The Principal Investigator proposes to continue his investigations into the structure of various classes of noncommutative rings, particularly quantum coordinate rings (quantized algebras of functions) and related algebras, with a focus on geometric aspects of noncommutative rings and ring-theoretic aspects of noncommutative algebraic geometry. The proposed projects, located within noncommutative algebra, will build up the infrastructure of and interconnections among active parts of several related areas -- noncommutative algebraic geometry, quantum groups, Poisson algebras, and ring theory. Roughly speaking, the long-term goals framing many of these projects stem from noncommutative algebraic geometry, and the basic examples come from quantum groups, while many lines of approach are recruited from ring theory. A major new drive directing the PI's research on the subject is the investigation of Poisson analogs in semiclassical limits of phenomena known or expected to hold in quantum algebras. The parallels that have arisen so far provide strong motivation for the strategy of linking up the two sides of the coin; at this point, knowledge of one side allows the formulation of more precise conjectures for the other. A major goal is to tighten these connections.
A pervasive theme in the mathematical study of geometric objects is that the properties of these objects are completely encoded in the functions on them, and are often more accessible via these functions than directly. Within algebraic geometry -- the study of geometric spaces defined by polynomial equations -- it is the polynomial functions on a space that determine it. These functions form a ring (a system endowed with compatible addition and multiplication operations) which is, moreover, commutative (fg = gf always). In the 1980s, researchers in the former Soviet Union, in the process of solving certain problems in theoretical quantum physics, discovered rings which appear to enjoy all the structure of rings of functions on geometric spaces, except that the multiplication is noncommutative. In honor of their origins in quantum theory, these rings are now called "quantized coordinate rings." It proved very useful to treat them as if they were rings of functions (except for the noncommutativity), and the guiding principle in their study became the search for "noncommutative versions of the geometry." Sufficiently many common phenomena (both geometric and algebraic) have been discovered in a wide range of quantized coordinate rings to lead one to conjecture that general, axiomatizable underpinnings within this class of rings are responsible for the parallels in their behavior. The main long-term thrust of the PI's research is to uncover such general structures and decode their geometric content. In the medium term, the proposal aims to extend the range of known shared phenomena within this class of rings, in order to gain better insight into their common base.
