Rigidity and Cohomology of Seaweed Lie Algebras

TL;DR

This paper characterizes the adjoint cohomology of seaweed Lie algebras, showing indecomposable ones are rigid while decomposable ones have cohomology described via their centers and zero-weight cohomology.

Contribution

It provides a complete description of the adjoint cohomology for seaweed Lie algebras, establishing indecomposability as the key to rigidity and detailing the cohomology structure.

Findings

Indecomposable seaweed Lie algebras are absolutely rigid with zero cohomology.

Decomposable seaweed Lie algebras have cohomology described by their centers and zero-weight cohomology.

The center is the unique source of nontrivial adjoint cohomology.

Abstract

Seaweed (biparabolic) subalgebras form a large and structurally rich class of subalgebras of simple Lie algebras. We determine their adjoint cohomology. If $\mathfrak{s}$ is an indecomposable seaweed subalgebra of a complex simple Lie algebra, then \[ H^\ast(\mathfrak{s},\mathfrak{s})=0, \] and hence $\mathfrak{s}$ is absolutely rigid. If $\mathfrak{s}$ is decomposable, then the Coll--Gerstenhaber decomposition for Lie semidirect products gives, for each $n\ge 0$, a canonical description of $H^n(\mathfrak{s},\mathfrak{s})$ in terms of exterior powers of $\mathcal{Z}(\mathfrak{s})^\ast$ and the zero-weight cohomology of $\mathfrak{s}/\mathcal{Z}(\mathfrak{s})$. In particular, the center is the unique source of nontrivial adjoint cohomology. These results identify indecomposability as the precise condition for cohomological rigidity and give a uniform description of adjoint cohomology for seaweed Lie algebras.