Hodge dualities and Laplacians on quantum spheres

In this talk I shall describe how it is possible to define Hodge dualities acting on a class of exterior algebras over the quantum group SU(2) and its quantum homogeneous sphere, and the class of corresponding Laplacians. I shall then compare it with the results coming from both similar (related to different definition of Hodge dualities) and different (related to the construction of a Dirac operator) approaches.