Lie algebras and cotriangular spaces

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Abstract

Let p (P,L) be a partial linear space in which any line contains three points and let K be a field. Then by LK(p) we denote the free K-algebra generated by the elements of P and subject to the relations xy = 0 if x and y are noncollinear elements from P and xy = z for any triple {x, y, z} ¿ L. We prove that the algebra LK(p) is a Lie algebra if and only if K is of even characteristic and p is a cotriangular space satisfying Pasch’s axiom. Moreover, if p is a cotriangular space satisfying Pasch’s axiom, then a connection between derivations of the Lie algebra LK(p) and geometric hyperplanes of p is used to determine the structure of the algebra of derivations of LK(p).
Original languageEnglish
Pages (from-to)209-221
JournalBulletin of the Belgian Mathematical Society : Simon Stevin
Volume12
Issue number2
Publication statusPublished - 2005

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