Journal of Lie Theory 19 (2009), No. 1, 185--207
Copyright Heldermann Verlag 2009
Diamonds of Finite Type in Thin Lie Algebras
Marina Avitabile
Dip. di Matematica ed Applicazioni, Università di Milano-Bicocca, Via Cozzi 53, 20125 Milano, Italy
marina.avitabile@unimib.it
Sandro Mattarei
Dip. di Matematica, Università degli Studi di Trento, Via Sommarive 14, 38050 Povo, Italy
mattarei@science.unitn.it
Borrowing some terminology from pro-p groups, thin Lie algebras are N-graded Lie algebras, generated in degree one, of width two and obliquity zero. In particular, their homogeneous components have degree one or two, and they are termed diamonds in the latter case. In one of the two main subclasses of thin Lie algebras the earliest diamond after that in degree one occurs in degree 2q-1, where q is a power of the characteristic. This paper is a contribution to an ongoing classification project of this subclass of thin Lie algebras. Specifically, we prove that the degree of the earliest diamond of finite type in such a Lie algebra can only attain certain values, which occur in explicit examples constructed elsewhere.
Keywords: Modular Lie algebras, graded Lie algebras, central extensions, finitely presented Lie algebras, loop algebras.
MSC: 17B50; 17B70, 17B65, 17B56
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