By David Mehrle
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Extra resources for 2-Kac-Moody Algebras
There is a 2-functor I : C Ñ C9 that is universal among 2-functors F : C Ñ D such that all idempotent 2-morphisms split under F. Proof. Define I : C Ñ C9 • on 0-cells by IpAq “ A; 9 • on 1-cells f : A Ñ B by Ip f q “ I A,B p f q, where I A,B : CpA, Bq Ñ CpA, Bq is the inclusion functor of the idempotent completion of 1-categories. f • On a 2-cell A α B, define Ipαq by I A,B pαq. g The fact that I is a genuine 2-functor follows from the way that composition is 9 Composition of 1-cells f : A Ñ B and g : B Ñ C is defined in C.
In particular, there are algebras Rpνq such that ˘ A Uq pgq – K0 pRpνq- projq. 3), we have that rEν,´ν1 , ek s is in the image of U pgq` b A Uq pgq´ under the composite map q A ` A Uq pgq b A Uq pgq´ 1µ pA U9 q pgqq1λ This outlines the proof of the following. 8]). γ is surjective. 64 γ K0 pU9q pgqpλ, µqq.
U9q pgq is nondegenerate if the set Bi,j,λ is a basis for grHompEi 1λ , Ej 1λ q for all i, j, λ. The graded dimension of the graded homs between pairs 2-morphisms can be used to define a semilinear form on K0 pU9q pgqpλ, µqq rEi 1λ , es , rEj 1λ , e1 s ÞÑ grdim grHomppEi 1λ , eq, pEj 1λ , e1 qq This can then be extended to all K0 pU9q pgqq by declaring that it takes the value zero on the pair X, Y unless X and Y have the same domain and codomain. By 62 imposing Qpqq-semilinearity, it further extends to a Qpqq-semilinear form on K0 pU9q pgqq bZrq,q´1 s Qpqq.