  We introduce an infinite-dimensional Lie superalgebra which is an extension
of the U-duality Lie algebra of maximal supergravity in D dimensions, for D
between 3 and 7. The level decomposition with respect to the U-duality Lie
algebra gives exactly the tensor hierarchy of representations that arises in
gauge deformations of the theory described by an embedding tensor, for all
positive levels p. We prove that these representations are always contained in
those coming from the associated Borcherds-Kac-Moody superalgebra, and we
explain why some of the latter representations are not included in the tensor
hierarchy. The most remarkable feature of our Lie superalgebra is that it does
not admit a triangular decomposition like a (Borcherds-)Kac-Moody
(super)algebra. Instead the Hodge duality relations between level p and D-2-p
extend to negative p, relating the representations at the first two negative
levels to the supersymmetry and closure constraints of the embedding tensor.
