Cohomological rigidity of solvable Lie algebras of maximal ran

TL;DR

This paper investigates the cohomological rigidity of a class of solvable Lie algebras arising from maximal solvable extensions of nilpotent Lie algebras, providing new criteria and explicit root configurations affecting their second cohomology.

Contribution

It extends classical rigidity conditions by offering a unified, computational framework for assessing cohomological rigidity in solvable Lie algebras derived from nilpotent structures.

Findings

Identifies conditions ensuring cohomological rigidity.

Explicit root configurations leading to non-trivial cohomology.

Provides a unified approach extending classical results.

Abstract

We study the second cohomology group with coefficients in the adjoint module for a class of solvable Lie algebras $\mathcal{R}_{\mathcal{T}}$ that arise as maximal solvable extensions of nilpotent Lie algebras $\mathcal{N}$ of maximal rank. Under suitable structural assumptions on the root system determined by the action of a maximal torus $\mathcal{T}$ on $\mathcal{N}$, we obtain sufficient conditions for the cohomological rigidity of $\mathcal{R}_{\mathcal{T}}$. Conversely, we identify explicit configurations of roots that force the second cohomology group to be non-trivial, thereby producing broad families of solvable Lie algebras that are not cohomologically rigid. Our results extend the classical sufficient conditions of Leger and Luks, and they provide a unified and computationally effective framework for determining the cohomological rigidity of a wide class of solvable Lie algebras, including several known results.