Cohomological rigidity of solvable Lie algebras of maximal ran

TL;DR

The work develops a framework to compute the second adjoint cohomology for solvable Lie algebras of maximal rank by leveraging Hochschild–Serre factorization and Leger–Luks-type root conditions. It provides explicit sufficient conditions on the root system that guarantee H^2(R_T,R_T)=0, extending classical Leger–Luks results and unifying several known rigidity proofs. It also identifies root configurations that ensure non-vanishing of H^2, supplies explicit cocycles to demonstrate non-rigidity, and proposes a lower bound conjecture for dim H^2 in this class. Collectively, the results yield a computable, geometry of root-system–driven criterion for cohomological rigidity across a broad family of maximal rank solvable Lie algebras, with concrete applications to model nilpotent and filiform extensions.

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.

Theorems & Definitions (37)

• Theorem 2.1
• Remark 2.2
• Definition 2.3
• Proposition 2.4
• proof
• Lemma 3.1
• Theorem 3.2
• proof
• Corollary 3.3
• Lemma 3.4