The purpose of the present text is to complete the вЂњtriangleвЂќ and provide a detailed derivation of the analogous results in the type IIAВ case. ДЄlthough quite distinct as far as the technical details are concerned, the M-theory and type IIB cases nonetheless have one thing in common - they both correspond to a certain вЂњminimalвЂќ class of exceptional algebroids, called exact. In both cases, a local classification result was proved, a method for constructing Leibniz parallelisable spaces was provided, and a general notion of the PoissonвЂ“Lie U-duality was studied, extending the exceptional Drinfeld algebra construction of.

The more detailed study of exceptional algebroids was the main focus of the works - the former focused on the M-theory setup, while the latter was devoted to the type IIB case. However, for simplicity (and in order to match the existing literature) we will simply stick to the terms M-theory/string theoryВ instead. 1 1 More precisely, these algebroids are used in the study of the corresponding low-energy effective theories (supergravities).

Taking G corresponding to non-compact exceptional Lie groups (instead of G-algebroids we then talk simply about exceptional algebroids or elgebroids), one reproduces in particular the Leibniz algebroids from, used in the study of M-theory and type II string theory compactifications. Taking G to be the orthogonal group, one recovers the Courant algebroids, while the general linear group leads to Lie algebroids. Recently, in a new class of geometric objects - so called G- algebroids - was introduced in order to provide a common ground for several types of structures describing the symmetries and dynamics of string and M-theory.

Complete subspace definition