By David Mehrle

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There is an action of Zrq, q´1 s on the 1-morphisms Uq pgq defined Àm 3 ´1 ` 1, by qt ¨ X “ Xttu and m ¨ X “ a“1 X. For example, if f pqq “ 2q ` q then f pqq ¨ Ei 1λ ttu “ Ei 1λ tt ` 3u ‘ Ei 1λ tt ` 3u ‘ Ei 1λ tt ´ 1u ‘ Ei 1λ ttu This extends to the idempotent completion U9q pgq by qt pX, eq “ pXttu, eq. 1) for all signed sequences i. We want to check that this extends to a homomorphism of Zrq, q´1 s-modules. To do that, it is enough to verify that the relations between elements of A U9 q pgq hold under the image of this map.

25) twice, to the j and k stands at the top to cross them over twice and to the j and k strands at the bottom to untangle them. 25) is applicable since j ‰ k. 6) in the case that j ‰ k. The other case is if i ‰ j. 25). 7). 4]). The following equations hold in Uq pgq for any λ and i. 6. 42 Proof. 30) when all dots are on the other side of the crossover. 10). In fact, the infinite Grassman equation holds for all bubbles in Uq pgq, not just the fake ones. 5]). For any λ P ΛW , the following equation holds in the power series ring Uq pgqp1λ , 1λ qrrTss.

3pα ,α q pαi ,αi q ‘ E`i,`i 1λ ´ 2i i , E “ E`i,`i 1λ 2 where E is the idempotent matrix of diagrams » λ 0 — i i — E“— – λ 0 i i fi ffi ffi ffi fl We want to show that r2sqi ! ¨ X`i,2 is isomorphic to ¸ ˜ E `i `i 1 λ , i i λ . This means that we want to find α : pE`i`i 1λ , 1E`i`i 1λ q Ñ r2sqi ! ¨ X`i,2 and β : r2sqi ! ¨ X`i,2 Ñ pE`i`i 1λ , 1E`i`i 1λ q such that αβ “ E and βα “ 1E`i`i 1λ . 18), one choice of α and β might be » fi « ff λ — i ffi i — ffi λ λ α“— β“ i ffi i , i i – fl λ ´ i i 60 so that βα “ 1E`i`i 1λ .

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2-Kac-Moody Algebras by David Mehrle

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