The structure of thin Lie algebras with characteristic two

Avitabile, Marina and Jurman, Giuseppe and Mattarei, Sandro (2010) The structure of thin Lie algebras with characteristic two. International Journal of Algebra and Computation, 20 (06). pp. 731-768. ISSN 0218-1967

Full content URL: http://dx.doi.org/10.1142/S0218196710005820

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Item Type:Article
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Abstract

Thin Lie algebras are graded Lie algebras with dim L_i ≤ 2 for all i, and satisfying a more stringent but natural narrowness condition modeled on an analogous condition for pro-p-groups. The two-dimensional homogeneous components of L, which include L_1, are named diamonds. Infinite-dimensional thin Lie algebras with various diamond patterns have been produced, over fields of positive characteristic, as loop algebras of suitable finite-dimensional simple Lie algebras, of classical or of Cartan type depending on the location of the second diamond. The goal of this paper is a description of the initial structure of a thin Lie algebra, up to the second diamond.

Specifically, if L_k is the second diamond of L, then the quotient L/L^k is a graded Lie algebras of maximal class. In odd characteristic p, the quotient L/L^k is known to be metabelian, and hence uniquely determined up to isomorphism by its dimension k, which ranges in an explicitly known set of possible values: 3, 5, a power of p, or one less than twice a power of p. However, the quotient L/L^k need not be metabelian in characteristic two. We describe here all the possibilities for L/L^k up to isomorphism. In particular, we prove that k + 1 equals a power of two.

Keywords:Modular Lie algebra, graded Lie algebra, graded Lie algebra of maximal class, thin Lie algebra
Subjects:G Mathematical and Computer Sciences > G110 Pure Mathematics
Divisions:College of Science > School of Mathematics and Physics
ID Code:18503
Deposited On:11 Dec 2015 09:17

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