ON THE DEFINITION OF KAC-MOODY 2-CATEGORY JONATHANBRUNDAN 5 1 0 Abstract. We show that the Kac-Moody 2-categories defined by Rouquier 2 andbyKhovanovandLaudaarethesame. n a J 2 1. Introduction ] Assume that we are given the following data: T • a (not necessarily finite) index set I; R • integers a for each i,j ∈ I such that a = 2, a ≤ 0 for all i 6= j, and ij ii ij h. aij =0 if and only if aji =0; t • positive integers d such that d a =d a for all i,j ∈I. a i i ij j ji m Thus A=(a ) is a symmetrizable generalizedCartanmatrix. Set d :=−a ij i,j∈I ij ij for short. Fix also the additional data: [ • a complex vector space h; 1 • linearly independent vectors α ∈h∗ for each i∈I called simple roots; v i 0 • linearly independent vectors hi ∈h for each i∈I such that hhi,αji=aij. 5 LetgbetheassociatedKac-MoodyalgebrawithChevalleygenerators{e ,h ,f } i i i i∈I 3 and Cartan subalgebra h. Let P :={λ∈h∗|hh ,λi∈Z for all i∈I} be its weight 0 i lattice. In [R], Rouquier has associated to g a certain 2-category A(g), which we 0 . will denote instead by U(g). It depends also on: 1 0 • a ground ring k; 5 • units tij ∈k× for i,j ∈I with i6=j such that tij =tji if dij =0; 1 • scalarsspq ∈k for i,j ∈I and 0≤p 0. Thinking of i (1.8) as a column vector of morphisms, its inverse is a row vector. We define the 2-morphisms σ′ and η′ so that −σ′ is the leftmost entry of this row vector and η′ is its rightmost entry: i i i hhi,λi−1• λ −1 − λ ⊕···⊕ λ :=(cid:18) λ ⊕···⊕ i (cid:19) . (1.14) i i Instead,ifhh ,λi<0,themorphism(1.9)isarowvectoranditsinverseisacolumn i vector. We define σ′ and ε′ so that −σ′ is the top entry of this column vector and ε′ is its bottom one: i λ i −1 i − λ ⊕···⊕ i :=(cid:18) λ ⊕···⊕ •−hhi,λi−1 (cid:19) . (1.15) λ i i To complete the definitions of σ′,η′ and ε′, it remains to set i i −1 λ :=− λ if hhi,λi=0, (1.16) (cid:18) (cid:19) i i λ :=− hhi,λi• if hhi,λi≥0, (1.17) i λ i i i := λ if hh ,λi≤0, (1.18) λ •−hhi,λi i i i −1 λ := λ if i6=j. (1.19) (cid:18) (cid:19) j j Now to prove the Main Theoremwe must show that allof the defining relations for the Khovanov-Lauda 2-category from [CL] are satisfied by the 2-morphisms in U(g) just introduced. First we will show that Lauda’s infinite Grassmannian relation holds inU(g). This assertsthat, aswellas the dottedbubble 2-morphisms r• λ , λ •s ∈ End(1 ) already defined for r,s ≥ 0, there are unique dotted λ i i bubble 2-morphisms for r,s<0 such that r• λ =0 if r 0. (1.22) i i rX,s∈Z r+s=t−2 KAC-MOODY 2-CATEGORIES 5 Using the infinite Grassmannian relation, we deduce that the inverses of the 2- morphisms from (1.8) and (1.9) are i hhi,λi−1 i − λ ⊕ r• λ (1.23) i Mn=0 Xr≥0 •−n−r−2 i and i −hhi,λi−1 i − λ ⊕ −n−r−2• λ , (1.24) i Mn=0 Xr≥0 •r i respectively. Severalother of the Khovanov-Laudarelations follow from this asser- tion;see(3.16)–(3.18)below. Afterthat,wewillshowthatthe2-morphismsη′ and ε′ define a unit and a counit making the right dual F 1 of E 1 also into its i λ+αi i λ left dual: i i = , = . (1.25) λ λ λ λ i i Consequently,U(g)isrigid, i.e. allofits1-morphismsadmitbothaleftandaright dual. Finally we willshow that the other generating2-morphismsare cyclic (up to scalars): j i i i j i j i • λ= • λ , λ = λ =t−ji1 λ . (1.26) Acknowledgements. The diagrams in this paper were created using Till Tantau’s TikZ package. 2. Chevalley involution In this section we introduce a useful symmetry. First though we record some of the most basic additional relations that hold in the 2-categoryU(g). Lemma 2.1. The following relations hold: i i λ λ • = • , • = • , (2.1) λ λ i i i i λ = λ , λ = λ . (2.2) i j i j j j j j j j j i •λ − i• λ = i• λ − i •λ = i λ if i=j, (2.3) 0 otherwise,  6 J.BRUNDAN i i i − i = r+srX=,sd≥i0j−1tijj r•k•s λ +00≤≤Xpq<0, λ = λ if hh ,λi<0. (3.3) ♠ i i hhi,λi−1 i i Also for all admissible values of n we have that n n i ♣ −λ ♣ λ i T λ = , T = −λ , (3.4) ♠ ♠ (cid:18) n (cid:19) i (cid:18) i (cid:19) n asfollowsonapplyingtheisomorphismTfromTheorem2.3tothedefinitions(3.1)– (3.2) and using (2.1). Lemma 3.1. The following relations hold: i hhi,λi−1 i i i −hhi,λi−1i i = ♠n λ − λ , = •nλ − λ , (3.5) λ nX=0 n• i λ nX=0 ♣n i i i i i i λ =0, n• λ =0, n•i λ =δn,hhi,λi−111λ all assuming 0≤nhhi,λi−1, (3.8) i  11λ if r =hhi,λi−1, 0 if r −hhi,λi−1, (3.9) i  11λ if s=−hhi,λi−1, 0 if s<−hh ,λi−1. i  Note by Theorem 2.3, (2.1) and (3.4) that T r• λ = −λ •r , T λ •r = r• −λ , (3.10) (cid:18) i (cid:19) i (cid:18) i (cid:19) i for all r,s∈Z. Given (3.6)–(3.9), the following theorem implies the infinite Grass- mannian relation as formulated in formulae (1.20)–(1.22) in the introduction. Theorem 3.2. The following holds for all t>0: r• λ •s =0. (3.11) i i rX,s∈Z r+s=t−2 Proof. We prove this under the assumption that hh ,λi ≥ 0; the result when i hh ,λi≤0 then follows using Theorem 2.3 and (3.10). We have that i r• λ •s (3=.8) hhi,λi n+t−1• λ •−n−1 + r• λ •s rX,s∈Z i i (3.9) nX=0 i i r≥−X1,s≥0 i i r+s=t−2 r+s=t−2 (3=.9) hhi,λi+t−1• −hhi,λi−1n+t−1• hhi,λii• λ + r• λ •s i nX=0 i ♠n r≥−X1,s≥0 i i r+s=t−2 i hhi,λi • (3=.5)− + r• λ •s λ i i r≥−X1,s≥0 t−1• r+s=t−2 i i (1.=17) λ + r• λ •s (2=.3) •λt−1 + −1• λ •t−1 . t−1• r≥−X1,s≥0 i i i i r+s=t−2 Itjustremainstoshowthatthe finalexpressionhereiszero. Whenhh ,λi>0this i follows by (3.6) and (3.8). Finally if hh ,λi=0 then i i i •t−1 •λt−1 + −1•i λ i•t−1 ((13.=.188)) λ + λ i•t−1 (3=.5)0. (cid:3) 10 J.BRUNDAN Corollary 3.3. The following relations hold: i i λ = r• if 0≤nn the left hand side of (3.15) can be rewritten as r• λ •s , i i rX,s∈Z r+s=m−n−2 which is zero by (3.11). (cid:3) Onsubstituting(3.12)–(3.13)intothedefinitions(3.1)–(3.2),thisestablishesthe assertions about the 2-morphisms (1.23)–(1.24) made in the introduction. Corollary 3.4. The following relations hold: i i hhi,λi−1 r• i λ λ = i •−n−r−2 − λ , (3.16) nX=0 Xr≥0 n• i i i

On the definition of Kac-Moody 2-category

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