arXiv subjects
Publications and source records attributed to P. Ma{ś}lanka.
The general framework of bicrossproduct Hopf algebras given by Majid is extended to $Z_2$-graded bicrossproduct Hopf superalgebras. As examples of bicrossproduct Hopf superalgebras we provide the graded algebras of functions on undeformed as well as $κ$-deformed $D=4$ Poincaré supergroups.
The $κ$-deformed $D=4$ Poincar{é} superalgebra written in Hopf superalgebra form is transformed to the basis with classical Lorentz subalgebra generators. We show that in such a basis the $κ$-deformed $D=4$ Poincare superalgebra can be written as graded bicrossproduct. We show that the $κ$-deformed $D=4$ superalgebra acts covariantly on $κ$-deformed chiral superspace.
The classical $r$-matrix for $N=1$ superPoincar{é} algebra, given by Lukierski, Nowicki and Sobczyk is used to describe the graded Poisson structure on the $N=1$ Poincar{é} supergroup. The standard correspondence principle between the even (odd) Poisson brackets and (anti)commutators leads to the consistent quantum deformation of the superPoincar{é} group with the deformation parameter $q$ described by fundamental mass parameter $κ\quad (κ^{-1}=\ln{q})$. The $κ$-deformation of $N=1$ superspace as dual to the $κ$-deformed supersymmetry algebra is discussed.