About the embedding of Moufang loops in alternative algebras II

2008/04/13 by N. I. Sandu, Sandu, N. I.
Mathematics · Social Sciences · #Advanced Topics in Algebra #Mathematics Education and Teaching Techniques #Mathematics and Applications #math.RA #msc:17D05 #msc:20N05

paper · pdf · doi:10.48550/arxiv.0804.2049

15 pages

arxiv created 2008/04/13 · arxiv updated 2009/12/01

Abstract

It is known that with precision till isomorphism that only and only loops M(F) = M₀(F)/<-1>, where M₀(F) denotes the loop, consisting from elements of all matrix Cayley-Dickson algebra C(F) with norm 1, and F be a subfield of arbitrary fixed algebraically closed field, are simple non-associative Moufang loops. In this paper it is proved that the simple loops M(F) they and only they are not embedded into a loops of invertible elements of any unitaly alternative algebras if char F ≠ 2 and F is closed under square root operation. For the remaining Moufang loops such an embedding is possible. Using this embedding it is quite simple to prove the well-known finding: the finite Moufang p-loop is centrally nilpotent.

Related

• The classification on simple Moufang loops
• About the embedding of Moufang loops in alternative algebras
• Radicals and embeddings of Moufang loops in alternative loop algebras
• Simple Moufang loops and alternative algebras
• Simple Moufang loops and Galois extensions