Cohomological rigidity of solvable Lie algebras of maximal ran

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.

💡 Research Summary

The paper investigates the second adjoint cohomology group (H^{2}(\mathfrak R_T,\mathfrak R_T)) for a class of solvable Lie algebras (\mathfrak R_T = \mathfrak N \rtimes \mathfrak T) that arise as maximal solvable extensions of nilpotent Lie algebras (\mathfrak N) of maximal rank. The authors begin by recalling that the vanishing of (H^{2}(\mathfrak g,\mathfrak g)) is equivalent to cohomological rigidity: every formal one‑parameter deformation of (\mathfrak g) is trivial. They emphasize the importance of this invariant both for deformation theory and for the geometry of the algebraic variety of Lie algebra laws, where rigid algebras correspond to Zariski‑open orbits.

A key structural feature of the algebras under consideration is the existence of a maximal torus (\mathfrak T) acting diagonally on (\mathfrak N). Consequently (\mathfrak N) decomposes into one‑dimensional weight spaces (\mathfrak N_\alpha) indexed by a set of weights (W). The authors adopt the Hochschild–Serre factorization theorem in the special situation where (\mathfrak T) is abelian and its representation on both (\mathfrak N) and any module is diagonalizable. This yields the isomorphism \