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 some known r-matrices for special parameter values.
- 3d gravity
- Chern-Simons theory
- Classical Yang-Baxter equation
- Fock-Rosly Poisson structure
- Poisson-Lie structures
ASJC Scopus subject areas
- Physics and Astronomy (miscellaneous)