Noncommutative Spaces

Alexander Rosenberg (Kansas State University)

In commutative algebraic geometry, "spaces" are usually understood as locally ringed spaces, and sometimes as sheaves of sets on the category of affine schemes with the flat (fpqc) topology. The flat topology plays a particularly important role due to the so-called descent property.

The known examples of noncommutative spaces appear as

locally ringed spaces, like D-schemes of Beilinson-Bernstein, "virtual" noncommutative formal spaces of Kapranov, and affine noncommutative schemes of P. Cohn,

categories (regarded as categories of quasi-coherent sheaves) over a base category, like the Proj of a graded noncommutative ring and the flag variety of a quantized enveloping algebra,

functors from the category Rings of associative unital rings to Sets which are sheaves on Rings^op with respect to a version of the flat "topology."

The talk will be mostly concentrated on examples of interest of the third kind. Among them are noncommutative Grassmannians and flag varieties.

Large Problems

D-modules on noncommutative spaces.

The development of D-module theory on noncommutative 'spaces' (in particular quantum D-module theory) and related topics of noncommutative geometry and representaion theory is one of the main purposes of our project with V. Lunts. Some problems in this direction which seem to be interesting:

(1) Study the extension of the localization construction to quantized enveloping algebras Kac-Moody Lie algebras. Note that completing this program should give a new insight on the localization of non-quantized Kac-Moody Lie algebras which was constructed by Kashiwara.

(2) Prove an analog of Bernstein's theorem for differential operators on quantum flag variety. Then we have a nice category of holonomic quantum D-modules. Via the localization this singles out a "good" category of modules over a quantized enveloping algebra.

(3) One of important tools of commutative D-module theory is Kashiwara's theorem (the one about the equivalence of the category of D-modules on a subvariety of a smooth variety and the subcategory of D-modules supported on this subvariety) In a work with V. Lunts, we have proved a version of Kashiwara's theorem for so called hyperbolic algebras which include, for instance, algebras of differential operators on (skew) affine spaces. This class of algebras sufficient in commutative algebraic geometry is not sufficient for the noncommutative geometry. The problem here:

Find an analog of the Kashiwara's theorem on noncommutative schemes, or at least on a class of noncommutative schemes which include quantum flag varieties.

Noncommutative local algebra and its applications.

Thanks to the relation of the spectrum of categories with the traditional objects of representation theory, it is natural to regard representation theory as a part of noncommutative algebraic geometry. In this approach, traditionally the main problem of representation theory -- classifying irreducible representations of an algebra R (say, R=U(

) or U_q(

) for some semisimple or Kac-Moody Lie algebra

) -- is replaced by the problem of describing the spectrum of the category of R-modules. It follows from already obtained results that the classical representation theory (Verma and Harish-Chandra modules) can be reconstructed from the (partial) description of the spectrum. And there are important classes of irreducible representations which are out of the reach of the classical theory, but within the reach of noncommutative spectral theory.

A challenging problem in this direction:

(4) Find a description of the spectrum of the category of D-modules on the quantum flag variety of a semisimple Lie algebra.

This should also provide a description of holonomic irreducible D-modules on a quantum flag variety. Thanks to D-affinity of the quantum flag varieties, such description should shed light on the structure of the spectrum (in particular irreducible representations which are closed points of the spectrum) of quantized enveloping algebras of simple Lie algebras.

Smooth noncommutative spaces.

(5) At the present stage, we have basic notions and facts and a number of examples and constructions undestood in different degree. It is clear that to be able to move futher we need to develop fundamentals and techniques (in particular, those of computing invariants) for this new area of mathematics. From what we know already, it is clear that the subject of smooth noncommutative geometry merits this effort.

(6) In a work with M. Kontsevich, we have found natural constructions which produce examples of 'compact' noncommutative smooth spaces. Some of them have no commutative analogs. We have products, blow-ups, flag varieties, different noncommutative versions of Grassmannians etc.. In all examples of 'compact' noncommutative smooth spaces we have studied, the category of coherent
sheaves has global homological dimension less than or equal to 1, and all Hom-spaces and Ext¹-spaces are finite dimensional. This leads to conjecture that noncommutative smooth spaces are, homologically, 'curves'. So far we were not able to prove or disprove this conjecture. But in any case, there is a number of important examples of noncommutative smooth spaces of homological dimension 1, abelian versions of which (-- singular spaces from noncommutative viewpoint) have higher dimensions.

Fall 2000 meeting of the PNGS