Rigidity and Cohomology of Seaweed Lie Algebras

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 $\mathfrak{s}\mathrm{/}\mathcal{Z}(\mathfrak{s})$. 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.