Plane Curves Which are Quantum Homogeneous Spaces

Let $\mathcal {C}$ C be a decomposable plane curve over an algebraically closed field k of characteristic 0. That is, $\mathcal {C}$ C is defined in k2 by an equation of the form g(x) = f(y), where g and f are polynomials of degree at least two. We use this data to construct three affine pointed Hopf algebras, A(x, a, g), A(y, b, f) and A(g, f), in the first two of which g [resp. f ] are skew primitive central elements, with the third being a factor of the tensor product of the first two. We conjecture that A(g, f) contains the coordinate ring $\mathcal {O}(\mathcal {C})$ O ( C ) of $\mathcal {C}$ C as a quantum homogeneous space, and prove this when each of g and f has degree at most five or is a power of the variable. We obtain many properties of these Hopf algebras, and show that, for small degrees, they are related to previously known algebras. For example, when g has degree three A(x, a, g) is a PBW deformation of the localisation at powers of a generator of the downup algebra A(− 1,− 1,0). The final section of the paper lists some questions for future work.

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 DEFORMATION OF THE BIG CELL INSIDE THE GRASSMANNIAN MANIFOLD G(r,n)

In this paper we construct a quantum analogue of the big cell inside the grassmannian manifold. Our deformation comes in tandem with a coaction of the upper parabolic subgroup in SLn(k), giving to the big cell the structure of quantum homogeneous space. At the end we give the De Rham complex of the quantum big cell and we define a ring of differential operators acting on the quantum big cell.