  We study the non-linear realisation of E11 originally proposed by West with
particular emphasis on the issue of linearised gauge invariance. Our analysis
shows even at low levels that the conjectured equations can only be invariant
under local gauge transformations if a certain section condition that has
appeared in a different context in the E11 literature is satisfied. This
section condition also generalises the one known from exceptional field theory.
Even with the section condition, the E11 duality equation for gravity is known
to miss the trace component of the spin connection. We propose an extended
scheme based on an infinite-dimensional Lie superalgebra, called the tensor
hierarchy algebra, that incorporates the section condition and resolves the
above issue. The tensor hierarchy algebra defines a generalised differential
complex, which provides a systematic description of gauge invariance and
Bianchi identities. It furthermore provides an E11 representation for the field
strengths, for which we define a twisted first order self-duality equation
underlying the dynamics.
