By David Mehrle
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Extra resources for 2-Kac-Moody Algebras
The idempotent completion (or Karoubi envelope) of a category C is the category C9 (also denoted KarpC q), where • objects are pairs pA, eq where A is an object of C and e : A Ñ A is idempotent; • morphisms f : pA, eq Ñ pA1 , e1 q are morphisms f : A Ñ B in C such that e1 f e “ f . It is not immediately obvious (at least not to me) that this is a category, but this is easy to check. 3. C9, as defined above, is a category with identity arrows given by 1p A,eq “ e : pA, eq Ñ pA, eq and composition inherited from composition in C Proof.
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. 1 Injectivity of γ The argument for injectivity of γ is essentially a dimension counting argument to show that the graded dimension of the space of graded Homs in U9q pgq matches the value of the semilinear form on U9 q pgq.
Here are some examples of idempotent 2-morphisms in Uq pgq. e`i,m “ m e`i,m “ p´1qp 2 q λ i i i i i λ i i i The proof that these are actually idempotents is given in [3, Lemma 5], and the specific equality that we seek is in particular the left hand side of Equation 11 on the bottom of page 6 (our idempotents are mirrored left-to-right from the ones in ). The proof is not particularly hard, but it is a tedious manipulation of diagrams and I’m getting sick of typesetting diagrammatic proofs at this point.