Comparison of two metaplectic cocycles
In my thesis I shall be investigating two distinct metaplectic extensions of the general linear group. The first of these was discovered by Matsumoto, it's existence intimately connected with the deep properties of the r-th order Hilbert symbol. His construction relies heavily on class field theory and algebraic K-theory. Having constructed his metaplectic group, which is known to be universal, Matsumoto was then able to define the cocycle representing this extension. The second of these metaplectic extensions was found recently by Dr Hill at University College London. In contrast, his construction is very elementary. He was able to prove the existence of a continuous cocycle resulting in the construction of a new non-trivial metaplectic extension. It has already been shown, by Hill, that these two metaplectic extensions are isomorphic if we restrict to the special linear group. However, little is known of this isomorphism. Throughout this thesis we shall investigate these two cocycles, finding explicit formulae in both cases. We shall then show that the isomorphism between the group extensions of Matsumoto and Hill may be defined via the discovery of the coboundary which splits the quotient of the corresponding cocycles. Having found this coboundary we shall then be able to prove that, in specific cases, the two extensions are in fact isomorphic over the full general linear group.