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Journal of Lie Theory 19 (2009), No. 3, 543--555
Copyright Heldermann Verlag 2009

Classifying Associative Quadratic Algebras of Characteristic not Two as Lie Algebras

Hermann Hähl
Institut für Geometrie und Topologie, Universität Stuttgart, 70550 Stuttgart, Germany

Michael Weller
Liliencronstr. 2, 70619 Stuttgart, Germany

We present an alternative to existing classifications [see L. Bröcker, Kinematische Räume, Geom. Dedicata 1 (1973) 241--268; H. Karzel, Kinematic spaces, Symposia Mathematica 11 (1973) 413--439] of those quadratic algebras (in the sense of Osborn) which are associative. The alternative consists in studying them as Lie algebras. This generalizes work of J. F. Plebanski and M. Przanowski [Generalizations of the quaternion algebra and Lie algebras, J. Math. Phys. 29 (1988) 529--535], where only algebras over the real and the complex numbers are considered, to algebras over arbitrary fields of characteristic not two; at the same time, considerable simplifications are obtained. The method is not suitable, however, for characteristic two.

Keywords: Associative quadratic algebra, Lie algebra, nilpotent Lie algebra, solvable Lie algebra, quaternion skew field, classification.

MSC: 6U99, 17B20, 17B30, 17B60

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