By David Mehrle
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Extra info for 2-Kac-Moody Algebras
The authors cite [2, 3], but I couldn’t find this identity in either of those papers. I couldn’t figure out a proof of this either, although not for lack of trying. In the idempotent completion C9 of ` a an ˘ additive category C , the direct sum 0 pA, aq ‘ pB, bq is the object pA ‘ B, 0 b q. Therefore, when we take the direct sum rmsqi ! ¨ X˘i,m , it appears that the idempotent associated to it is the diagonal matrix E with the diagonal composed of idempotents e˘i,m . An isomorphism rmsqi ! ¨ X˘i,m “ prmsqi !
S — — .. — – . 31). 31) “ i i “ idE`i´i 1λ It is significantly harder to show that αβ is the identity, not only because the product αβ is a matrix of diagrams, but also because it’s harder to deduce that the elements of this matrix are either zero or identities. To show that αβ is the identity matrix, there are several things that we need to show, each of which has been outsourced to a lemma. λ • ´ λ . 4. “ i i i i • The off-diagonal terms in the first column vanish. 6. • The off-diagonal terms in the first row vanish.
Then f ‘ h % g ‘ k. Proof. Let a : X ùñ A an b : X ùñ b be 2-cells. Denote by HomC px, yq the collection of 2-cells between 1-cells x and y. The fact that f % g gives a bijection HomC p f a, bq – HomC pa, gbq, and similarly, h % k gives a bijection HomC pha, bq – HomC pa, kbq. Then we may compose natural bijections to get a natural bijection HomC pp f ‘ hqa, bq – HomC p f a ‘ ha, bq – HomC p f a, bq ‘ HomC pha, bq – HomC pa, gbq ‘ HomC pa, kbq – HomC pa, gb ‘ kbq – HomC pa, pg ‘ kqbq. 7, an adjunction f ‘ h % g ‘ k.