E11, Borcherds algebras and maximal supergravity

The dynamical p-forms of torus reductions of maximal supergravity theory have been shown some time ago to possess remarkable algebraic structures. The set ("dynamical spectrum") of propagating p-forms has been described as a (truncation of a) real Borcherds superalgebra V-D that is characterized concisely by a Cartan matrix which has been constructed explicitly for each spacetime dimension 11 >= D >= 3. In the equations of motion, each differential form of degree p is the coefficient of a (super-) group generator, which is itself of degree p for a specific gradation (the V-gradation). A slightly milder truncation of the Borcherds superalgebra enables one to predict also the "spectrum" of the non-dynamical (D-1) and D-forms. The maximal supergravity p-form spectra were reanalyzed more recently by truncation of the field spectrum of E-11 to the p-forms that are relevant after reduction from 11 to D dimensions. We show in this paper how the Borcherds description can be systematically derived from the split ("maximally non compact") real form of E-11 for D >= 1. This explains not only why both structures lead to the same propagating p-forms and their duals for p <= (D - 2), but also why one obtains the same (D-1)-forms and "top" D-forms. The Borcherds symmetries V-2 and V-1 are new too. We also introduce and use the concept of a presentation of a Lie algebra that is covariant under a given subalgebra.