Duflo-Serganova homology for exceptional modular Lie superalgebras with Cartan matrix

Abstract

For the exceptional finite-dimensional modular Lie superalgebras \(\mathfrak{g}(A)\) with indecomposable Cartan matrix \(A\), and their simple subquotients, we computed non-isomorphic Lie superalgebras constituting the homologies of the odd elements with zero square. These homologies are~key ingredients in the Duflo--Serganova approach to the representation theory. There were two definitions of defect of Lie superalgebras in the literature with different ranges of application. We suggest a third definition and an easy-to-use way to find its value. In positive characteristic, we found out one more reason to consider the space of roots over reals, unlike the space of weights, which should be considered over the ground field. We proved that the rank of the homological element (decisive in calculating the defect of a given Lie superalgebra) should be considered in the adjoint module, not the irreducible module of least dimension (although the latter is sometimes possible to consider, e.g., for \(p=0\)). We also computed the above homology for the only case of simple Lie superalgebras with symmetric root system not considered so far over the field of complex numbers, and its modular versions: \(\mathfrak{psl}(a|a+pk)\) for \(a\) and \(k\) small, and \(p=2, 3, 5\).