# T-Duality and Doubling of the Isotropic Rigid Rotator

## TL;DR

This paper explores the application of T-duality and doubled geometry to the isotropic rigid rotator, revealing new dual models and their relations within Poisson-Lie and generalized geometric frameworks.

## Contribution

It introduces a novel dual model on the group $SB(2, ext{C})$ within the Drinfel'd double structure of $SL(2, ext{C})$, connecting Poisson-Lie duality with generalized geometry.

## Key findings

- Constructed dual models on $SU(2)$ and $SB(2, ext{C})$
- Developed a doubled generalized action with $TSL(2, ext{C})$
- Demonstrated relations between Poisson-Lie symmetry and generalized brackets

## 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 $SL(2,\mathbb{C})$ starting from a non-Abelian deformation of the natural carrier space of its Hamiltonian description on $T^*SU(2) \simeq SU(2) \ltimes \mathbb{R}^3$. A new model is then introduced on the dual group $SB(2,\mathbb{C})$, within the Drinfel'd double description of $SL(2,\mathbb{C})=SU(2) \bowtie SB(2,\mathbb{C})$. The two models are analyzed from the Poisson-Lie duality point of view, and a doubled generalized action is built with $TSL(2,\mathbb{C})$ 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 $T \oplus T^*$ 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.