Cohomology of $\mathfrak{psl}(n|n)$ in trivial and in adjoint representations

Abstract

We compute the cohomology of the projective special linear Lie superalgebra $\mathfrak{psl}(n|n)$ 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 $\mathfrak{psu}(n|n)$ --- in particular, $\mathfrak{psu}(2,2|4)$ for AdS/CFT. We also find the cohomology of $\mathfrak{sl}(m|n)$ 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 $n\leqslant4$.