Graded Nilpotent Algebras and Eggert's Conjecture

Abstract

Let N be a commutative nilpotent algebra over a field of characteristic p > 0 and N[p] be the subalgebra generated by all pth powers of elements of N. Without further conditions on N, Eggert's Eggert 1971 conjecture, that dim N ≥ p·dim N[p], has been proved only when dim N[p] ≤ 4. Now impose the condition that N is graded. Let N[p] have n generators and index of nilpotence s. A previous article, McLean (2004), showed that, when s = 2, the conjecture is true for all values of dim N[p]. Here I extend this to each of the following cases: when s = 3, when n < 3p and 3 ≤ s − 1 ≤ p, and when p ≥ 3 and N[p] is any free nilpotent algebra. I also proved that dim N > (p − 1) dim N[p] and use this to answer the commutative graded cases of two problems recently posed by Amberg and Kazarin.