April 25, 2010

Noncommutative algebraic geometry: II geometrization

Posted in Uncategorized at 10:10 am by noncommutativeag

In fact, in the real noncommutative algebraic geometry, we do not have the real space. What we do, we take a category as category of quasi coherent sheaves on some presume space. This comes from Grothendieck. This observation gave the strong tools to geometric representation theory.

Borel-Weil-Bott gave me deep impression. This theorem talked about the following things: if we consider line bundles on flag variety of Lie algebra. Then consider the line bundles parametrized by dominant highest weights, the global section of the line bundle on the flag variety will be isomorphic to irreducible finite dimensional highest weight representation. This theorem connects representation theory and algebraic geometry,further more, it gives us a new point of view to some problems. i.e, we can construct a geometric object from representation theory without assuming the existence of this geometric object!

To be precisely, let us consider the direct sum of irreducible highest and dominant representations. What we get, we get an graded algebra.The multiplication is given by R_{\lambda}\otimes R_{\nu}\rightarrow R_{\lambda+\nu}, which follows from Weyl character formular),we denote this algebra by R,the we consider the proj-category associated to this algebra, then according to Serre[1], we get a category which is category of quasi coherent sheaves on some projective variety and the coordinate ring of this projective variety is just R. On the other hand, if we consider the line bundles indexed by dominant weights on flag variety of Lie algebra,  we know these line bundles are ample, then follows from Hartshorne[2],we have a closed embedding to projective space. Then this flag variety is indeed a projective variety and according to Borel-Weil and Serre theorem[2], the coordinate ring is exactly R. Therefore, here, we reconstruct the flag variety without assuming its existence from a category. Actually, this is a very special case of reconstruction theorem [3],[4]. This kind of construction has very big advantage for it is very easy to work using Grothendieck machinery. Beilinson-Bernstein has told us that the flag variety of Lie algebra is D-affine which means the category of D-modules on flag variety is equivalent to representation category of Lie algebra[5]. This is a great example for geometric representation theory.

Therefore, we can try to extend this formalism to similar problems in representation theory. Such as representation theory of quantum groups 【1】. What we need to pay attention is in this situation, we are completely living in noncommutative world, which means that we do not have real space at all. However, we can still consider the direct sum of finite dimensional irreducible representation of quantum enveloping algebra U_{q}(g). We get an algebra again,but noncommutative this time. Then we also construct the proj-category associated with this algebra and define it as coordinate ring of quantized flag variety, then follow the same line as classical case, we consider the quotient category and define it as quantized flag variety. Then we consider the category of quantum D-modules on quantized flag variety. Lunts-Rosenberg-Tanisaki showed to us that this category is equivalent to representation category of quantum group[6],[7]. Moreover, we still have quantum version of Borel-Weil-Bott[6]【3】.

Here, what I want to talk about is not only the geometrization, but also want to point out what Borel-Weil really is from the point of view of noncommutative algebraic geometry, we know that at least,it survives in quantum case. In fact, Borel Weil gives a shift on projective scheme(or equivalently, proj-category). We denote it by  L(\lambda)\otimes -, it is an auto-equivalence functor, and all the functors like this form a Picard group of this category.[7](or in some cases, the automorphism group). This observation told us that some important facts in representation theory naturally give us components of algebraic geometry, I prefer to say the components of Grothendieck machine. 【4】.This phenomenon is not accidental, actually we have more. For example, Weyl character formula is essentially Grothendieck Riemann Roch for flag variety[8]【5】, Kadzdan-Lusztig formula is essentially Grothendieck Riemman Roch for category O[9]. (This  observation comes from A.Rosenberg) .Kirillov character formula can also be interpreted as Grothendieck Riemman Roch【6】. Verma modules(and correspondence irreducible quotients and related induction process) are closely related to the morphism between spectrum and Beck’s theorem [10],[11]in noncommutative algebraic geometry.

My opinion(of course, highly non-original and obvious) is that all the similar formulas in representation theory(finite dimensional, infinite dimensional, quantum,super)are the “same”. They are instances of “components” of Grothendieck machine. More philosophical, I intend to think the similarities across the various branches of mathematics is due to the hidden machinery behind rather than take it as some mystery in mathematics.I think representation theory and algebraic geometry are the two sides of a coin.

Reference:

[1]:J.P.Serre Faisceaux Algébriques Cohérents (FAC)
[2]:Robin Hartshorne: Algebraic Geometry
[3]:P.Gabriel:Des Catégories Abéliennes
[4]:A.Rosenberg:The spectrum of abelian categories and reconstructions of schemes
[5]:Alexander A. Beilinson and Joseph N. Bernstein:A proof of Jantzen conjectures, Preprint, 1986
[6]:V. A. Lunts1 and A. L. Rosenberg:Localization for quantum group
[7]:A.Rosenberg: Differential Calculus in noncommutative algebraic geometry II:D- Calculus in the braided case. The localization of quantized enveloping algebras. Tanisaki:Beilinson-Bernstein correspondence for quantized flag manifolds
[8]:Bernhard Koeck:The Grothendieck-Riemann-Roch theorem for group scheme actions
[9]:M.Kontsevich and A.Rosenberg: Noncommutative schemes and spaces.monograph coming soon.
[10]:J.Beck:http://ncatlab.org/nlab/show/monadicity+theorem
[11]:A.Rosenberg: spectral,associtated points and representations
Remark:
【1】：V.Kac and D.H.Peterson wrote a paper ” Regular functions on certain infinite dimensional group”. They had some “noncommutative mind”. They considered the direct sum of irreducible integral contragradient modules and defined it as the coordinate ring of flag variety of Kac-Moody algebra without assuming the existence of it.
【2】： In some not very noncommutative case, we can still expect that there is real commutative space whose category of quasi coherent sheaves on it is equivalent to proj-category of noncommutative homogeneous ring. (for example, PI-ring). Look:J.T. Stafford; M. Van den Bergh  Noncommutative curves and noncommutative surfaces.
【3】：In fact, Borel-Weil and Bott are two works, Borel Weil discussed the zero cohomology case, which latter considered about higher cohomology of line bundles with non-dominant weight, it is essentially Serre duality
【4】：
【5】：Hirzebruch knew this observation in fifties last century but    he did not write it out.
【6】：A.Rosenberg will prove this in the course next semester using his machinery.