Z2×Z2 \mathbb{Z}_2 \times \mathbb{Z}_2 generalizations of N=1{\cal N} = 1 superconformal Galilei algebras and their representations

We introduce two classes of novel color superalgebras of $\mathbb{Z}_2 \times \mathbb{Z}_2$ grading. This is done by realizing members of each in the universal enveloping algebra of the ${\cal N}=1$ supersymmetric extension of the conformal Galilei algebra. This allows us to upgrade any representation of the super conformal Galilei algebras to a representation of the $\mathbb{Z}_2 \times \mathbb{Z}_2$ graded algebra. As an example, boson-fermion Fock space representation of one class is given. We also provide a vector field realization of members of the other class by using a generalization of the Grassmann calculus to $\mathbb{Z}_2 \times \mathbb{Z}_2$ graded setting.Comment: 17 pages, no figur

Maps and twists relating U(sl(2))U(sl(2)) and the nonstandard Uh(sl(2))U_{h}(sl(2)): unified construction

A general construction is given for a class of invertible maps between the classical $U(sl(2))$ and the Jordanian $U_{h}(sl(2))$ algebras. Different maps are directly useful in different contexts. Similarity trasformations connecting them, in so far as they can be explicitly constructed, enable us to translate results obtained in terms of one to the other cases. Here the role of the maps is studied in the context of construction of twist operators between the cocommutative and noncocommutative coproducts of the $U(sl(2))$ and $U_{h}(sl(2))$ algebras respectively. It is shown that a particular map called the `minimal twist map' implements the simplest twist given directly by the factorized form of the ${\cal R}_{h}$-matrix of Ballesteros-Herranz. For other maps the twist has an additional factor obtainable in terms of the similarity transformation relating the map in question to the minimal one. The series in powers of $h$ for the operator performing this transformation may be obtained up to some desired order, relatively easily. An explicit example is given for one particularly interesting case. Similarly the classical and the Jordanian antipode maps may be interrelated by a similarity transformation. For the `minimal twist map' the transforming operator is determined in a closed form.Comment: LaTeX, 13 page