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

TL;DR

This paper investigates the relationship between gradings of nilpotent Lie algebras and the Toral Rank Conjecture, showing that certain bounds on cohomology dimensions can be achieved through specific gradings, but not all.

Contribution

It proves that if a nilpotent Lie algebra admits a grading with a polynomial sum exceeding a bound, it can be refined to another grading with the same property, but this does not hold for step-wise gradings.

Findings

Answer to (A) is yes, such a grading exists.

Answer to (B) is no, such a grading does not necessarily exist.

Provides insights into grading structures related to the Toral Rank Conjecture.

Abstract

If $\mathfrak{n}$ is a $\mathbb{Z}^d_+$-graded nilpotent finite dimensional Lie algebra over a field of characteristic zero, it is well known that $\dim H^{\ast }(\mathfrak{n})\geq 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 Deninger and Singhof derived the Toral Rank Conjecture (TRC) for 2-step nilpotent Lie algebras. An algebraic version of the TRC states that $\dim H^{\ast }(\mathfrak{n})\geq 2^{\dim (\mathfrak{z)}} $ for any finite dimensional Lie algebra $\mathfrak{n}$ with center $\mathfrak{z}$. The TRC is more that 25 years old and remains open even for $\mathbb{Z}^d_+$-graded 3-step nilpotent Lie algebras. Investigating to what extent the above bound for $\dim H^{\ast }(\mathfrak{n})$ could help to prove the TRC in this case, we considered the following two questions regarding a nilpotent Lie algebra $\mathfrak{n}$ with center $\mathfrak{z}$: (A) If $\mathfrak{n}$ admits a $\mathbb{Z}_+^d$-grading $\mathfrak{n}=\bigoplus_{\alpha\in\mathbb{Z}_+^d} \mathfrak{n}_\alpha$, such that its associated polynomial $p$ satisfies $L(p)>2^{\dim\mathfrak{z}}$, does it admit a grading $\mathfrak{n}=\mathfrak{n}'_{1}\oplus \mathfrak{n}'_{2}\oplus \dots\oplus \mathfrak{n}'_{k}$ such that its associated polynomial $p'$ satisfies $L(p')>2^{\dim\mathfrak{z}}$? (B) If $\mathfrak{n}$ is $r$-step nilpotent admitting a grading $\mathfrak{n}=\mathfrak{n}_{1}\oplus \mathfrak{n}_{2}\oplus \dots\oplus \mathfrak{n}_{k}$ such that its associated polynomial $p$ satisfies $L(p)>2^{\dim\mathfrak{z}}$, does it admit a grading $\mathfrak{n}=\mathfrak{n}'_{1}\oplus \mathfrak{n}'_{2}\oplus \dots\oplus \mathfrak{n}'_{r}$ such that its associated polynomial $p'$ satisfies $L(p')>2^{\dim\mathfrak{z}}$? In this paper we show that the answer to (A) is yes, but the answer to (B) is no.