By David Mehrle
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Extra info for 2-Kac-Moody Algebras
If pB, dq Ñ pC, cq is another morphism of C9, then we want to show that composition of f and g is well-defined, or that cg f e “ g f . But notice cg f e “ cpcgdqpd f eqe “ pcgdqpd f eq “ g f . Finally, composition inherits associativity from C . We also want to know that C9 is actually idempotent complete. 4. Every idempotent splits in C9. 45 Proof. Suppose e : pA, aq Ñ pA, aq is an idempotent in C9. Then we have that aea “ e and ee “ e, aa “ a. Furthermore, e “ 1p A,eq e “ ae and similarly, e “ ea.
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. , we can rewrite the quantum Serre relations as 1´aij ÿ k “0 ˜ p´1qk k E˘ i rksqi ! ¸ ¨ E˘ j ˝ 58 1´aij ´k E˘i r1 ´ aij ´ ksqi ! ˛ ‚“ 0 And finally, we rearrange both sides so that all of the terms on each side are positive. Z 1´aij 2 ^ ˜ ÿ 2k E˘ i k “0 r2ksqi !
So the lack of orientations on strands in the proofs given in [2, 3] doeesn’t matter. Finally, we can prove the existence of the homomorphism γ. 1. 1). 2. 3. 12. 3 9 A Uq pgq Ñ K0 pU9q pgqq. 1; we first show that γ is injective, and then that γ is surjective. As always, let’s start with the easier thing to prove, which is in this case injectivity. 2), while the proof of surjectivity doesn’t need it at all. 1. 4], it is shown that when g “ slpnq the nondegeneracy condition holds. According to , Webster  showed that nondegeneracy in fact holds for any symmetrizable Kac-Moody algebra.
2-Kac-Moody Algebras by David Mehrle