The tensor hierarchy of maximal supergravity in D dimensions is known to be closely related to a Borcherds (super) algebra that is constructed from the global symmetry group E11-D. We here explain how the Borcherds algebras in different dimensions are embedded into each other and can be constructed from a unifying Borcherds algebra. The construction also has a natural physical explanation in terms of oxidation. We then go on to show that the Hodge duality that is present in the tensor hierarchy has an algebraic counterpart. For D > 8 the Borcherds algebras we find differ from the ones existing in the literature although they generate the same tensor hierarchy.
Until the end of September 2012, the work by JP was performed at Universite Libre de Bruxelles, and supported by IISN - Belgium (conventions 4.4511.06 and 4.4514.08) by the Belgian Federal Science Policy Office through the Interuniversity Attraction Pole P6/11.