By David Mehrle
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Extra resources for 2-Kac-Moody Algebras
This is likely the least obvious of the five things we need to show. Consider the elements down the first column of the matrix αβ. 16. 6. 7. αs β s “ 11λ . Proof. 27). 28). 8. αs β t “ 0 for s ‰ t. Proof. 27). 17. 3, categorifying the commutator relation of A U9 q pgq. 2) . This is where we need to work within the idempotent complete category U9q pgq as opposed to the category Uq pgq. As we did when categorifying the commutator relations, we need to massage the quantum Serre relations to be in a form more amenable to categorification.
For any morphisms X, Y with the same domain and codomain, we define à grHompX, Yq “ HomUq pgq pXttu, Yq. t PZ The graded dimension of one of these graded homs is the generating function for the dimensions of the direct summands. ÿ grdim grHompX, Yq “ dim HomUq pgq pXttu, Yq ¨ qt tPZ By analogy to the situation of representations of finite groups, this graded dimension should match the value of the U9 q pgq semilinear form on X and Y; this is what Lauda was trying to achieve when constructing the categorification of slp2q in .
But it doesn’t matter because we will be taking the Grothendieck groups of the hom-categories of U9 q pgq where the direct sum is formal and exact sequences don’t exist. As with most definitions that we make on 1-categories, we can lift the Grothendieck group construction up to the setting of 2-categories by taking Grothendieck groups of the hom-categories. However, instead of a Grothendieck group, we get a Grothendieck additive category. 3. If C is an additive 2-category, then K0 pCq is the additive category with the same objects as C, and morphisms A Ñ B given by K0 pCpA, Bqq.
2-Kac-Moody Algebras by David Mehrle