Cohomology of $\mathfrak{p}\mathfrak{s}\mathfrak{l}(n\mathrm{\mid }n)$ in trivial and in adjoint representations

We compute the cohomology of the projective special linear Lie superalgebra $\mathfrak{p}\mathfrak{s}\mathfrak{l}(n\mathrm{\mid }n)$ over a field of characteristic zero in trivial and in adjoint coefficients in full detail through its associated Hochschild–Serre spectral sequences. Applications in physics exist for its real form, the projective special unitary Lie superalgebra $\mathfrak{p}\mathfrak{s}\mathfrak{u}(n\mathrm{\mid }n)$ — in particular, $\mathfrak{p}\mathfrak{s}\mathfrak{u}(2,2\mathrm{\mid }4)$ for AdS/CFT. We also find the cohomology of $\mathfrak{s}\mathfrak{l}(m\mathrm{\mid }n)$ in trivial coefficients along the way and we point out a discrepancy with a previous statement in the literature. We present the explicit cohomology groups of low degree for $n⩽4$.