User:ILearn/Quantum groups

Classical story
We can define algebraic group as a group object in the category of algebraic varieties, that is some algebraic variety with algebraic maps
 * $$ \mu: G\times G\to G$$
 * $$ {}^{-1}: G\to G $$

satisfying appropriate axioms. Note that even though these groups 'consist' of points, this definition is already stated just in terms of functions. Examples are GLn, the general linear group of invertible matrices over a field (since the law of multiplication is clearly algebraic there) and its algebraic subgroup SLn.

Representations of group $$G$$ (which is the same $$G$$-modules) naturally make category Rep(G) with the following additional structures: Here's the name for something satisfying these properties: symmetric tensor category with fiber functor.
 * it's an abelian category: there is the notion of exact sequences
 * tensor product: for any $$ M,N \in \mbox{Ob}$$ there is $$ M\otimes N$$ satisfying some axioms (in particular, it's a functor by both arguments)
 * there is 'flip': $$ M\otimes N \to N \otimes M$$ (which just sends $$ a\otimes b$$ to $$ b\otimes a$$), it's a functor and its square is identity
 * there is a a forgetful functor to vector spaces (by forgetting the action of G)

Now when you have constructed the category Rep(G) you ask whether it's possible to reconstruct G back. Tannakian duality says that G is the group of automorphisms of powers of fiber functor, and that this construction would work for any abovementioned category. The bottom line is that instead of studying groups you can study their representation categories (in line with Pontryagin duality and Langlands program).

What is quantum group?
According to algebraic geometry, affine variety corresponds to its ring of functions. We can write this as an equivalence of categories:


 * affine varieties/C $$ \leftrightarrow $$ (commutative C-algebras)op

Putting group structure on the LHS gives extra structure on the RHS, see Hopf algebra:


 * affine algebraic groups/C $$ \leftrightarrow $$ (commutative C-Hopf algebras)op

Commutativity of group law would translate to cocommutativity of Hopf algebra.

Let's make a guess for what are noncommutative varieties:


 * NC affine varieties/C $$ \leftrightarrow $$ (all C-algebras)op

the group object in this category is what we call quantum group:


 * quantum groups/C $$ \leftrightarrow $$ (all C-Hopf algebras)op

Examples

 * For any Lie algebra $$g$$ the universal enveloping algebra $$ U(g)$$ provides an example of noncommutative Hopf algebra, so producing interesting quantum groups. All these are cocommutative though.
 * (Drinfeld, Jimbo, mid-80s) $$\mathcal O_q(SL_n)$$ --- deformation of Hopf algebra of functions on $$SL_n$$. Here $$q = e^\hbar\in\mathbb C^*$$ and the algebra reduces to classical one  for $$q=1$$. When $$q$$ is a root of unity things become special ---  algebras become closer to commutative, representations become more finite-dimensional, etc. This can also be done for any semisimple algebra.

$$\hbar$$ here comes from quantum physics, since it was the first time where mathematicians encountered deformations of commutative algebras

Braided tensor categories
Representations of Hopf algebra H are H-comodules. Fot the same reasons they form a category Rep(H) with a tensor product and fiber functor, but the flip
 * $$ \sigma_{M,N} : M\otimes N \to N \otimes M$$

now doesn't satisfy $$ \sigma^2 = 1$$.

Deligne has observed that one can obtain n-braids as fundamental group of configuration space of n distinct points on a plane. In this manner one obtains all braids whose map from start to endpoints is trivial (so-called pure braids). The pure braid group is the thing that acts on tensor products $$V_1\otimes \dots \otimes V_n$$ in Rep(H) with $$\sigma_{V_i, V_{i+1}}$$ as generators of elementary braid twists satisfying braid relations. That's why the category is called braided tensor category.

Related examples came from statistical physics where elementary braid twists allow to exactly solve some spin chain models and go under the name of R-matrices.

Compactifying the plane, we can view this configuration space as moduli space of $$n+1$$ marked points on $$\mathbb CP^1$$, the thing sometimes denoted by $$ \mathcal M_{0,n+1}$$. Representations of fundamental group are then flat connections on vector bundles (something traditionally called local systems) over that space. Some particular connections coming from conformal field theory are Knizhnik-Zamolodchikov connections. Bundles over this space give rise to (genus 0) modular functors by associating a vector space to any curve of genus 0 with n+1 points.

Geometry and representation theory
We're temporary back to classical groups. It turns out that representation theory of group G is closely connected to some particular geometry, that of flag variety F of group G. This space generalizes $$P^1$$ for $$SL_2$$ and space of flags in vector space for $$SL_n$$ and its points are all Borel subgroups (generalizations of uppertriangular subgroups of $$SL_n$$).

Borel-Weil-Bott theorem

 * You can get all simple G-modules as global sections of G-equivariant line bundles on F
 * Example: global sections of line bundles $$\mathcal O(n)$$ on $$\mathbb CP^1$$ are polynomials of degree n in two variables that furnish weight n (n+1-dimensional) representations of $$SL_2(\mathbb C)$$. For negative n nontrivial representations happen in the H1 instead

Beilinson-Bernstein localization

 * Category of D-modules on the flag variety is equivalent to category of $$\mathfrak g$$-modules
 * Original paper is short and good

What will be done

 * 1) Define a quantum flag variety for quantum  groups
 * 2) Prove a quantum version of Borel-Weil-Bott
 * 3) Construct D-modules on quantum flag variety and prove quantum localization

Links
Useful books: Physical perspective:
 * Kassel. Quantum groups
 * Jantzen. Quantum groups
 * Chazi-Grossey. Quantum groups
 * Etingof, Shiffman have book that emphasizes deformation point of view
 * Classic Baxter's book
 * Jimbo. Algebraic Analysis of Solvable Lattice Models

To appreciate the vastness of the subject, note that arXiv has whole subsection math.QA for quantum algebra (some of it is Alain Connes' Noncommutative geometry, which is a different, but somewhat related approach)