  This paper begins by introducing the concept of a quasi-hom-Lie algebra which
is a natural generalization of hom-Lie algebras introduced in a previous paper
by the authors. Quasi-hom-Lie algebras include also as special cases (color)
Lie algebras and superalgebras, and can be seen as deformations of these by
homomorphisms, twisting the Jacobi identity and skew-symmetry. The natural
realm for these quasi-hom-Lie algebras is as a generalization-deformation of
the Witt algebra $\Witt$ of derivations on the Laurent polynomials
$\C[t,t^{-1}]$. We also develop a theory of central extensions for qhl-algebras
which can be used to deform and generalize the Virasoro algebra by centrally
extending the deformed Witt type algebras constructed here. In addition, we
give a number of other interesting examples of quasi-hom-Lie algebras, among
them a deformation of the loop algebra.
