Poisson-Lie plurals of Bianchi cosmologies and Generalized Supergravity Equations

Poisson-Lie T-duality and plurality are important solution generating techniques in string theory and (generalized) supergravity. Since duality/plurality does not preserve conformal invariance, the usual beta function equations are replaced by Generalized Supergravity Equations containing vector $\mathcal{J}$. In this paper we apply Poisson-Lie T-plurality on Bianchi cosmologies. We present a formula for the vector $\mathcal{J}$ as well as transformation rule for dilaton, and show that plural backgrounds together with this dilaton and $\mathcal{J}$ satisfy the Generalized Supergravity Equations. The procedure is valid also for non-local dilaton and non-constant $\mathcal{J}$. We also show that $Div\,\Theta$ of the non-commutative structure $\Theta$ used for non-Abelian T-duality or integrable deformations does not give correct $\mathcal{J}$ for Poisson-Lie T-plurality.