2-Kac-Moody Algebras by David Mehrle

By David Mehrle

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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 ! ¨ ¸ E˘ j ˝ 1´aij ´2k E˘i r1 ´ aij ´ 2ksqi ! Ya ] ij ˛ 2 ÿ ˜ ‚“ k “0 ¸ 2k`1 E˘ i r2k ` 1sqi ! ¨ E˘ j ˝ aij ´2k E˘i raij ´ 2ksqi ! This is the form of the quantum Serre relations that we will categorify. The first step here is to find the appropriate element of U9q pgq that categorifies the elements a X “ E˘ i {rasqi !.

The two cases are symmetric; we only deal with the case that xαi , λy ě 0. In this case, we are trying to establish the isomorphism E `i ´i 1 λ – E ´i `i 1 λ ‘ xαià ,λy´1 1λ t´2s ` xαi , λy ´ 1u. s “0 We will define two 2-cells α : E`i´i 1λ ùñ E´i`i 1λ ‘ xαià ,λy´1 1λ t´2s ` xαi , λy ´ 1u s “0 β : E ´i `i 1 λ ‘ xαià ,λy´1 1λ t´2s ` xαi , λy ´ 1u ùñ E`i´i s “0 À À as below. Recall that in any additive category, a morphism in“1 Ai Ñ m j“1 Bm in represented by an m ˆ n matrix F “ p f ij q with f ij : Ai Ñ Bj .

Since the prototypical example of a 2-category is the category Cat of small categories, it makes sense to generalize from Cat to arbitrary 2-categories. One of the ideas that is particularly important for categorification of quantum groups is the notion of adjunctions internal to a 2-category. The idea is to replace categories by 0-cells, functors by 1-cells, and natural transformations by 2-cells in the definition of an adjunction. In most cases, facts about adjunctions in Cat are also true for adjunctions internal to 2-categories.

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