Some remarks on graded nilpotent Lie algebras and 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 $\dim H^{\ast }(\mathfrak{n})\ge L(p)$ 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 $\dim H^{\ast }(\mathfrak{n})\ge 2^{\dim (\mathfrak{z})}$ 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 $\mathfrak{n}={⨁}_{\alpha \in {\mathbb{Z}}_{+}^d}{\mathfrak{n}}_{\alpha }$, such that its associated polynomial $p$ satisfies $L(p)>2^{\dim \mathfrak{z}}$, does it admit a grading $\mathfrak{n}={\mathfrak{n}}_1^{\prime }\oplus {\mathfrak{n}}_2^{\prime }\oplus \cdots \oplus {\mathfrak{n}}_k^{\prime }$ such that its associated polynomial $p^{\prime }$ satisfies $L(p^{\prime })>2^{\dim \mathfrak{z}}$? (B) If $\mathfrak{n}$ is $r$-step nilpotent admitting a grading $\mathfrak{n}={\mathfrak{n}}_1\oplus {\mathfrak{n}}_2\oplus \cdots \oplus {\mathfrak{n}}_k$ such that its associated polynomial $p$ satisfies $L(p)>2^{\dim \mathfrak{z}}$, does it admit a grading $\mathfrak{n}={\mathfrak{n}}_1^{\prime }\oplus {\mathfrak{n}}_2^{\prime }\oplus \cdots \oplus {\mathfrak{n}}_r^{\prime }$ such that its associated polynomial $p^{\prime }$ satisfies $L(p^{\prime })>2^{\dim \mathfrak{z}}$? In this paper we show that the answer to (A) is yes, but the answer to (B) is no.

arXiv commentRevised version, major modifications