T-duality and doubling of the isotropic rigid rotator

Abstract

After reviewing some of the fundamental aspects of Drinfel’d doubles and Poisson-Lie T-duality, we describe the three-dimensional isotropic rigid rotator on SL(2,C) starting from a non-Abelian deformation of the natural carrier space of its Hamiltonian description on T*SU(2) ≃ SU(2) n ℝ3. A new model is then introduced on the dual group SB(2,C), within the Drinfel’d double description of SL(2,C) = SU(2) ./ SB(2,C). The two models are analyzed from the Poisson-Lie duality point of view, and a doubled generalized action is built with TSL(2,C) as carrier space. The aim is to explore within a simple case the relations between Poisson-Lie symmetry, Doubled Geometry and Generalized Geometry. In fact, all the mentioned structures are discussed, such as a Poisson realization of the C-brackets for the generalized bundle T ⊗T* over SU(2) from the Poisson algebra of the generalized model. The two dual models exhibit many features of Poisson-Lie duals and from the generalized action both of them can be respectively recovered by gauging one of its symmetries.