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2024-09-11 quiver hecke algebra
After so many years, let me read up on the 'convolution algebra' presentation for quiver hecke algebra, because after all, hecke algebra originates from convolution on flag variety, and quiver hecke algebra is a generalization.
2024-08-24 B-side for knots
Mina has the whole categorifications on A-side and B-side, whereas Cautis-Kamnitzer also had some earlier B-side construction. I need to understand the relations.
2024-07-23
- Goal: understand Yetter-Drinfeld module, Drinfeld double and then Andy Manion's comment paper
2024-07-21
reading Etingof on quantum group
2024-05-20
What's new?
- What is the central charge formula on the B-side? What's the relationship with stability condition? SOD?
- What's Xin Jin's story? How does the open torus embed?
2024-01-14
a category with a notion of 'equal', or 'isomorphism', or 'quasi-isomorphism'.
- category of set? then isomorphism, bijection
- category of vector space? isomorphism
- of chain complex of abelian groups? Well, quasi-isomorphism may not be.
- the category of dg categories. we really just want to do equivalence of category, not isomorphism of categories. that requires that we have two ways functors.
Let's go simple. Suppose you have a sheaf on $\R$, and you want to test whether it is locally constant, so you do restriction. what's the condition? to test $(x,\xi)$ is not in the SS, it means there is an open set of $U$, such that $F(x-\epsilon, x+\epsilon) \to F(x-\epsilon, x)$ is a nice arrow.
Are we working in the homotopy category of dg categories?
Suppose the category $C$ have the notion of isomorphisms. Like set, or homotopy category of dg categories.
Then, we need to say, what does locally constant mean, indeed restriction to smaller open induces isomorphism in $C$.
Does Ho(dg-cat) admits arbitrary limit and colimit? Should be. The cat dg-cat embed into Ho(dg-cat), since it is a localization.
2024-01-05
In the simplest setting, we have mirror symmetry for $Coh(\C^* \times \C^2)$.
Next, we are going to take symmetric power.
Do you remember what happens when two eigenvalues collide? No, don't do Hermitian matrices, that will never be nilpotent. What if you have a matrix that looks like $( (1,1), (0, 1+x) )$. What is the eigenvector for e.v. $1+x$? How about $(1,x)$? Eigenvector for $\lambda = 1$, is $(1,0)$. So you see, when $x \to 0$, the two eigenspaces also collide!
The key question is: what is the superKLRW algebra?
From the A-side, let me guess, we have 'fiber product' of two MC. So, when y1 = y2, in the fiber, we need to have x1 = x2, and z1 = z2 (the new pair of fiber coord). And we need to remember the ratio of (x1-x2)/(y1-y2), and (z1-z2)/(y1-y2).