Construction of nilpotent Lie algebras over arbitrary fields

Abstract

In this paper we present a general description of a computationally efficient algorithm for constructing every n-dimensional nilpotent Lie algebra as a central extension of a nilpotent Lie algebra of dimension less than n. As an application of the algorithm, we present a complete list of all real nilpotent six-dimensional Lie algebras. Since 1958, four such lists have been developed: namely, those of Morozov [2], Shedler [3], Vergne [5] and Skjelbred and Sund [4]. No two of these lists agree exactly. Our list resolves all the discrepancies in the other four lists. Moreover, it contains each earlier list as a subset.