Marina Avitabile

Some loop algebras of Hamiltonian Lie algebras

We consider a class of thin Lie algebras with second diamond in weight a power of the characteristic of the underlying field. We identify these Lie algebras with loop algebras of a graded Hamiltonian algebra or loop algebras of an extension of the Hamiltonian algebra by an outer derivation. We also prove that the Lie algebras considered are not finitely presented.

Thin loop algebras of Albert–Zassenhaus algebras

Thin Lie algebras are Lie algebras over a field, graded over the positive integers and satisfying a certain narrowness condition. In particular, all homogeneous components have dimension one or two, and are called diamonds in the latter case. The first diamond is the component of degree one, and the second diamond can only occur in degrees 3, 5, q or 2q−1, where q is a power of the characteristic of the underlying field. Here we consider several classes of thin Lie algebras with second diamond in degree q. In particular, we identify the Lie algebras in one of these classes with suitable loop algebras of certain Albert–Zassenhaus Lie algebras. We also apply a deformation technique to recover other thin Lie algebras previously produced as loop algebras of certain graded Hamiltonian Lie algebras.

Laguerre polynomials of derivations

We introduce a grading switching for arbitrary non-associative algebras of prime characteristic p, aimed at producing a new grading of an algebra from a given one. We take inspiration from a fundamental tool in the classification theory of modular Lie algebras known as toral switching, which relies on a delicate adaptation of the exponential of a derivation. Our grading switching is achieved by evaluating certain generalized Laguerre polynomials of degree p − 1, which play the role of generalized exponentials, on a derivation of the algebra. A crucial part of our argument is establishing a congruence for them which is an appropriate analogue of the functional equation ex · ey = ex+y for the classical exponential. Besides having a wider scope, our treatment provides a more transparent explanation of some aspects of the original toral switching, which can be recovered as a special case.

THE STRUCTURE OF THIN LIE ALGEBRAS WITH CHARACTERISTIC TWO

Thin Lie algebras are graded Lie algebras

THE OTHER GRADED LIE ALGEBRA ASSOCIATED TO THE NOTTINGHAM GROUP

L = \oplus_{i = 1}^{\infty}L_{i}

Thin Lie algebras with diamonds of finite and infinite type

with dim Li ≤ 2 for all i, and satisfying a more stringent but natural narrowness condition modeled on an analogous cond...

Diamonds in Thin Lie Algebras

ABSTRACT The graded Lie algebra L associated to the Nottingham group with respect to its natural filtration is known to be a loop algebra of the first Witt algebra W 1 . The fact that the Schur multiplier of W 1 , in characteristic p > 3, is one-dimensional implies that L is not finitely presented. Consider the universal covering Ŵ 1 of W 1 and the corresponding loop algebra M of Ŵ 1 . In this paper we prove that M itself is finitely presented for p > 3. In characteristic p > 11 the algebra M turns out to be presented by two relations.