Representations of centrally extended Lie superalgebra $\mathfrak{p}\mathfrak{s}\mathfrak{l}(2\mathrm{\mid }2)$

The symmetries provided by representations of the centrally extended Lie superalgebra $\mathfrak{p}\mathfrak{s}\mathfrak{l}(2\mathrm{\mid }2)$ are known to play an important role in the spin chain models originated in the planar anti-de Sitter/conformal field theory correspondence and one-dimensional Hubbard model. We give a complete description of finite-dimensional irreducible representations of this superalgebra thus extending the work of Beisert which deals with a generic family of representations. Our description includes a new class of modules with degenerate eigenvalues of the central elements. Moreover, we construct explicit bases in all irreducible representations by applying the techniques of Mickelsson–Zhelobenko algebras.