NCG&T Prague

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... more