Classical r-matrices for the generalised Chern-Simons formulation of 3d gravity
Prince K Osei, Bernd J Schroers
TL;DR
This paper investigates the conditions under which classical r-matrices are compatible with the generalized Chern-Simons formulation of 3D gravity, providing new solutions and a systematic construction method.
Contribution
It introduces a novel construction of compatible r-matrices through generalized complexification and derives equations characterizing the most general solutions.
Findings
New families of r-matrices compatible with 3D gravity
A systematic construction method via generalized complexification
Known solutions recovered as special cases
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
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