Simple Lie Algebras of Type A

Abstract

In a recent paper3 the author discussed the Lie algebras of characteristic 0 obtainable as the set of skew elements of an involutorial normal simple associative algebra. The present paper gives an extension of these results to simple associative algebras of second kind as defined by Albert.4 The resulting Lie algebras in this case constitute Landherr's class Al, .5 We reduce the problem of classifying these Lie algebras, as in our earlier work on Lie algebras of types B, C, D,6 to two standard problems in associative algebra, namely, classification of involutorial simple algebras and of generalized hermitian matrices relative to cogredience. In this sense a complete determination of normal simple Lie algebras of characteristic 0 except those of a finite number of orders7 results from Landherr's work on the algebras of type A, and our own on types Al1, B, C, D.