Uncountably Many Nonisomorphic Nilpotent Lie Algebras

Chong-Yun Chao
1962 Proceedings of the American Mathematical Society
Throughout this note, L denotes a Lie algebra over the real number field R. We shall define L* and L, inductively. L = L° = L0, Li = [L*-1,1/-1], and L,= [L, Lv_i] for all integers *2ï 1. Thus, L* is the space of all finite sumsEI*. j\> x> yCL'~l. Similarly, Lt is the space of all finite sumsEl*. y]> xEL and y£L,_i. If Lr = 0 and Lr~19¿0, L is said to be solvable of index r. If L¡ = 0 and Lt-i^O, L is said to be nilpotent of length t. Definition. Let F be a subfield of R. A Lie algebra L over R
more » ... is said to be an F-algebra if its structure constants with respect to some basis of L lie in F. Malcev [l ] showed that for each integer w^ 16 there is a nilpotent Lie algebra of length 2 and dimension n which is not a rational algebra. The purpose of this note is to prove the following theorem which contains an improvement of Malcev's result: Theorem. There exist uncountably many nonisomorphic nilpotent Lie algebras of length 2 for any given dimension A7^ 10. Following from the theorem we can easily show: Corollary 1. There exist uncountably many solvable not nilpotent Lie algebras of index 3 for any given dimension M 2; 11. Received by the editors November 13, 1961.