Generalized type IIB supergravity equations and non-Abelian classical r-matrices

We study Yang–Baxter deformations of the ${AdS}_5\times S^5$ superstring with non-Abelian classical r-matrices which satisfy the homogeneous classical Yang–Baxter equation. By performing a supercoset construction, we can get deformed ${AdS}_5\times S^5$ backgrounds. While this is a new area of research, the current understanding is that Abelian classical r-matrices give rise to solutions of type IIB supergravity, while non-Abelian classical r-matrices lead to solutions of the generalized supergravity equations. We examine here some examples of non-Abelian classical r-matrices and derive the associated backgrounds explicitly. All of the resulting backgrounds satisfy the generalized equations. For some of them, we derive 'T-dualized' backgrounds by adding a linear coordinate dependence to the dilaton and show that these satisfy the usual type IIB supergravity equations. Remarkably, some of the 'T-dualized' backgrounds are locally identical to undeformed ${AdS}_5\times S^5$ after an appropriate coordinate transformation, but this seems not to be generally the case. (paper)