Commun.Math.Phys. 201 (1999) 35-60",
"title": "Moebius Transformations in Noncommutative Conformal Geometry",
"url": "https://arxiv.org/abs/physics/9710005"
},
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