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
haehl@mathematik.uni-stuttgart.de
Michael Weller
Liliencronstr. 2, 70619 Stuttgart, Germany
micha-weller@t-online.de
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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