| Summary: | It has been proposed that the low energy e↵ective action of the theory of strings and branes possesses a large symmetry described by the Kac-Moody algebra E11. The non-linear realisation of this algebra and its vector representation determines the fields and coordinates of the theory, as well as the equations that describe their dynamics. In order to construct the generators of E11 algebra it is split into representations of its GL(d) ⇥ E11−d subalgebra. Here d is an integer that determines the dimension of the corresponding E11 theory. The low levels of the non-linear realisation contain the set of equations of the supergravity theory in corresponding space-time dimension, while the higher levels introduce an infinite number of fields that are connected to the supergravity ones via a chain of duality relations, as well as standalone fields that have no counterparts in standard supergravity theory. In this thesis we derive the set of commutators of E11 algebra and its vector representation up to a certain level in five and ten-dimensional cases. We use the non-linear realisation approach to construct the generalised vielbein and the Cartan forms of the E11 theory in four, five, ten and eleven dimensions. We then build a set of E11 invariant equations in five and eleven-dimensional theories from the non-linear realisation of E11. The low level equations, when appropriately truncated, are shown to perfectly reproduce the dynamics of the standard supergravity theories in corresponding dimensions. The dynamics of certain higher level fields are considered, including the dual graviton field and an eleven-dimensional field that, when reduced to ten dimensions, gives rise to the Romans mass term in type IIA theory. Lastly, we describe the non-linear realisation of very extended A1 algebra, called A+++ 1 , together with its commutators, Cartan forms and generalised vielbein.
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