  We construct generalised diffeomorphisms for E$_9$ exceptional field theory.
The transformations, which like in the E$_8$ case contain constrained local
transformations, close when acting on fields. This is the first example of a
generalised diffeomorphism algebra based on an infinite-dimensional Lie algebra
and an infinite-dimensional coordinate module. As a byproduct, we give a simple
generic expression for the invariant tensors used in any extended geometry. We
perform a generalised Scherk--Schwarz reduction and verify that our
transformations reproduce the structure of gauged supergravity in two
dimensions. The results are valid also for other affine algebras.

High Energy Physics - Theory