On Low-Dimensional Complex ω-Lie Superalgebras

Let (g, [−, −], ω) be a finite-dimensional complex ω-Lie superalgebra. In this paper, we introduce the notions of derivation superalgebra Der(g) and the automorphism group Aut(g) of (g, [−, −], ω). We study Derω (g) and Autω (g), which are superalgebra of Der(g) and subgroup of Aut(g), respectively. For any 3-dimensional or 4-dimensional complex ω-Lie superalgebra g, we explicitly calculate Der(g) and Aut(g), and obtain Jordan standard forms of elements in the two sets. We also study representation theory of ω-Lie superalgebras and give a conclusion that all nontrivial non-ω-Lie 3-dimensional and 4-dimensional ω-Lie superalgebras are multiplicative, as well as we show that any irreducible respresentation of the 4-dimensional ω-Lie superalgebra P2,k(k 6= 0, −1) is 1-dimensional.

THE GENERALIZED WITT MODULAR LIE SUPERALGEBRA OF CARTAN TYPE

We construct the generalized Witt modular Lie superalgebra $\tilde {W}$ of Cartan type. We give a set of generators for $\tilde {W}$ and show that $\tilde {W}$ is an extension of a subalgebra of $\tilde {W}$ by an ideal $\overline {J} $. Finally, we describe the homogeneous derivations of Z-degree of $\tilde {W}$ and we determine the derivation superalgebra of $\tilde {W}$.

A Class of Finite-dimensional Lie Superalgebras of Hamiltonian Type

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].

The Finite-dimensional Modular Lie Superalgebra Γ

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.