Vertex embeddings of quantum groups via quivers

Question

Let U_q be a quantised enveloping algebra of type affine ADE (untwisted). By the loop presentation of U_q, we see that for each vertex of the finite Dynkin diagram, there is an inclusion U_q($\hat{sl_2}$)→U_q.

Now let us restrict to the positive part U⁺ of U_q. It is well known how to construct U⁺ from the representations of the affine Dynkin diagram (given some orientation so that it becomes a quiver). This proceeds either by the Hall algebra approach or Lusztig's geometric approach with perverse sheaves.

My question is: Can we see the appearance of the vertex embeddings discussed in the first paragraph via the quiver perspective?

The obvious approach of trying to choose an orientation of our affine quiver so that it has a full subcategory (with objects of the correct dimensions) equivalent to the category of representations of the Kronecker quiver doesn't seem to work (eg look at E7 and the vertex of valence 1 closest to the central node).

Answer 1

Let me first point out, as it confused me initially, that one expects a Dynkin diagram's worth of these embeddings, and the quiver involved is affine. So we can't have a bijection between vertices of $Q$ and these embeddings.

Given $Q$ an oriented extended Dynkin diagram, I can describe an appealing set of abelian categories, each equivalent to the representations of a Kronecker quiver, and which seem (in some sense) to be arranged in a Dynkin configuration. I have not checked whether or not they behave nicely in the Hall algebra.

Let $Q$ have $n+1$ vertices (so $n$ indexes the Dynkin type). For convenience, let me assume that $Q$ is not itself a Kronecker quiver, so $n>1$.

The subcategories of rep $Q$ which are isomorphic to the representations of a Kronecker quiver are obtained as follows.

Inside rep $Q$, there are one, two, or three "inhomogeneous tubes" of regular representations. (If you like to think about rep $Q$ as being derived equivalent to coherent sheaves on a weighted $\mathbb P^1$, the inhomogeneous tubes correspond to the points on the $\mathbb P^1$ with non-trivial weight.)

Number the tubes $\mathcal T_1, \dots, \mathcal T_N$ (with $1\leq N \leq 3$). At the bottom of each tube, you have quasi-simple objects (indecomposable objects not admitting an injection from any other object in the tube). Number the quasi-simples at the bottom of tube $i$ as $S_{i1}, S_{i2},\dots, S_{ir_i}$.

It is a fact that $\sum_i r_i = n + N-1$.

Let $\mathcal T$ consist of the $N$-tuples $(t_1,\dots,t_N)$ with $1\leq t_i \leq r_i$. This is going to index the embeddings of the representations of the Kronecker quiver.

For $t \in \mathcal T$, define $X_t$ as follows: take the direct sum of the quasi-simples at the bottom of each tube, except $S_{1t_1}, S_{2t_2},\dots$, i.e., except the ones indexed by $t$.

Define the subcategory $\mathcal Y_t$ to consist of the representations $Z$ of $Q$ such that Hom$(Z,X_t)=0$ and Ext${}^1(Z,X_t)=0$. (We say $\mathcal Y_t$ is left perpendicular to $X_t$.)

Now $\mathcal Y_t$ is equivalent to the representations of the Kronecker quiver, and in fact, these subcategories are the only ones for which this is true. I can provide more details if you want; the quick version is that such an abelian subcategory can necessarily be realized as a left perpendicular of something, and that something must be regular (or the resulting category would be too small) but it must also be big enough to knock all the inhomogeneous tubes down to being homogeneous.

This defines $\prod_i r_i$ subcategories, which is more than we wanted. A reasonably Dynkin choice would be to take a fixed $t\in \mathcal T$, and then consider the elements of $\mathcal T$ which you get by varying only one of the indices. This gives you $1+\sum (r_i -1) = 1+ (n+N-1) -N=n$ embeddings, and they are naturally arranged in a tree with branches of length $r_i$, which is a Dynkin diagram of the appropriate type.