A bstract Over-extended Kac-Moody algebras contain so-called gradient structures — a gl (d) gl d -covariant level decomposition of the algebra contains strings of modules at different levels that can be interpreted as spatial gradients. We present an algebraic origin for this phenomenon, based on the recently introduced Lie algebra extension of an over-extended Kac-Moody algebra by its fundamental module 1, appearing in tensor hierarchy algebra super-extensions of over-extended Kac-Moody algebras. The extensions are described in terms of Lie algebra cohomology, vanishing for finite-dimensional simple Lie algebras, but non-vanishing in relevant infinite-dimensional cases. The extension is described in a few different gradings, where it is given a covariant description with respect to different subalgebras. We expect the results to be important for the connection between extended geometry and cosmological billiards.
Cederwall et al. (Mon,) studied this question.