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A class of finite-dimensional Cartan-type Lie superalgebras H(n,m) over a field of prime characteristic is studied in this paper. We first determine the derivation superalgebra of H(n,m). Then we obtain that H(n,m) is restrictable and it is an extension of the Lie superalgebra [Formula: see text]. Finally, we prove that H(n,m) is isomorphic to a subalgebra of the restricted Hamiltonian Lie superalgebra [Formula: see text].

A new family of finite-dimensional modular Lie superalgebras Γ is defined. The simplicity and generators of Γ are studied and an explicit description of the derivation superalgebra of Γ is given. Moreover, it is proved that Γ is not isomorphic to any known Lie superalgebra of Cartan type.

Let 𝔽 be an algebraically closed field and char 𝔽 = p > 3. In this paper, we determine the second cohomology group of the finite-dimensional Contact superalgebra K(m,n,t).

The natural filtrations of infinite-dimensional modular Lie superalgebras W and S are proved to be invariant under their automorphism groups by means of investigating the ad-nilpotent elements and determining certain subalgebras generated by ad-nilpotent elements. As an application, we obtain an intrinsic characterization of W and S, and give a property of the automorphisms of these modular Lie superalgebras.