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Bracket ideals and Hilbert polynomial of filiform Lie algebras

F. J. Castro-Jiménez, M. Ceballos

TL;DR

This work studies the bracket bifiltration and the bivariate Hilbert polynomial ${\rm HP}_{\frak g}(t,s)$ of complex filiform Lie algebras, linking the polynomial to invariants $z_1,z_2$ and to the $\theta$-vector that encodes when bracket ideals collapse. It proves a homogeneous behavior of the structure constants under adapted-basis scalings, derives detailed descriptions of algebras with $z_2(\frak g)=n-2$, and shows that ${\rm HP}_{\frak g}$ can distinguish among isomorphism classes not separable by the invariants $z_1,z_2$ or the $\theta$-vector. The paper then computes explicit HPs for several families, including sporadic cases: for $(4,5,8)$ HP distinguishes two classes, for $(5,6,9)$ HP does not distinguish, and for $(5,7,10)$ HP separates six classes across three subfamilies. Overall, ${\rm HP}$ provides finer, constructive invariants for classifying filiform Lie algebras and reveals growth in distinguishable classes as the dimension increases relative to the invariants.

Abstract

For a complex finite-dimensional filiform Lie algebra $\mathfrak g$, we first study the bifiltration given by the bracket ideals $[C^k\mathfrak g,C^\ell\mathfrak g]$ and then the behavior of its associated bivariate Hilbert polynomial. This behavior depends in particular on two numerical invariants that measure, on one hand, certain properties of the centralizers in $\mathfrak g$ of the ideals in the lower central sequence and, on the other hand, the dimension of the largest abelian ideal that appears in the lower central series. We give examples proving that the Hilbert polynomial can distinguish isomorphism classes of filiform Lie algebras that cannot be distinguished by the two aforementioned invariants.

Bracket ideals and Hilbert polynomial of filiform Lie algebras

TL;DR

This work studies the bracket bifiltration and the bivariate Hilbert polynomial of complex filiform Lie algebras, linking the polynomial to invariants and to the -vector that encodes when bracket ideals collapse. It proves a homogeneous behavior of the structure constants under adapted-basis scalings, derives detailed descriptions of algebras with , and shows that can distinguish among isomorphism classes not separable by the invariants or the -vector. The paper then computes explicit HPs for several families, including sporadic cases: for HP distinguishes two classes, for HP does not distinguish, and for HP separates six classes across three subfamilies. Overall, provides finer, constructive invariants for classifying filiform Lie algebras and reveals growth in distinguishable classes as the dimension increases relative to the invariants.

Abstract

For a complex finite-dimensional filiform Lie algebra , we first study the bifiltration given by the bracket ideals and then the behavior of its associated bivariate Hilbert polynomial. This behavior depends in particular on two numerical invariants that measure, on one hand, certain properties of the centralizers in of the ideals in the lower central sequence and, on the other hand, the dimension of the largest abelian ideal that appears in the lower central series. We give examples proving that the Hilbert polynomial can distinguish isomorphism classes of filiform Lie algebras that cannot be distinguished by the two aforementioned invariants.
Paper Structure (13 sections, 18 theorems, 128 equations)

This paper contains 13 sections, 18 theorems, 128 equations.

Table of Contents

  1. Previous results
  2. Main results
  3. Bracket bifiltration and Hilbert polynomial
  4. The $\theta$ vector and the support of ${\rm HP}$
  5. Homogeneity of the structure constants
  6. Description of filiform Lie algebras $\mathfrak g$ with $z_2(\mathfrak g)=n-2$
  7. Filiform Lie algebras associated with $(n-2,n-2,n)$
  8. Filiform Lie algebras associated with $(z_1,n-2,n)$ where $z_1<n-2$
  9. Hilbert polynomial of a filiform Lie algebra with $z_2(\mathfrak g)=n-2$
  10. Some sporadic examples of Hilbert polynomials
  11. ${\rm HP}$ of FLA associated with $(4,5,8)$
  12. ${\rm HP}$ for FLA associated with $(5,6,9)$
  13. ${\rm HP}$ for FLA associated with $(5,7,10)$

Key Result

Theorem 1

CCN Let $\mathfrak{g}$ be a $n$-dimensional filiform Lie algebra with associated triple $(z_1,z_2,n)$ and let $\{e_1, \ldots, e_n\}$ be an adapted basis of $\mathfrak g$. Then, there exist complex numbers $\alpha_i$, $\gamma_j$ and $\beta_{k \ell}$, with $1 \leq i \leq z_2-z_1+1$, $1 \leq j \leq n-z

Theorems & Definitions (45)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Definition 1
  • Lemma 1
  • proof
  • Remark 3
  • Proposition 1
  • proof
  • Remark 4
  • ...and 35 more