Hom-Lie algebra (superalgebra) structure appeared naturally in $q$-deformations, based on $\sigma $-derivations of Witt and Virasoro algebras (superalgebras). They are a twisted version of Lie algebras (superalgebras), obtained by deforming the Jacobi identity by a homomorphism. In this paper, we discuss the concept of $\alpha ^k$-derivation, a representation theory, and provide a cohomology complex of Hom-Lie superalgebras. Moreover, we study central extensions. As application, we compute derivations and the second cohomology group of a twisted ${\rm osp}(1,2)$ superalgebra and $q$-deformed Witt superalgebra.
The aim of this note, which raises more questions than it answers, is to study natural operations acting on the cohomology of various types of algebras. It contains a lot of very surprising partial results and examples.
The external derivative d on differential manifolds inspires graded operators on complexes of spaces Λr g ∗ , Λr g ∗ ⊗ g, Λr g ∗ ⊗ g ∗ stated by g ∗ dual to a Lie algebra g. Cohomological properties of these operators are studied in the case of the Lie algebra g = se(3) of the Lie group of Euclidean motions.
Let T : X → X be a continuous selfmap of a compact metrizable space X. We prove the equivalence of the following two statements: (1) The mapping T is a Banach contraction relative to some compatible metric on X. (2) There is a countable point separating family F ⊂ C(X) of non-negative functions f ∈ C(X) such that for every f ∈ F there is g ∈ C(X) with f = g − g ◦ T.