Nichols Algebras and Quantum Principal Bundles

A general procedure for constructing Yetter-Drinfeld modules from quantum principal bundles is introduced. As an application a Yetter-Drinfeld structure is put on the cotangent space of the Heckenberger-Kolb calculi of the quantum Grassmannians. For the special case of quantum projective space the associated braiding is shown to be non-diagonal and of Hecke type. Moreover, its Nichols algebra is shown to be finite-dimensional and equal to the anti-holomorphic part of the total differential calculus.Comment: Updated grant details. arXiv admin note: text overlap with arXiv:1611.0796

Quantum Groups and Noncommutative Complex Geometry.

PhDNoncommutative Riemannian geometry is an area that has seen intense activity over the past 25 years. Despite this, noncommutative complex geometry is only now beginning to receive serious attention. The theory of quantum groups provides a large family of very interesting potential examples, namely the quantum flag manifolds. Thus far, only the irreducible quantum flag manifolds have been investigated as noncommutative complex spaces. In a series of papers, Heckenberger and Kolb showed that for each of these spaces, there exists a q-deformed Dolbeault double complex. In this thesis a comprehensive framework for noncommutative complex geometry on quantum homogeneous spaces is introduced. The main ingredients used are covariant differential calculi and Takeuchi's categorical equivalence for faithfully at quantum homogeneous spaces. A number of basic results are established, producing a simple set of necessary and sufficient conditions for noncommutative complex structures to exist. It is shown that when applied to the quantum projective spaces, this theory reproduces the q-Dolbeault double complexes of Heckenberger and Kolb. Furthermore, the framework is used to q-deform results from Borel{Bott{ Weil theory, and to produce the beginnings of a theory of noncommutative Kahler geometry

Curvature of positive relative line modules over the quantum projective spaces

We show that the curvature of a positive relative line module over quantum projective space is given by $q$-integer deformation of its classical curvature. This generalises a result of Majid for the Podle\'s sphere.Comment: 18 pages. arXiv admin note: substantial text overlap with arXiv:1912.0880