Classification of infinite dimensional irreducible modules for type I Lie superalgebras (Q1803211)

Lie superalgebras have been studied for nearly 20 years, since they were first introduced in the middle 1970's as symmetry algebras in supersymmetric theories of elementary particles. Structurally, the simple superalgebras are somewhat akin to the ordinary simple Lie algebras. Their representations, however, turns out to be more difficult to handle. On the one hand there are the so-called typical representations, for which a reasonably complete constructive theory exists. The remainder, the atypical representations, are much more complex, and so far there is only theory which covers some atypical representations of some algebras. In this paper, the author specialises to type I basic classical Lie superalgebras and restricts attention to irreducible modules (finite or infinite dimensional). In this context, it is proved that all irreducible modules admit an infinitesimal character and are uniquely characterised (up to isomorphism) by their maximal \(Z\)-graded component which, it turns out, is an irreducible module for the even part of the algebra. It seems likely that this work will provide a starting point for further, more detailed, research.