Abstract. Hopf algebra structures on the extended q-superplane and its differential algebra are defined. An algebra of forms which is obtained from the generators of the extended q-superplane is introduced and its Hopf algebra structure is given.
Super-Hopf algebra structure on the function algebra on the extended quantum superspace has been defined. It is given a bicovariant differential calculus on the superspace. The corresponding (quantum) Lie superalgebra of vector fields and its Hopf algebra structure are obtained. The dual Hopf algebra is explicitly constructed. A new quantum supergroup that is the symmetry group of the differential calculus is found.
We present a differential calculus on the extension of the quantum plane obtained by considering that the (bosonic) generator x is invertible and by working with polynomials in ln x instead of polynomials in x. We construct the quantum Lie algebra associated with this extension and obtain its Hopf algebra structure and its dual Hopf algebra.
Abstract. The differential calculus on the quantum supergroup GL q (1|1) was introduced by Schmidke et al. (1990 Z. Phys. C 48 249). We construct a differential calculus on the quantum supergroup GL q (1|1) in a different way and we obtain its quantum superalgebra. The main structures are derived without an R-matrix. It is seen that the found results can be written with help of a matrixR
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