The central aim of this research project is the study of various classes of quantum algebras (particularly quantized algebras of functions), with strong focus on both algebraic structure and geometric aspects. These algebras were discovered in the 1980s in the process of solving problems in theoretical quantum physics. They are mathematical systems that model rings of functions on geometric structures as in algebraic geometry (the study of geometric spaces defined by polynomial equations) except that the multiplication is noncommutative. The internal structure of quantum algebras will be intensively investigated, particularly those substructures that correspond to points and geometric subspaces in the classical analogs, together with relations between quantum and classical algebras that are connected via a semiclassical limit (a process in which the noncommutativity tends to zero). These commutative semiclassical limit algebras carry a remnant of the noncommutativity of the associated quantum algebras, recorded in a bilinear form called a Poisson bracket. The work will also pursue the closely related questions about Poisson algebras, as well as the classical-quantum interplay between Poisson semiclassical limits and their quantum counterparts. The research aims both to advance, and to tighten the links among, several highly active mathematical areas -- quantum groups, Poisson algebra, noncommutative algebraic geometry, and cluster theory. In particular, the work will furnish tools to pass insights from one area to the others.
Specific projects include the exploitation of cluster algebra structures on large classes of quantum and Poisson algebras towards the ideal theory of these algebras. A related effort will go into the construction of more viable bridges over which to shift information back and forth between quantum algebras and their semiclassical limits. Along this line, the investigator conjectures that the prime and primitive spectra of a quantum algebra are homeomorphic to the Poisson-prime and Poisson-primitive spectra of a semiclassical limit, and he will devote major effort towards a confirmation. Other projects target the classical-quantum interplay. The goal is to develop noncommutative versions of properties enjoyed by commutative coordinate rings, prominent among them being catenarity of their prime spectra. The work will explore the conjecture that all quantized coordinate rings enjoy catenarity. These efforts are related to the goal of elucidating the global topological structures of the prime and primitive spectra of quantum algebras. This is aimed at gluing the (already secured) classical pieces of these spectra via auxiliary spaces and maps that are not required in the classical setting. Establishing a specific, detailed format for this gluing and auxiliary data, together with similar, matching formats for the spectra of quantum algebras and semiclassical limits, would yield explicit homeomorphisms between these spectra, and thus establish the conjectured quantum-Poisson interplay.
