[Paper Review] Nonfiliform characteristically nilpotent Lie algebras

Jose Maria Ancochea-Bermudez, Otto Rutwig Campoamor-Stursberg|ArXiv.org|Jul 14, 2000

Advanced Topics in Algebra12 references3 citations

TL;DR

This paper constructs nonfiliform characteristically nilpotent Lie algebras via deformations of the graded Lie algebra $σ_{(m,m-1)}^4$, a central naturally graded extension of the filiform algebra $L_n$. By analyzing graded cohomology and prolonging cocycles, the authors explicitly construct, for any $m \geq 4$, characteristically nilpotent Lie algebras of dimension $2m+2$ with characteristic sequence $(2m-1,2,1)$, demonstrating compatibility with central extensions and identifying rigid and complete solvable models.

ABSTRACT

We construct large families of characteristically nilpotent Lie algebras by considering deformations of the Lie algebra g_{m,m-1}^{4} of type Q_{n},and which arises as a central extension fo the filiform Lie algebra L_{n}. By studying the graded cohomology spaces we obtain that the sill algebras are isomorphic to the nilradicals of solvable, complete Lie algebra laws. For extremal cocycles these laws are also rigid. Considering supplementary cocycles we construcy, for dimensions n>8, nonfiliform characteristically nilpotent Lie algebras and show that for certain deformations these are compatible with central extensions.

Motivation & Objective

• To construct explicit families of nonfiliform characteristically nilpotent Lie algebras, a class less studied than filiform examples.
• To analyze deformations of the graded Lie algebra $\mathfrak{g}_{(m,m-1)}^4$, a central naturally graded extension of the filiform algebra $L_n$, using cohomological techniques.
• To determine which deformations yield nilradicals of solvable, complete, or rigid Lie algebras.
• To investigate compatibility of deformations with central extensions of degree one, leading to new nilpotent algebras of higher characteristic sequence.
• To establish the existence of characteristically nilpotent Lie algebras with characteristic sequence $(2m-1,2,1)$ in dimension $2m+2$ for $m \geq 4$.

Proposed method

• Deformation theory of the Lie algebra $\mathfrak{g}_{(m,m-1)}^4$ is analyzed using graded cohomology spaces $H^2(\mathfrak{g}, \mathbb{C})$, partitioned to classify extensions.
• Cocycles $\psi_{2,k}$ are extended to higher-dimensional algebras $\mathfrak{g}_{(m,m-1)}^{4,1}$ via prolongation by zeros, preserving structure under specific index conditions.
• The Cartan-Maurer equations of the extended algebras are explicitly defined, incorporating additional generators $\omega_{2m+1}$ and $\omega_{2m+2}$ with specific wedge product relations.
• Linear expandability of extended cocycles $\tilde{\psi}_{2,k}$ is determined by closure conditions on $d\omega_{2m+2}$, restricting valid $k$ to $2m-5$ and $2m-4$.
• Rigid and complete solvable Lie algebras are constructed by introducing a new generator $\theta$ with nontrivial differential $d\theta = 0$ and scaled action on other forms.
• The nilradical of the resulting solvable algebra is identified as $\mathfrak{g}_{(m,m-1)}^{4,1} + \tilde{\psi}_{2,2m-5}$, confirming its role as a nilradical of a rigid, complete Lie algebra.

Experimental results

Research questions

• RQ1 Can nonfiliform characteristically nilpotent Lie algebras be systematically constructed using deformation theory of known nilpotent models?
• RQ2 Which cohomology classes of $\mathfrak{g}_{(m,m-1)}^4$ yield nilradicals of solvable, complete, or rigid Lie algebras upon deformation?
• RQ3 Under what conditions are deformations of $\mathfrak{g}_{(m,m-1)}^4$ compatible with central extensions of degree one?
• RQ4 What is the characteristic sequence of the resulting algebras after deformation and extension, and can it be explicitly computed?
• RQ5 For which indices $k$ is the prolongation $\tilde{\psi}_{2,k}$ linearly expandable in the extended algebra $\mathfrak{g}_{(m,m-1)}^{4,1}$?

Key findings

• For any $m \geq 4$, the paper constructs a characteristically nilpotent Lie algebra $e_1(\mathfrak{g}_{(m,m-1)}^4 + \psi_{2,2m-4})$ of dimension $2m+2$ with characteristic sequence $(2m-1,2,1)$.
• The nilradical of the rigid, complete solvable Lie algebra $\mathfrak{r}_{(m,m-1)}^{4,1,2m-5}$ is isomorphic to $\mathfrak{g}_{(m,m-1)}^{4,1} + \tilde{\psi}_{2,2m-5}$, confirming its role as a nilradical of a rigid law.
• The prolongation $\tilde{\psi}_{2,k}$ is linearly expandable if and only if $k = 2m-5$ or $k = 2m-4$, which corresponds to specific modifications of $d\omega_{2m}$ and $d\omega_{2m-1}$.
• The extended algebra $\mathfrak{g}_{(m,m-1)}^{4,1}$ has characteristic sequence $(2m-1,2,1)$ and is $(m-1)$-abelian, though it loses its natural grading.
• The construction yields a family of nonfiliform characteristically nilpotent Lie algebras, providing one of the few explicit methods to generate such algebras beyond the filiform case.
• The results confirm that certain deformations of $\mathfrak{g}_{(m,m-1)}^4$ are compatible with central extensions of degree one, enabling the construction of new nilpotent algebras with controlled characteristic sequences.