Presenting the Graded Lie Algebra Associated to the Nottingham Group

Abstract

The graded Lie algebra L associated to the Nottingham group is a loop algebrâ of the Witt algebra W . The universal covering W of W has one-dimensional 1 1 1 centre, so that the corresponding loop algebra M of W has an infinite-dimen-H. Neumann that L is not finitely presented. However, we are able to show that M itself is finitely presented.We work more generally with the Zassenhaus algebras W . In the group context, n examples of finitely presented groups whose centre is not finitely generated were given

Some thin Lie algebras related to Albert-Frank algebras and algebras of maximal class

“…If there are other diamonds, then we refer to the finite sequence of the homogeneous components of dimension 1 between two consecutive diamonds as a chain. [5] Thin Lie algebras 161…”

“…In [5] we studied the graded Lie algebra associated to the Nottingham group with respect to its lower central series, and more generally Lie algebras in which the second diamond occurs in class which is a power of p, the characteristic of the underlying field.…”

Gradings of non-graded Hamiltonian Lie algebras

“…We recall the definitions of these algebras in Sections 2 and 3, and point out in Remark 3.1 other notations in use for them. In particular, infinite-dimensional thin Lie algebras with second diamond in degree q have been constructed as loop algebras of Zassenhaus algebras (which have dimension [5] Gradings of non-graded Hamiltonian Lie algebras 403 a power of p, see [16,17,18]), and Hamiltonian algebras of the types H(2 : n;co 0 ) = H(2 : n) (the graded simple Hamiltonian algebras, of dimension two less than a power of p, see [3]) and H (2 : n; u>\) (which are Albert-Zassenhaus algebras and have dimension a power of p, see [6]). A preliminary version of the present paper, which predated and inspired some of the other papers cited here, had as its main goal the construction of some infinitedimensional thin Lie algebras with second diamond in degree 2q -1 as loop algebras of Hamiltonian algebras //(2 : n; co 2 ), which have dimension one less than a power of p. (In fact, a construction for those thin Lie algebras had already been given in [20], but as loop algebras of certain finite-dimensional Lie algebras defined 'ad hoc'.)…”

Nottingham Lie Algebras With Diamonds of Finite Type

“…The graded Lie algebra associated to the Nottingham group over the field with p elements, with respect to its lower central series, has been studied in [8]. This is a thin Lie algebra, with second diamond in weight p, and can be described as (the positive part of) a twisted loop algebra of the smallest Zassenhaus algebra W (1 : 1).…”

“…(We use a standard notation for simple Lie algebras of Cartan type, as used, for example, in [24] or [5].) More generally, in [8] certain twisted loop algebras of the Zassenhaus algebras W (1 : n) over the field with p elements have been studied. These are thin Lie algebras, with diamonds occurring exactly in the weights congruent to 1 modulo q − 1, where q = p n .…”