Parabolic noncommutative geometry
Fries, Magnus LU
; Goffeng, Magnus
LU
and Masters, Ada
LU
(2026)
In Advances in Mathematics
499.
- Abstract
We introduce to spectral noncommutative geometry the notion of tangled spectral triple, which encompasses the anisotropies arising in parabolic geometry as well as the parabolic commutator bounds arising in so-called “bad Kasparov products”. Tangled spectral triples incorporate anisotropy by replacing the unbounded operator in a spectral triple that mimics a Dirac operator with several unbounded operators mimicking directional Dirac operators. We allow for varying and dependent orders in different directions, controlled by using the tools of tropical combinatorics. We study the conformal equivariance of tangled spectral triples as well as how they fit into K -homology by means of producing higher order spectral triples. Our main... (More)
We introduce to spectral noncommutative geometry the notion of tangled spectral triple, which encompasses the anisotropies arising in parabolic geometry as well as the parabolic commutator bounds arising in so-called “bad Kasparov products”. Tangled spectral triples incorporate anisotropy by replacing the unbounded operator in a spectral triple that mimics a Dirac operator with several unbounded operators mimicking directional Dirac operators. We allow for varying and dependent orders in different directions, controlled by using the tools of tropical combinatorics. We study the conformal equivariance of tangled spectral triples as well as how they fit into K -homology by means of producing higher order spectral triples. Our main examples are hypoelliptic spectral triples constructed from Rockland complexes on parabolic geometries; we also build spectral triples on nilpotent group C⁎-algebras from the dual Dirac element and crossed product spectral triples for parabolic dynamical systems.
(Less)
- author
- Fries, Magnus
LU
; Goffeng, Magnus
LU
and Masters, Ada
LU
- organization
- publishing date
- 2026-07
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Differential complexes, Kasparov product, Nilpotent groups, Noncommutative geometry, Parabolic geometry, Spectral triples, Unbounded KK-theory
- in
- Advances in Mathematics
- volume
- 499
- article number
- 111057
- publisher
- Academic Press
- external identifiers
-
- scopus:105039846270
- ISSN
- 0001-8708
- DOI
- 10.1016/j.aim.2026.111057
- language
- English
- LU publication?
- yes
- id
- 15670250-eae7-4190-ba86-ce5d35365c91
- date added to LUP
- 2026-08-13 09:53:53
- date last changed
- 2026-08-13 09:54:26
@article{15670250-eae7-4190-ba86-ce5d35365c91,
abstract = {{<p>We introduce to spectral noncommutative geometry the notion of tangled spectral triple, which encompasses the anisotropies arising in parabolic geometry as well as the parabolic commutator bounds arising in so-called “bad Kasparov products”. Tangled spectral triples incorporate anisotropy by replacing the unbounded operator in a spectral triple that mimics a Dirac operator with several unbounded operators mimicking directional Dirac operators. We allow for varying and dependent orders in different directions, controlled by using the tools of tropical combinatorics. We study the conformal equivariance of tangled spectral triples as well as how they fit into K -homology by means of producing higher order spectral triples. Our main examples are hypoelliptic spectral triples constructed from Rockland complexes on parabolic geometries; we also build spectral triples on nilpotent group C<sup>⁎</sup>-algebras from the dual Dirac element and crossed product spectral triples for parabolic dynamical systems.</p>}},
author = {{Fries, Magnus and Goffeng, Magnus and Masters, Ada}},
issn = {{0001-8708}},
keywords = {{Differential complexes; Kasparov product; Nilpotent groups; Noncommutative geometry; Parabolic geometry; Spectral triples; Unbounded KK-theory}},
language = {{eng}},
publisher = {{Academic Press}},
series = {{Advances in Mathematics}},
title = {{Parabolic noncommutative geometry}},
url = {{http://dx.doi.org/10.1016/j.aim.2026.111057}},
doi = {{10.1016/j.aim.2026.111057}},
volume = {{499}},
year = {{2026}},
}