By David Mehrle
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Additional info for 2-Kac-Moody Algebras
Let’s check that F9 On arrows f : pA, aq Ñ pB, bq, define Fp respects composition and identities. 9 9 Fp1 p A,aq q “ Fpaq “ f a Fpaqga “ f a ga f a ga “ 1Xe “ 1 F9 p A,eq Given s : pA, aq Ñ pB, bq and t : pB, bq Ñ pC, cq, note that t “ t1pB,bq “ tb. Then, 9 Fpsq 9 Fptq “ f c Fptqgb f b Fpsqga “ f c FptqFpbqFpsqga “ f c Fptbsqga “ f c Fptsqga 9 “ Fptsq. 46 9 “ F by construction. Finally, FI If F, G : C Ñ D both split idempotents, there is also a correspondence 9 between natural transformations F ùñ G and F9 ùñ G.
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. We need to check that composition is well-defined and associative, and moreover that the proposed identity morphisms are actually identities. Given f : pA, eq Ñ pB, dq in C9, we have 1pB,dq f “ d f “ dd f e “ d f e “ f and f 1p A,eq “ f e “ d f ee “ d f e “ f . 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 .
First, let’s define a Zrq, q´1 s-module structure on each K0 pU9q pgqpλ, µqq for each λ, µ P ΛW . These Grothendieck groups are already Z-modules, so we need only define the effect of multiplying by qt for t P Z. This is degree-shifting. qt rEi 1λ tsu, es “ rEi 1λ ts ` tu, es 52 Now I need only to define a system of orthogonal idempotents for the À Zrq, q´1 s-module λ,µPΛW K0 pU9q pgqpλ, µqq. 2 tells us that the multiplication on this module is given by composition if the two 1-cells can be composed, or zero otherwise.
2-Kac-Moody Algebras by David Mehrle