Bicovariant differential calculus on the quantum superspace ℝq(1|2)

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.

A Two-parameter bicovariant differential calculus on the (1 + 2)-dimensional q-superspace

We construct a two-parameter bicovariant differential calculus on [Formula: see text] with the help of the covariance point of view using the Hopf algebra structure of [Formula: see text]. To achieve this, we first use the consistency of calculus and the approach of [Formula: see text]-matrix which satisfies both ungraded and graded Yang–Baxter equations. In particular, based on this differential calculus, we investigate Cartan–Maurer forms for this [Formula: see text]-superspace. Finally, we obtain the quantum Lie superalgebra corresponding the Cartan–Maurer forms.

CALCULI, HODGE OPERATORS AND LAPLACIANS ON A QUANTUM HOPF FIBRATION

We describe Laplacian operators on the quantum group SUq(2) equipped with the four-dimensional bicovariant differential calculus of Woronowicz as well as on the quantum homogeneous space [Formula: see text] with the restricted left covariant three-dimensional differential calculus. This is done by giving a family of Hodge dualities on both the exterior algebras of SUq(2) and [Formula: see text]. We also study gauged Laplacian operators acting on sections of line bundles over the quantum sphere.

A NEW APPROACH IN BICOVARIANT DIFFERENTIAL CALCULUS ON SUq(2)

When reviewing the work of Aschieri–Castellani concerning the bicovariant differential calculus on GLq(2), we remarked that a peculiar basis within the space of quantum one-forms over SUq(2) can lead to a new bicovariant differential calculus on SUq(2) easy to handle than those known in the literature. New results have been obtained for the exterior product, the exterior differential, the C-structure constants, the Cartan–Maurer equations, and the q-Lie algebra.

INHOMOGENEOUS QUANTUM GROUPS IGLq,r(N): UNIVERSAL ENVELOPING ALGEBRA AND DIFFERENTIAL CALCULUS

A review of the multiparametric linear quantum group GL q,r(N), its real forms, its dual algebra U [ gl q,r(N)] and its bicovariant differential calculus is given in the first part of the paper. We then construct the (multiparametric) linear inhomogeneous quantum group IGL q,r(N) as a projection from GL q,r(N+1) or, equivalently, as a quotient of GL q,r(N+1) with respect to a suitable Hopf algebra ideal. A bicovariant differential calculus on IGL q,r(N) is explicitly obtained as a projection from that on GL q,r(N+1). Our procedure unifies in a single structure the quantum plane coordinates and the q group matrix elements [Formula: see text], and allows one to deduce without effort the differential calculus on the q plane IGL q,r(N)/ GL q,r(N). The general theory is illustrated on the example of IGL q,r(2).