On some almost quadratic algebras coming from twisted derivations

This paper explores the quasi-deformation scheme devised in [1, 3] as applied to the simple Lie algebra sl(2)(F) for specific choices of the involved parameters and underlying algebras. One of the main points of this method is that the quasi-deformed algebra comes endowed with a canonical twisted Jacobi identity. We show in the present article that when the quasi-deformation method is applied to sl(2)(F) one obtains multiparameter families of almost quadratic algebras, and by choosing parameters suitably, sl(2)(F) is quasi-deformed into three-dimensional and four-dimensional Lie algebras and algebras closely resembling Lie superalgebras and colour Lie algebras, this being in stark contrast to the classical deformation schemes where sl(2)(F) is rigid