Simple, locally finite dimensional Lie algebras in positive\n characteristic

2013/11/20 by Johanna Hennig, Hennig, Johanna
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.1311.5299

openalex publication_date 2013/11/20 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28

Abstract

We prove two structure theorems for simple, locally finite dimensional Lie\nalgebras over an algebraically closed field of characteristic p which give\nsufficient conditions for the algebras to be of the form [R^(-), R^(-)] /\n(Z(R) \∩ [R^(-), R^(-)]) or [K(R, *), K(R, *)] for a simple, locally\nfinite dimensional associative algebra R with involution *. The first\nproves that a condition we introduce, known as locally nondegenerate, along\nwith the existence of an ad-nilpotent element suffice. The second proves that a\nuniformly ad-integrable Lie algebra is of this type if the characteristic of\nthe ground field is sufficiently large. Lastly we construct a simple, locally\nfinite dimensional associative algebra R with involution * such that K(R,\n*) \≠ [K(R, *), K(R, *)] to demonstrate the necessity of considering the\ncommutator in the first two theorems.\n

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