Alex Hoffnung

Things I need to work out

  • Centers

I can not seem to figure out what the idea of Drinfeld center is supposed to be about over at the geometric infinity-function theory journal club page. There are probably tons of other things I should be trying to really understand instead, but I am going to be stubborn and fixate on this one.

I want to know what the Drinfeld center is and also why “the Drinfeld center is not a “sub” in any reasonable sense (except in the loose sense that it’s a categorical limit).”

So maybe I should just start listing off different notions of center and trying to understand these and just hope for the best.

What types of centers do I know about?

A monoid MM has a center.

A group GG has a center.

A ring/algebra AA has a center.

These can all be interpreted as 11-object categories where a morphism is in the center iff it commutes with all other morphisms. This, of course, makes sense since all morphisms have the same source and target.

Each of these algebraic objects 𝒞\mathcal{C} has an injection from the center Z(𝒞)𝒞Z(\mathcal{C})\hookrightarrow \mathcal{C}.

Bruce, can we finish our conversation here?

“center of Rep(G) = Rep (loop groupoid of G)” which seems to summarize everything

In this case, it’s more like we have Rep(G) embedding inside Rep (/\G)

Alex, I remember Jim Dolan’s telling me stuff about Drinfeld doubles a while back. My memories of the conversation are not vivid, but I thought I took away the memory that we are dealing with the ordinary center of a monoidal category MM, which is not a sub-thing of MM. Rather, the objects are objects xx of MM together with an isomorphism xyyxx \otimes y \to y \otimes x natural in yy … well maybe you know all this already, and you’re driving at something else? – Todd

Hi Todd, I know a little about the center of a monoidal category. Your comment is still helpful to me. My comments came from trying to understand part of the Ben-Zvi/Francis/Nadler paper. I was just getting a little overwhelmed and decided to take a step back and see if a big picture would dawn on me. I can’t really say if that has happened yet. Maybe I should bug Jim a bit.


In these cases, we interpeted the composition in the category as the binary operation which we could query for commutativity. As these are all cases of monoids with extra structure, we really want to think of the composition as a monoidal product. This leads us to a more general notion of the center of a category.

We consider the periodic table of kk-tuply monoidal nn-categories described here This generalized center takes us vertically down the columns of the periodic table. Taking one object versions of the gadget in a certain square takes us southwest down the chart.

From poking around a bit it seems that having these ideas down we are supposed now start thinking about Hopf algebras.

Gotta go eat and for a walk…be back soon. I had a meatball sub, tea, orange juice and now I am back.

Braided Hopf algebras provide solutions to the braid equation. The Drinfel’d quantum double construction is a way to get our hands on these starting with finite dimensional Hopf algebras with invertible antipodes.

The quantum double D(H)D(H) is the bicrossed product of the Hopf algebra HH and its dual H *H^*. See Kassel for definitions. The quantum double contains HH and its dual as Hopf subalgebras. Given a finite group GG there is a double construction for its group algebra.

Theorem: Left D(H)D(H)-modules are the same as crossed HH-bimodules.

(Need to say what this means and possible relationship to Drinfel’d double.)

See for the categorification of the Drinfel’d double of a finite group.

Things I will forget to keep reading if I do not write down some notes here.

commutative rings \rightarrow categorical rings

Keep track of extra information (e.g. quotients) by adding extra isomorphisms to the obvious discrete categorification of the ring

What is Serre’s intersection formula?

To understand the entire formula we need commutative ring structures on \infty-groupoids

Two approaches: 1. topological spaces (or simplicial sets) w. comm ring structure axioms satisfied on nose and addition, multiplication are given by continuous maps

This forms the \infty-category SCR (see section 4.1)

  1. commutative ring structures on \infty-groupoids, axioms hold up to coherent homotopy

This gives theory of (connective E E_\infty-rings)

For a topological ring we care about homotopy structure not topology

What is correct notion of “generalized rings”?

  1. E E_\infty-ring spectra - apply algebraic topology to commutative algebra

  2. simplicial commutative rings - apply commutative algebra to algebraic topology (in particular, stable homotopy theory)

K-theory and other cohomology theories can be endowed with E E_\infty-structures


scheme - “locally looks like Spec AA, for some commutative ring AA

derived scheme - “locally looks like Spec AA, for some commutative simplicial ring AA

Given a geometry GG and a topological space XX, there is an associated theory of GG-structures on XX.

“A GG-structure on XX is a sheaf O XO_X endowed with some operations.”

For appropriate GG, we recover the classical theory of ringed and locally ringed spaces.

Page 6 gives some general overview for the paper which will be helpful.

Go back here to see different correspondences related to the Hecke algebra

Borel-Weil Example

Following Joel Kamnitzer’s notes from the summer school on Geometric Representation Theory in Ottawa, we attempt an example.

Consider the highest weight λ=ω 1=(1,0,,0)\lambda = \omega_1 = (1,0,\ldots,0). Then we have the standard representation V λV_\lambda.

According to the line bundle construction L(λ)L(\lambda) on the flag variety Fl( n)Fl(\mathbb{C}^n):

L(λ) V =(V n/V n1) λ 1(V n1/V n2) λ 2(V 1/V 0) λ n L(\lambda)_{V_\bullet} = (V_n/V_{n-1})^{\otimes \lambda_1} \otimes (V_{n-1}/V_{n-2})^{\otimes \lambda_2} \otimes\cdots \otimes (V_1/V_0)^{\otimes\lambda_n}

we have L(λ) V L(\lambda)_{V_\bullet} = V n/V n1V_n/V_{n-1}. We can construct a map from the flag variety to the projective space n1\mathbb{P}^{n-1} = {H ndimH=n1}\{ H\subset \mathbb{C}^n | dim\; H = n-1\} taking a flag to the (n1)(n-1)-dimensional component.

We then have the following pullback diagram:

L(λ) Q Fl( n) f n1, \array{ L(\lambda) &\to& Q \\ \downarrow && \downarrow \\ Fl(\mathbb{C}^n) &\stackrel{f}{\to}& \mathbb{P}^{n-1} } \,,

where ff is the map descried above. Here Q has the same fibers as the line bundle in the sense that L(λ) x=Q f(x)L(\lambda)_x = Q_{f(x)}.

Now to see the Borel-Weil theorem at work we consider the map

Γ( n1),Q)Γ(Fl( n),L(λ)) \Gamma(\mathbb{P}^{n-1}),Q)\to \Gamma(Fl(\mathbb{C}^n),L(\lambda))

This takes a section ss to the section

xs(f(x)). x \mapsto s(f(x)).

We have the following isomorphisms:

nΓ( n1),Q)Γ(Fl( n),L(λ))=V λ \mathbb{C}^n \cong \Gamma(\mathbb{P}^{n-1}),Q)\cong \Gamma(Fl(\mathbb{C}^n),L(\lambda)) = V_\lambda

The cool part is that these isomorphisms are intertwiners.

Ginzburg Construction

To understand the Ginzburg construction we first need to understand partial flag varieties, Springer fibers and Borel-Moore homology. But doing things in order would be boring. So we skip to the main construction and fill in the details as we go:

OK, so we already need to know about partial flag varieties and Springer fibers. Let’s define these:

We first fix n,Nn,N\in \mathbb{N} and define the nn-step flags in N\mathbb{C}^N:

Fl n( N)={0=V 0V 1V n= N} Fl_n(\mathbb{C}^N) = \{0 = V_0\subseteq V_1\subseteq \ldots \subseteq V_n = \mathbb{C}^N\}

For partial flags we consider μ n\mu \in \mathbb{N}^n such that μ 1++μ n=N\mu_1 + \cdots + \mu_n = N, then define

Fl μ( N)={0=V 0V ndimV idimV i1=μ i}. Fl_\mu(\mathbb{C}^N) = \{0 = V_0 \subset \cdots \subset V_n | dim\;V_i-dim\;V_{i-1} = \mu_i\}.

To define Springer fibers we need to consider a map from the cotangent bundle of the flag variety to the nn-nilpotent operators on n\mathbb{C}^n. First we take a look at the cotangent bundle:

T *(Fl n( N)={(X,V )V Fl n( N),XEnd( N),X(V i)V i1,i} T^*(Fl_n(\mathbb{C}^N) = \{(X,V_\bullet)|V_\bullet \in Fl_n(\mathbb{C}^N), X\in End(\mathbb{C}^N), X(V_i)\subset V_{i-1}, \forall i \}

Then the fibers of the obvious map below are the Springer fibers:

T *(Fl n( N))N n( N)={xEnd( n)x n=0} T^*(Fl_n(\mathbb{C}^N)) \to N_n(\mathbb{C}^N) = \{x\in End(\mathbb{C}^n)|x^n = 0\}

The fibers are denoted:

Fl n( N) X={V X(V i)V i1} Fl_n(\mathbb{C}^N)^X = \{ V_\bullet | X(V_i) \subset V_{i-1} \}

This breaks up over partitions as:

Fl n( N) X= μFl μ( N) X Fl_n(\mathbb{C}^N)^X = \bigcup_\mu Fl_\mu(\mathbb{C}^N)^X

Moving on:

We want to construct a representation on a certain vector space. This comes from Borel-Moore homology. This means my dreams of pushing the technical stuff to the side have been completely dashed. So to construct a representation on H (Fl( N) X),XN n( N)H_\bullet(Fl(\mathbb{C}^N)^X), X\in N_n(\mathbb{C}^N) we need to describe this homology theory.

Borel-Moore homology?

We want to construct a homology theory for topological spaces and/or complex algebraic varieties Y. This functor is concerned with locally compact spaces and there are several ways to define this. We will take a first stab at it by locally finite chains. Here we will sketch the idea and eventually should formalize this on the page for Borel-Moore homology. In the meantime the Wikipedia article plus its references and the Chriss-Ginzburg book Representation Theory and Complex Geometry provide plenty of information.

H (Y)=(possiblyunbounded)ichainsinY(withcoefficientsin) H_\bullet(Y) = (possibly\; unbounded)\; i-chains\; in\; Y\;(with\; coefficients\; in \;\mathbb{C})
H i(Y)=idimsubspaceofYwithoutboundary(possiblysingularorunbounded) H_i(Y) = i-dim\; subspace\;of\; Y \;without\; boundary \;(possibly \;singular\; or\; unbounded)

The construction starts by triangulating the space and considering the vector space of formal sums of ii-simplices.

Of course, one wants independence of triangulation so we need to take a direct limit of these vectors spaces over all refinements of the triangulation. The boundary map then comes from simplicial homology and we have a chain complex. The homology of this complex is the Borel-Moore homology.

To anyone reading this: These are notes from the University of Ottawa/Fields Institute summer school which took place last week (if you are reading this before Friday July 3). Adam Katz and I are trying to make our way through the notes from several of the classes. Anyone is welcome to add comments, corrections, insights or just join our discussion. Actually it would be great if we had some more people involved. I think the plan is something like: 1) Understand Kamnitzer’s lectures on Borel-Weil, Ginzburg construction and geometric Satake, then move to Savage’s lectures relating the geometry from Kamnitzer and the combinatorics from Kang’s lectures. So Kang’s lectures will hopefully serve as background reading to understand Savage’s lectures.


Other projects:

1) Understand q-Schur algebras and relationship to representation theory of Hecke algebras

2) Learn lots of things in Ginzburg-Chriss.

3) Write my thesis!

4) Write everything else I should be writing.

5) Get a job! :)

I guess 3) and 5) are taken care of now.

If anyone wants to talk about the first item that would be great. I guess this is my new recruiting station to get people to talk to me about math.

category: people

Revised on July 2, 2010 04:50:42 by Alex Hoffnung (