Classifying associative quadratic algebras of characteristic not two as Lie algebras. (English) Zbl 1233.17006

Hähl, Hermann; Weller, Michael

J. Lie Theory 19, No. 3, 543-555 (2009); biographical note 20, No. 1, 213 (2010).

Authors’ summary: We present an alternative to existing classifications [L. Bröcker, Kinematische Räume, Geom. Dedicata 1, 241–268 (1973; Zbl 0249.50014), H. Karzel, Kinematic spaces, Algebra Commut., Geometria, Convegni 1971/1972, Symp. Math. 11, 413–439 (1973; Zbl 0284.50017)], of those quadratic algebras (in the sense of Osborn) which are associative. The alternative consists in studying them as Lie algebras. This generalizes [J. F. Plebánski and M. Przanowski, J. Math. Phys. 29, 529–535 (1988; Zbl 0652.17016)], 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.
In an additional biographical note the authors correct the record of this paper: The results are contained in the previously published article by A. Elduque [Quadratic alternative algebras, J. Math. Phys 31, No. 1, 1–5 (1990; Zbl 0716.17002)].

MSC:

17A45	Quadratic algebras (but not quadratic Jordan algebras)
16S99	Associative rings and algebras arising under various constructions
17B60	Lie (super)algebras associated with other structures (associative, Jordan, etc.)

Keywords:

associative quadratic algebra; Lie algebra; nilpotent Lie algebra; solvable Lie algebra; quaternion skew field; classification