Classical r-matrices for the generalised Chern-Simons formulation of 3d gravity

Abstract

We study the conditions for classical r-matrices to be compatible with the generalised Chern-Simons action for 3d gravity. Compatibility means solving the classical Yang-Baxter equations with a prescribed symmetric part for each of the real Lie algebras and bilinear pairings arising in the generalised Chern-Simons action. We give a new construction of r-matrices via a generalised complexification and derive a non-linear set of matrix equations determining the most general compatible r-matrix. We exhibit new families of solutions and show that they contain known solutions for special parameter values

20 pages, minor corrections and comments added in v2

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• Quantum symmetries in 2+1 dimensions: Carroll, (a)dS-Carroll, Galilei and (a)dS-Galilei
• Minimal Factorization of Chern-Simons Theory -- Gravitational Anyonic Edge Modes
• Semidual Kitaev lattice model and tensor network representation