Nilpotent Lie algebras in which all proper subalgebras have class at most n

Abstract

Let L be a finitely generated nilpotent Lie algebra over a field K and let d be the smallest integer such that L can be generated by d elements. Let n≥d be a positive integer and suppose that every proper subalgebra of L has class at most n. It is not difficult to show that the class of L is at most n+q where q=⌊n/(d−1)⌋. Our main result shows that there exist such Lie algebras of class (exactly) n+q whenever q≥3 and K has characteristic 0 or prime characteristic p such that p does not divide (q2−1)q/2.