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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Item Status: | Live Archive |

## 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 |
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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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