Generalised fluxes, Yang-Baxter deformations and the O(d,d) structure of non-abelian T-duality

Based on the construction of Poisson-Lie T-dual $\sigma$-models from a common parent action we study a candidate for the non-abelian respectively Poisson-Lie T-duality group. This group generalises the well-known abelian T-duality group O(d,d) and we explore some of its subgroups, namely factorised dualities, B- and $\beta$-shifts. The corresponding duality transformed $\sigma$-models are constructed and interpreted as generalised (non-geometric) flux backgrounds. We also comment on generalisations of results and techniques known from abelian T-duality. This includes the Lie algebra cohomology interpretation of the corresponding non-geometric flux backgrounds, remarks on a double field theory based on non-abelian T-duality and an application to the investigation of Yang-Baxter deformations. This will show that homogeneously Yang-Baxter deformed $\sigma$-models are exactly the non-abelian T-duality $\beta$-shifts when applied to principal chiral or coset $\sigma$-models.