2-Kac-Moody Algebras by David Mehrle

By David Mehrle

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4). 36 We need only check one of these identities; checking the other triangular identity is dual. We have that 1 pε ¨ 1 f q ˝ p1 f ¨ η A q “ pΦ´ g,1 p1 g q ¨ 1 f q ˝ p1 f ¨ Φ1 A , f p1 f qq A 1 “ Φ´ g f , f p1 g f q ˝ p1 f ¨ ηq 1 “ Φ´ 1 , f pη ˝ 1 g f q A 1 “ Φ´ 1 , f pηq A “ 1 Φ´ 1 A , f pΦ1 A , f p1 f qq “ 1f And this is one of the triangular identities. 7. Let f : A Ñ B and g : B Ñ A be 1-cells in C. Then the following are equivalent: (a) an adjunction f % g with unit η and counit ε; (b) a bijection between 2-cells f a ùñ g and a ùñ gb, natural in both a : X Ñ A and b : X Ñ B.

19. 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.

8. 4. 6, so we have the desired equality. 5. Aside from the element in the upper left corner of the matrix αβ, all of the elements of the first row of this matrix vanish. Proof. Consider the elements along the first row of the matrix αβ. 16. 6). 6. Aside from the element in the upper left corner of the matrix αβ, all of the elements of the first column of this matrix vanish. Proof. This is likely the least obvious of the five things we need to show. Consider the elements down the first column of the matrix αβ.

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