Noncommutative Geometry and Topology in Prague

Cross Product Exterior Algebras

The first step in noncommutative Riemannian geometry by constructive approach, as develop by Beggs and Majid and many other collaborators, is to determine an exterior algebra of a given algebra in which we can use to build the elements of noncommutative geometry. In this talk, we will see that a strongly bicovariant exterior algebra of cross product Hopf algebras is the cross product of exterior algebras. As an example, we find a strongly bicovariant exterior algebra on \mathbb{C}_q[GL_2]\ltimes \mathbb{C}_q^2, which is a quantum deformation of maximal parabolic P \subset SL_3 and isomorphic to a quotient of \mathbb{C}_q[SL_3]. Moreover, from Manin, we know that the structure of \mathbb{C}_q[GL_2] is largely determined from its coaction on quantum plane \mathbb{C}_q^2. By requiring that this coaction is differentiable, we find that the structure of 4D strongly bicovariant \Omega(\mathbb{C}_q[GL_2]) is largely determined by its coaction on \Omega(\mathbb{C}_q^2). I will also talk a little bit about more problems that arise from this results