I will explain two different ways of extending the U-duality algebra
of maximal supergravity in D dimensions to an infinite-dimensional Lie
superalgebra with a corresponding level decomposition, such that the
dynamical p-form fields transform in the representations at level p.
One way, which leads to a Borcherds algebra, is to add an odd null
root to the set of simple roots of the U-duality algebra. The other
way is to consider the embedding tensor, which describes gauge
deformations of the theory, as the components of an element at level
-1. This leads to a new Lie superalgebra with a non-triangular
structure, which gives exactly the tensor hierarchy of representations
arising in the embedding tensor formalism. I will prove that these
representations are always contained in those coming from the
Borcherds algebra, and explain why some of the latter are not
included in the tensor hierarchy. Furthermore, I will show that the
different Borcherds algebras for different D can be unified into a
single one. The talk is based on 1301.1346 (with A. Kleinschmidt) and
work in progress.
