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

G. AMES; L. CAGLIERO,; M. CRUZ

JOURNAL OF ALGEBRA AND ITS APPLICATIONS

WORLD SCIENTIFIC PUBL CO PTE LTD

Lugar: London, UK; Año: 2015 vol. 14

0219-4988

If$mathfrak{n}$ is a $Z^d_+$-graded nilpotent finite dimensional Lie algebra over a field of characteristic zero,a well known result of Deninger and Singhof states 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 they 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 nilpotent Lie algebra $ $ with center $z$.The TRC is more than 25 years old and remains openeven for $Z^d_+$-graded 3-step nilpotent Lie algebras.Investigating to what extent the bound given by Deninger and Singhofcould help to prove the TRC in this case, we considered the following two questionsregarding a nilpotent Lie algebra $ $ with center $z$:(A) If $ $ admits a $Z_+^d$-grading $ =igoplus_{alphainZ_+^d} _alpha$,such that its associated polynomial $p$ satisfies $L(p)>2^{dimz}$,does $ $ admit a $Z_+$-grading$mathfrak{n}=mathfrak{n}´_{1}oplus mathfrak{n}´_{2}oplus dotsoplus mathfrak{n}´_{k}$such that its associated polynomial $p´$ satisfies $L(p´)>2^{dimz}$?(B)  If $ $ is $r$-step nilpotent admitting a grading$mathfrak{n}=mathfrak{n}_{1}oplus mathfrak{n}_{2}oplus dotsoplus mathfrak{n}_{k}$such that its associated polynomial $p$ satisfies $L(p)>2^{dimz}$,does $ $ admit a grading$mathfrak{n}=mathfrak{n}´_{1}oplus mathfrak{n}´_{2}oplus dotsoplus mathfrak{n}´_{r}$such that its associated polynomial $p´$ satisfies $L(p´)>2^{dimz}$?In this paper we show that the answer to (A) is yes, but the answer to (B) is no.