Atiyah sequences of braided Lie algebras and their splittings

2023/12/01 by Paolo Aschieri, Aschieri, Paolo, Giovanni Landi +3
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.2312.00495

openalex publication_date 2023/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Associated with an equivariant noncommutative principal bundle we give an Atiyah sequence of braided derivations whose splittings give connections on the bundle. Vertical braided derivations act as infinitesimal gauge transformations on connections. For the SU(2)-principal bundle over the sphere S⁴_θ an equivariant splitting of the Atiyah sequence recovers the instanton connection. An infinitesimal action of the braided conformal Lie algebra so_θ(5,1) yields a five parameter family of splittings. On the principal SO_θ(2n,ℝ)-bundle of orthonormal frames over the sphere S²ⁿ_θ, the splitting of the sequence leads to the Levi-Civita connection for the `round' metric on the S²ⁿ_θ. The corresponding Riemannian geometry of S²ⁿ_θ is worked out.

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