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With the Poincar\'e group $\mathbf{R}^{3,\,1}\rtimes O(3,\,1)$ as the modelof departure, we focus, for $n=p+q$, $p+q\ge 4$, on the affine indefiniteorthogonal group $\mathbf{R}^{p,q}\rtimes SO(p,\,q)$. Denote by$\mathfrak{h}_{p,\,q}$ the Lie algebra of the affine indefinite orthogonalgroup. We compute the Leibniz cohomology of $\mathfrak{h}_{p,\,q}$ with adjointcoefficients, written $HL^*(\mathfrak{h}_{p,\,q};\,\mathfrak{h}_{p,\,q})$. Wecalculate several indefinite orthogonal invariants, and $\mathfrak{h}_{p,\,q}$-invariants and provide the Leibniz cohomology in terms of these invariants.

Leibniz Cohomology with Adjoint Coefficients