Atiyah sequences of braided Lie algebras and their splittings
Given an equivariant noncommutative principal bundle, we construct an Atiyah sequence of braided derivations whose splittings give connections on the bundle. Vertical braided derivations act as infinitesimal gauge transformations on connections. In the case of the principal \mathrm{SU}(2) -bundle over the sphere S^{4}_{\theta} an equivariant splitting of the Atiyah sequence recovers the instanton connection. An infinitesimal action of the braided conformal Lie algebra \mathrm{so}_{\theta}(5,1) yields a five parameter family of splittings. On the principal \mathrm{SO}_\theta(2n,\mathbb{R}) -bundle of orthonormal frames over the sphere S^{2n}_\theta , a splitting of the sequence leads to the Levi-Civita connection for the ‘round’ metric on S^{2n}_\theta . The corresponding Riemannian geometry of S^{2n}_{\theta} is worked out.
Authors
- Paolo Aschieri (ORCID: https://orcid.org/0000-0001-5477-6654)
- Chiara Pagani (ORCID: https://orcid.org/0000-0002-6962-4466)
- Giovanni Landi (ORCID: https://orcid.org/0000-0003-4293-8454)
Institutions
- Università degli Studi del Piemonte Orientale “Amedeo Avogadro” (IT)
- University of Trieste (IT)
- Istituto Nazionale di Fisica Nucleare, Sezione di Torino (IT)
- Istituto Nazionale di Fisica Nucleare, Sezione di Trieste (IT)
- University of Naples Federico II (IT)
- University of Bologna (IT)
Publication Details
- Journal
- Journal of Noncommutative Geometry
- Published
- 2024-09-06
- DOI
- https://doi.org/10.4171/jncg/591
- Citations
- 1
- Primary Topic
- Algebraic structures and combinatorial models
- Type
- article
- Field-Weighted Citation Impact
- 1.01
Funders
- Università degli Studi di Napoli Federico II
- Istituto Nazionale di Alta Matematica "Francesco Severi"
- Dipartimenti di Eccellenza
- Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni
- Gruppo Nazionale per la Fisica Matematica