Leibniz Cohomology with Adjoint Coefficients

With the Poincaré group ${\mathbf{R}}^{3,1}⋊O(3,1)$ as the model of departure, we focus, for $n=p+q$, $p+q\ge 4$, on the affine indefinite orthogonal group ${\mathbf{R}}^{p,q}⋊SO(p,q)$. Denote by ${\mathfrak{h}}_{p,q}$ the Lie algebra of the affine indefinite orthogonal group. We compute the Leibniz cohomology of ${\mathfrak{h}}_{p,q}$ with adjoint coefficients, written $H{L}^{\ast }({\mathfrak{h}}_{p,q};{\mathfrak{h}}_{p,q})$. We calculate several indefinite orthogonal invariants, and ${\mathfrak{h}}_{p,q}$-invariants and provide the Leibniz cohomology in terms of these invariants.