Jantzen filtration and strong linkage principle for modular Lie superalgebras

In this paper, we introduce super Weyl groups, their distinguished elements and properties for basic classical Lie superalgebras. Then we formulate Jantzen filtration for baby Verma modules in graded restricted module categories of basic classical Lie superalgebras over an algebraically closed field of odd characteristic, and obtain a sum formula in the corresponding Grothendieck groups. Consequently, we formulate a strong linkage principle for such categories.

Generators of Simple Modular Lie Superalgebras

Let X be one of the finite-dimensional graded simple Lie superalgebras of Cartan type W, S, H, K, HO, KO, SHO or SKO over an algebraically closed field of characteristic p > 3. In this paper we prove that X can be generated by one element except the ones of type W, HO, KO or SKO in certain exceptional cases in which X can be generated by two elements. As a subsidiary result, we prove that certain classical Lie superalgebras or their relatives can be generated by one or two elements.

Restricted Envelopes of Lie Superalgebras

Certain important results concerning p-envelopes of modular Lie algebras are generalized to the super-case. In particular, any p-envelope of the Lie algebra of a Lie superalgebra can be naturally extended to a restricted envelope of the Lie superalgebra. As an application, a theorem on the representations of Lie superalgebras is given, which is a super-version of Iwasawa's theorem in Lie algebra case. As an example, the minimal restricted envelopes are computed for three series of modular Lie superalgebras of Cartan type.

The classification of modular Lie superalgebras of type M

The natural filtration of the infinite-dimensional simple modular Lie superalgebra M over a field of characteristic p > 2 is proved to be invariant under automorphisms by discussing ad-nilpotent elements. Moreover, an intrinsic property is obtained and all the infinite-dimensional simple modular Lie superalgebras M are classified up to isomorphisms. As an application, a property of automorphisms of M is given.

Simple Modules for Modular Lie SuperalgebrasW(0∣n),S(0∣n), andK(n)

This paper constructs a series of modules from modular Lie superalgebrasW(0∣n),S(0∣n), andK(n)over a field of prime characteristicp≠2. Cartan subalgebras, maximal vectors of these modular Lie superalgebras, can be solved. With certain properties of the positive root vectors, we obtain that the sufficient conditions of these modules are irreducibleL-modules, whereL=W(0∣n),S(0∣n), andK(n).