THE OTHER GRADED LIE ALGEBRA ASSOCIATED TO THE NOTTINGHAM GROUP

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.

Key Words:

• Associated Lie structures
• Graded Lie algebras
• Infinite-dimensional Lie algebras
• Loop algebras
• Modular Lie algebras
• Nottingham group
• Pro-p-groups
• Witt algebra

2000 Mathematics Subject Classification:

• 20F40
• 17B50
• 17B70
• 17B65

ACKNOWLEDGMENT

The author is grateful to A. Caranti for suggesting the problem and to A. Caranti, N. Gavioli, S. Mattarei, and C. M. Scoppola for several useful discussions.

Notes

Communicated by E. Zelmanov.

‡The author is a member of INdAm-GNSAGA; Italy.