We analyse the very-extended Kac-Moody algebras as representations in terms of certain A d subalgebras and find the generators at low levels. Our results for low levels agree precisely with the bosonic field content of the theories containing gravity, forms and scalars which upon reduction to three dimensions can be described by a non-linear realisation. We explain how the Dynkin diagrams of the very-extended algebras encode information about the field content and generalised T-duality transformations.

We formulate the bosonic sector of IIB supergravity as a non-linear realisation. We show that this non-linear realisation contains the Borel subalgebras of SL(11) and E 7 and argue that it can be enlarged so as to be based on the rank eleven Kac-Moody algebra E 11

A description of the bosonic sector of massive IIA supergravity as a non-linear realisation is given. An essential feature of this construction is that the momentum generators have non-trivial commutation relations with the generators associated with the gauge fields.Igor Schnakenburg is financially supported by DAAD (D/00/09914).

A description of the bosonic sector of ten-dimensional N = 1 supergravity as a non-linear realisation is given. We show that if a suitable extension of this theory were invariant under a Kac-Moody algebra, then this algebra would have to contain a rank eleven Kac-Moody algebra, that can be identified to be a particular real form of very-extended D 8 . We also describe the extension of N = 1 supergravity coupled to an abelian vector gauge field as a non-linear realisation, and find the Kac-Moody algebra governing the symmetries of this theory to be very-extended B 8 . Finally, we discuss the related points for the N = 1 supergravity coupled to an arbitrary number of abelian vector gauge fields.

This is a review article of eleven dimensional supergravity in which we present all necessary calculations, namely the Noether procedure, the equations of motion (without neglecting the fermions), the Killing spinor equation, as well as some simple and less simple supersymmetric solutions to this theory. All calculations are printed in much detail and with explicit comments as to how they were done. Also contained is a simple approach to Clifford algebras to prepare the grounds for the harder calculations in spin space and Fierz identities.