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Cartan structure equations and Levi-Civita connection in braided geometry (2006.02761v2)
Published 4 Jun 2020 in math.QA, hep-th, math-ph, and math.MP
Abstract: We study the differential and Riemannian geometry of algebras $A$ endowed with an action of a triangular Hopf algebra $H$ and noncomutativity compatible with the associated braiding. The modules of one forms and of braided derivations are modules in a compact closed category of $H$-modules $A$-bimodules, whose internal morphisms correspond to tensor fields. Different approaches to curvature and torsion are proven to be equivalent by extending the Cartan calculus to left (right) $A$-module connections. The Cartan structure equations and the Bianchi identities are derived. Existence and uniqueness of the Levi-Civita connection for arbitrary braided symmetric pseudo-Riemannian metrics is proven.