2-Kac-Moody Algebras by David Mehrle PDF

By David Mehrle

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If we were to ignore the grading on 1-morphisms and omit degree shifts of the 2-morphisms, then we obtain a categorification of the non9 quantum idempotented universal enveloping algebra Upgq. As with most categorifications, the addition of a grading corresponds to moving from a categorification of algebras over k to algebras over kpqq. 3 Working with Uq pgq In this section, we record properties of Uq pgq and some more relations among the 2-cells that will be used to prove that U9q pgq categorifies U9 q pgq.

38 Proof. 11. 3 Identities, isomorphisms, and more relations In this section, we collect a few more facts about the 2-category Uq pgq which will either prove useful later or illustrate how to work with the objects in this 2-category. First, notice that the relations defining Uq pgq give relations between certain configurations only when these 2-cells have all strands oriented up. 16) to turn over the crossings and define the same relations upside-down. This is recorded in the following proposition.

1. 1). 2. 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 [6], Webster [12] 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.

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2-Kac-Moody Algebras by David Mehrle

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