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Over a field of characteristic p > 2, the first cohomology of the special linear Lie superalgebra sl(2,1) with coefficients in all \c{hi}-reduced Kac modules and simple modules is determined by use of the weight space decompositions of these modules relative to the standard Cartan subalgebra of sl(2,1).

Let $\Gamma$ be a finite group and $V$ a finite-dimensional $\Gamma$-graded space over an algebraically closed field of characteristic not equal to 2. In the sense of conjugation, we classify all the so-called pre-nil or nil maximal abelian subalgebras for the general linear Lie color algebra $\frak{gl}(V,\Gamma)$. In the situation of $\Gamma$ being a cyclic group, we determine the minimal dimensions of pre-nil or nil faithful representations for any finite-dimensional abelian Lie color algebra.

Over an algebraically closed field of characteristic $p>2$, the 0-dimensional and 1-dimensional cohomology of the queer Lie superalgebra $\frak{q}(2)$ with coefficients in all baby Verma modules and all the simple modules are determined.

Let $\frak{g}$ be a contact Lie superalgebra of odd type or special contact Lie superalgebra of odd type over an algebraically closed field of characteristic $p>3$. In this paper we study non-restricted representations of $\frak{g}$. By using induced Kac modules, we characterize all simple $\frak{g}$-modules with nonsingular or $\Delta$-invertible $p$-characters. We also obtain all simple $\frak{g}$-modules with regular semisimple $p$-characters.

Over a field of characteristic p > 3, the first cohomology group of Lie superalgebra D(2,1;\alpha) with coefficients in baby Verma modules is determined by calculating the outer superderivations of D(2,1;\alpha).

In this paper, all (super)algebras are over a field $\mathbb{F}$ of characteristic different from $2, 3$. We construct the so-called 5-sequences of cohomology for central extensions of a Lie superalgebra and prove that they are exact. Then we prove that the multipliers of a Lie superalgebra are isomorphic to the second cohomology group with coefficients in the trivial module for the Lie superalgebra under consideration., Comment: 20 pages

In this paper, we study the cup products and Betti numbers over cohomology superspaces of two-step nilpotent Lie superalgebras with coefficients in the adjoint modules over an algebraically closed field of characteristic zero. As an application, we prove that the cup product over the adjoint cohomology superspaces for Heisenberg Lie superalgebras is trivial and we also determine the adjoint Betti numbers for Heisenberg Lie superalgebras by means of Hochschild-Serre spectral sequences., Comment: 13 pages

In this paper we define the so-called twisted Heisenberg superalgebras over the complex number field by adding derivations to Heisenberg superalgebras. We classify the fine gradings up to equivalence on twisted Heisenberg superalgebras and determine the Weyl groups of those gradings., Comment: 30 pages. arXiv admin note: text overlap with arXiv:1405.4093 by other authors

Let $L$ be a Lie superalgebra over a field of characteristic different from $2,3$ and write $\mathrm{ID}^{*}(L)$ for the Lie superalgebra consisting of superderivations mapping $L$ to $L^{2}$ and the central elements to zero. In this paper we first give an upper bound for the superdimension of $\mathrm{ID}^{*}(L)$ by means of linear vector space decompositions. Then we characterize the $\mathrm{ID}^{*}$-superderivation superalgebras for the nilpotent Lie superalgebras of class 2 and the model filiform Lie superalgebras by methods of block matrices., Comment: 12 pages

Suppose the ground field $\mathbb{F}$ is an algebraically closed field of characteristic different from 2, 3. We determine the Betti numbers and make a decomposition of the associative superalgebra of the cohomology for the model filiform Lie superalgebra. We also describe the associative superalgebra structures of the (divided power) cohomology for some low-dimensional filiform Lie superalgebras., Comment: 30 pages

Suppose that the underlying field is of characteristic different from $2, 3$. In this paper we first prove that the so-called stem deformations of a free presentations of a finite-dimensional Lie superalgebra $L$ exhaust all the maximal stem extensions of $L$, up to equivalence of extensions. Then we prove that multipliers and covers always exist for a Lie superalgebra and they are unique up to Lie superalgebra isomorphisms. Finally, we describe the multipliers, covers and maximal stem extensions of Heisenberg superalgebras of odd centers and model filiform Lie superalgebras., Comment: 16 pages

In this paper, we introduce the concept of (super-)multiplier-rank for Lie superalgeras and classify all the finite-dimensional nilpotent Lie superalgebras of multiplier-rank $\leq 2$ over an algebraically closed field of characteristic zero. In the process, we also determine the multipliers of Heisenberg superalgebras., Comment: 15 pages

In this paper, the Hom-Lie superalgebra structures on exceptional simple Lie superalgebras of vector fields are studied. Taking advantage of the Z-grading structures and the transitivity, we prove that there is only the trivial Hom-Lie superalgebra structures on exceptional simple Lie superalgebras. Our proof is obtained by studying the Hom-Lie superalgebra structures on their 0-th and -1-st Z-components., Comment: 14 pages