Classical Dynamical r-matrices for the Chern–Simons Formulation of Generalized 3d Gravity

Classical dynamical r-matrices arise naturally in the combinatorial description of the phase space of Chern–Simons theories, either through the inclusion of dynamical sources or through a gauge fixing procedure involving two punctures. Here we consider classical dynamical r-matrices for the family of Lie algebras which arise in the Chern–Simons formulation of 3d gravity, for any value of the cosmological constant. We derive differential equations for classical dynamical r-matrices in this case and show that they can be viewed as generalized complexifications, in a sense which we define, of the equations governing dynamical r-matrices for \(\mathfrak {su}(2)\) $\mathfrak{su}(2)$ and \(\mathfrak {sl}(2,{\mathbb {R}})\) $\mathfrak{sl}(2,\mathbb{R})$ . We obtain explicit families of solutions and relate them, via Weierstrass factorization, to solutions found by Feher, Gabor, Marshall, Palla and Pusztai in the context of chiral WZWN models.