Exceptional algebroids and type IIA superstrings

We study exceptional algebroids in the context of warped compactifications of type IIA string theory down to $n$ dimensions, with $n\le 6$. In contrast to the M-theory and type IIB case, the relevant algebroids are no longer exact, and their locali moduli space is no longer trivial, but has $5$ distinct points. This relates to two possible scalar deformations of the IIA theory. The proof of the local classification shows that, in addition to these scalar deformations, one can twist the bracket using a pair of $1$-forms, a $2$-form, a $3$-form, and a $4$-form. Furthermore, we use the analysis to translate the classification of Leibniz parallelisable spaces (corresponding to maximally supersymmetric consistent truncations) into a tractable algebraic problem. We finish with a discussion of the Poisson-Lie U-duality and examples given by tori and spheres in $2$, $3$, and $4$ dimensions.