Presenting the Graded Lie Algebra Associated to the Nottingham Group

Abstract

The graded Lie algebraLassociated to the Nottingham group is a loop algebra of the Witt algebraW1. The universal coveringW1ofW1has one-dimensional centre, so that the corresponding loop algebraMofW1has an infinite-dimensional centreZ(M). AsM/Z(M) is isomorphic toL, it follows from a result of B. H. Neumann thatLis not finitely presented. However, we are able to show thatMitself is finitely presented.We work more generally with the Zassenhaus algebrasWn. In the group context, examples of finitely presented groups whose centre is not finitely generated were given by V. N. Remeslennikov and H. Abels.