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--- blog:2025-01-01 [2025/01/01 08:18] – created pzhou
+++ blog:2025-01-01 [2025/01/02 07:23] (current) – pzhou
@@ Line 6: / Line 6: @@
 ===== CK's construction =====
+I am reading their old paper, almost 20 years old, https://arxiv.org/pdf/math/0701194
+First they give a quick review of the Reshtikhin-Turaev theory,  that assigns to tangle  $T$  a linear map  $\psi_T: V^{\otimes n} \to V^{\otimes m}$ .
+Then they say what a weak categorification is, which is a graded dg category, that assigns to a tangle  $T$  some functor  $\Psi_T: D_n \to D_m$ , (why they say this only upto isomorphism?)
+Bernstein-I.Frenkel-Khovanov conjectured a weak categorification, and proved by Stroppel. Here  $D_n$  is some direct sum of categories associated to category  $O$  for  $gl_n$ . Khovanov's  $D_n$  is as graded module over graded algebra, combinatorial approach.
+In this paper, CK uses  $D_n = D(Y_n)$ . They also construct, from a tangle  $T$ , a functor  $\Psi(T)$ , by composing the elementary functors: merging  $F_n^i$ , splitting  $G_n^i$ , and braiding  $T$ . Here merge and split between  $Y_n$  and  $Y_{n-2}$  are realized by a correspondence  $X_n^i$
+What is the space  $Y_n$ ? First, we fix an  $(N, N)$  nilpotent element  $z \in End(\C^{2N})$ . (From this data, we can build an  $N$ -step flag, by taking kernel of  $z^k$ . Hold that thought.) Then, we build a 'weed',  $L_1 \In L_2 \cdots \In L_n$ , where  $\dim L_i=i$  inside  $\C^N$ . Such that  $z L_i \In L_{i-1}$ .
+Let's think a bit. Can we take the limit  $N$  goes to  $\infty$ ? Yes, say, the polynomial ring  $\C[x]$  as a vector space is the limit of $\C[x]/(x^N) $. Here, we can say, rank$ 2 $vector bundle over an artin disk$ Spec \C[x]/(x^N) $. But, what are those$ L_i $? We can start by thinking about$ L_1 $, we need$ z L_1=0 $, so that means$ L_1 $needs to be in$ ker(z) $. I would like to say$ z = \d_x $, so that$ L_1$ is some 'flat' section. OK. What is  $L_2$ ? We can parametrize  $L_2$  by saying, choose a generator  $e_1(x)$  for  $L_1$ , such that  $\d_x e_1(x)=0$ , then choose a section  $e_2$ , so that  $\d_x e_2 = e_1$ , then  $L_2$  is generated by  $e_1, e_2$ .  $L_3$ , we want something  $e_3$ , such that  $\d_x e_3 = e_2$ . No, this is not what it should be.
+There are two models for infinite dimensional vector space where an operator acts locally nilpotently, one is  $\d_x$  on  $\C[x]$ , another is  $z\cdot$  on  $\C[z,1/z] / \C[z]$ . One can certainly take bundles over this. The second one seems more amicable.
+Question: if  $z$  is an nilpotent endomorphism of  $V$ , and  $W \In V$  is a subspace invariant under  $z$ , how do I know how large is  $ker(z: V/W \to V/W)$ ? OK, not sure.
+In our case, for a generic element in  $Y_n$ , an generic element in  $L_i$  takes  $i$  step to die under action by  $z$ . I want to believe that,  $L_i$  is just a lattice, no better and no worse than any other lattices in $\C[t,t^{-1}]^2 $. This should be the best description. If that is the case, then$ Y_n $is an iterated$ \P^1$-bundle. Let's see if that is true.
+Yes, that is true, see http://arxiv.org/abs/0710.3216v2 their second paper on  $sl(m)$  case.
+So why did Cautis-Kamnitzer only deal with  $sl(m)$ ? What's so hard about general case?