Quantum Groups, Quantum Symmetries, and Non-Commutative Geometry

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Noncommutative geometry is a field of pure mathematics that traces its origins to problems in mathematical physics motivated by the discovery of quantum phenomena. The typical mathematical framework in which geometric entities (such as the 4-dimensional space-time that underlies the general theory of relativity) can be studied algebraically needs to be enlarged and generalized if it is to be reconciled with quantum phenomena. In the resulting setup, the symmetries of a physical system (in the sense of structure-preserving transformations) sometimes need to be discarded and replaced with new and more exotic notions of symmetry. The mathematical embodiment of these exotic symmetries are known as "quantum groups," and they are the main focus of this project. Much of what can be taken for granted in the context of "plain" geometry and actions of ordinary transformations on ordinary spaces becomes problematic in the noncommutative setting. This research project aims to shed light on a number of these problems, in a range of subfields within the larger realm of noncommutative geometry. This project investigates several aspects of noncommutative geometry that revolve around the notion of quantum symmetry. This involves studying quantum groups, their representation theory, and their actions on algebraic and geometric structures, as well as attendant problems in non-commutative algebraic geometry. One goal is to further understand the phenomenon of quantum rigidity, whereby certain structures admit no truly quantum symmetries. What this means is that whenever a sufficiently well-behaved Hopf algebra (which is the algebraic embodiment of a quantum group) coacts in a structure-preserving manner, the coaction factors through one by the function algebra on an ordinary group. Many special cases of this are known (integral affine algebraic varieties, certain smooth non-commutative projective algebraic varieties, compact connected smooth manifolds, certain classes of metric spaces, etc.), but the general phenomenon is poorly understood. Another goal is to attempt to transport tools and concepts specific to discrete or reductive algebraic groups (such as Borel subgroups, maximal tori, weight systems, compactifications, residual finiteness, linearity) over to quantum groups in order to further elucidate their structure and representation theory. Finally, symmetry considerations allow for the construction of new examples of smooth noncommutative projective schemes that in some sense behave generically within the moduli spaces that classify such schemes. The representation theory of the corresponding algebras would then shed light into the nature of these moduli spaces that are at the moment not well understood.