Some remarks on graded nilpotent Lie algebras and the Toral Rank Conjecture

Abstract

If is a Zd+-graded nilpotent finite-dimensional Lie algebra over a field of characteristic zero, a well-known result of Deninger and Singhof states that dimH∗() ≥ L(p) where p is the polynomial associated to the grading and L(p) is the sum of the absolute values of the coefficients of p. From this result they derived the Toral Rank Conjecture (TRC) for 2-step nilpotent Lie algebras. An algebraic version of the TRC states that dimH∗() ≥ 2dim() for any finite-dimensional nilpotent Lie algebra with center. The TRC is more than 25 years old and remains open even for Zd+-graded 3-step nilpotent Lie algebras. Investigating to what extent the bound given by Deninger and Singhof could help to prove the TRC in this case, we considered the following two questions regarding a nilpotent Lie algebra with center : (A) If admits a Z+d-grading n = Z+d nα, such that its associated polynomial p satisfies L(p) > 2dim, does admit +-grading n = n1 ⊕ n2 ⊕ nk such that its associated polynomial p′ satisfies L(p′) > 2dim (B) If is r-step nilpotent admitting a grading n = n1• n2 Š• ⋯ nk such that its associated polynomial p satisfies L(p) > 2dim, does admit a grading n= n1 ⊕ n2 ⊕ ⊕ nr such that its associated polynomial p′ satisfies L(p′) > 2dim? In this paper we show that the answer to (A) is yes, but the answer to (B) is no.