A Multiparametric Quantum Superspace and Its Logarithmic Extension

TL;DR

This paper constructs a multiparametric quantum superspace with a Hopf superalgebra structure, introduces its logarithmic extension generalizing $k$-Minkowski superspace, and develops associated differential calculus and algebraic structures.

Contribution

It presents a new multiparametric quantum superspace with a Hopf superalgebra and introduces a logarithmic extension that generalizes $k$-Minkowski superspace, along with differential calculus.

Findings

Established a bicovariant differential calculus on the superspace

Defined a logarithmic extension with nonhomogeneous relations

Reduced to standard superalgebra when deformation parameters are set to one

Abstract

We introduce a multiparametric quantum superspace with $m$ even generators and $n$ odd generators whose commutation relations are in the sense of Manin such that the corresponding algebra has a Hopf superalgebra. By using its Hopf superalgebra structure, we give a bicovariant differential calculus and some related structures such as Maurer-Cartan forms and the correspoinding vector fields. It is also shown that there exists a quantum supergroup related with these vector fields. Morever, we introduce the logarithmic extension of this quantum superspace in the sense that we extend this space by the series expansion of the logarithm of the grouplike generator, and we define new elements with nonhomogeneous commutation relations. It is clearly seen that this logarithmic extension is a generalization of the $\kappa-$Minkowski superspace. We give the bicovariant differential calculus and the related algebraic structures on this extension. All noncommutative results are found to reduce to those of the standard superalgebra when the deformation parameters of the quantum (m+n)-superspace are set to one.