Vol. 37, No. 3, 1971

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Lie algebras of genus one and genus two

James W. Bond

Vol. 37 (1971), No. 3, 591–616

The genus of a finite dimensional algebra is the difference between its dimension and the number of elements in a minimal generating set for the algebra. In this paper the classification of finite dimensional Lie algebras of genus one and genus two is accomplished in four steps. First, it is shown that every such algebra is either solvable or contains a simple subalgebra. Second, the algebras containing a simple subalgebra are determined. Third, nonminimal genus one and two Lie algebras are shown to have a one, two dimensional ideal, J, respectively, with genus zero quotient. Fourth, the different possible L module structures for J are analyzed and completely determined except for the genus two index two nilpotent algebras.

Mathematical Subject Classification 2000
Primary: 17B05
Received: 23 January 1967
Revised: 19 October 1970
Published: 1 June 1971
James W. Bond