GSQS · Geometry and Symmetry of Quantum Spaces

Geometry and Symmetry of Quantum Spaces

Study of spaces and structures built on them (like bundles of vectors, measures of distances) is one of the main aims of contemporary mathematics and mathematical physics. Classification of structures is usually achieved by means of computing invariants: they are designed to distinguish the inequivalent classes of structures. They are indispensable but hard to compute. For this reasons, new methods and new techniques are desired and required. One of the main lines of approach is based on symmetries, which were always fundamental in solving problems both in mathematics and in physics. In the studies of quantum spaces, new types of symmetries arise and serve as a crucial organising principle allowing one to handle highly complicated situations. Our project, in precise mathematical terms, concentrates around noncommutative index theory, Connes' spectral geometry, and appropriate adaptations to coalgebraic or Hopf-algebraic setting of cyclic (co)homology and the Chern-Connes character as our general tools. As a specific venue of investigation, we plan Hopf-algebra equivariant fiber products of algebras, where Mayer Vietoris methods can be applied. A key technical feature of this project is a standing assumption that the algebra describing a compact quantum fibration corresponds to the algebra of functions continuous along the base space and bounded-degree polynomial along the fibers. This is a strategy to take the best of two different worlds: our fixed-point algebras are C*-algebras offering advantages of functional calculus and computability of K-theory, whereas the total space algebras are comodule algebras allowing us to take advantage of tools and methods of the well-developed noncommutative Galois theory of ring extensions. Within the project, we developed and applied verifiable criteria to distinguish different isomorphism classes of principal extensions of noncommutative algebras and K0-classes of modules associated to them. Also, we achieved a new guiding principle in computing K-groups of multi-pullbacks of C*-algebras.

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