A class of quadratic deformations of Lie superalgebras

We study certain Z₂-graded, finite-dimensional polynomial algebras of degree 2 which are a special class of deformations of Lie superalgebras, which we call quadratic Lie superalgebras. Starting from the formal definition, we discuss the generalized Jacobi relations in the context of the Koszul property, and give a proof of the Poincar´e–Birkhoff–Witt basis theorem. We give several concrete examples of quadratic Lie superalgebras for low-dimensional cases, and discuss aspects of their structure constants for the ‘type I’ class. We derive the equivalent of the Kac module construction for typical and atypical modules, and a related direct construction of irreducible modules due to Gould. We investigate in detail one specific case, the quadratic generalization gl₂(in/i/1) of the Lie superalgebra sl(n/1). We formulate the general atypicality conditions at level 1, and present an analysis of zero-and one-step atypical modules for a certain family of Kac modules.