ON THE DEFINITION OF KAC-MOODY 2-CATEGORY

Transcription

1 ON THE DEFINITION OF KAC-MOODY 2-CATEGORY JONATHAN BRUNDAN Abstract. We show that the Kac-Moody 2-categores defed by Rouquer ad by Khovaov ad Lauda are the same. 1. Itroducto Assume that we are gve the followg data: a (ot ecessarly fte) dex set I; tegers a for each, I such that a 2, a 0 for all, ad a 0 f ad oly f a 0. Thus A (a ), I s a geeralzed Carta matrx. Set d : a for short. Fx also the addtoal data: a complex vector space h; learly depedet vectors α h for each I called smple roots; learly depedet vectors h h for each I such that h, α a. Let g be the assocated Kac-Moody algebra wth Chevalley geerators {e, h, f } I ad Carta subalgebra h. Let P : { h h, Z for all I} be ts weght lattce. I [R], Rouquer has assocated to g a certa 2-category A(g), whch we wll deote stead by U(g). It depeds also o a commutatve groud rg k; uts t k for, I such that t 1 ad t t f d 0; scalars s pq k for, I ad 0 p < d, 0 q < d such that s pq sqp. The followg s the defto from [R, 4.1.3] formulated dagrammatc terms. Defto 1.1. The Kac-Moody 2-category U(g) s the strct addtve k-lear 2-category wth obect set P, geeratg 1-morphsms E 1 : + α ad F 1 : α for each I ad P, ad geeratg 2-morphsms x : E 1 E 1, τ : E E 1 E E 1, η : 1 F E 1 ad ε : E F 1 1, subect to certa relatos. To record these, we adopt a dagrammatc formalsm lke [KL], represetg the detty 2-morphsms of E 1 ad F 1 by +α ad, respectvely, ad the other geerators by α x, τ, η, ε. (1.1) We stress that our dagrams are smply shorthads for algebrac expressos bult by horzotally ad vertcally composg geerators; they do ot satsfy ay topologcal varace other tha the rectlear sotopy mpled by the terchage 2010 Mathematcs Subect Classfcato: 17B10, 18D10. Research supported part by NSF grat DMS

2 2 J. BRUNDAN law. Frst, we have the quver Hecke relatos: f, 0 otherwse, 0 f, t k k d t f d 0, + t d t k r+sd 1 + s pq 0 p<d 0 q<d r s + p q 0 p<d 0 q<d r+sp1 s pq otherwse, r s k (1.2) (1.3) q f k, 0 otherwse. (1.4) (Note the above relatos that we represet powers of x by decoratg the dot wth a multplcty.) Next we have the rght aducto relatos,, (1.5) whch mply that F 1 +α s the rght dual of E 1. Fally there are some verso relatos. To formulate these, we frst troduce a ew 2-morphsm σ : : E F 1 F E 1. (1.6) The we requre that the followg 2-morphsms are somorphsms: h, 1 0 h, 1 0 : E F 1 F E 1 f, (1.7) : E F 1 F E 1 1 h, f h, 0, (1.8) : E F 1 1 h, F E 1 f h, 0. (1.9) (Ths meas formally that there are some addtoal as yet uamed geerators whch serve as two-sded verses to the 2-morphsms (1.7) (1.9).) Our ma theorem detfes the 2-category U(g) ust defed wth the Khovaov- Lauda 2-category from [KL]. Actually Khovaov ad Lauda worked ust wth the choce of parameters whch t 1 ad s pq 0 always. Subsequetly, Cauts

3 KAC-MOODY 2-CATEGORIES 3 ad Lauda [CL] geeralzed the defto to corporate more geeral choces of these parameters as above. By the Khovaov-Lauda 2-category we really mea the more geeral verso from [CL]. Ma Theorem. Rouquer s Kac-Moody 2-category U(g) s somorphc to the Khovaov-Lauda 2-category. The proof s a elemetary relato chase. To expla the strategy, recall that the Khovaov-Lauda 2-category has the same obects ad 1-morphsms as U(g). The there are geeratg 2-morphsms represeted by the same dagrams as x, τ, η ad ε above, plus addtoal geeratg 2-morphsms x : F 1 F 1, τ : F F 1 F F 1, η : 1 E F 1 ad ε : F E 1 1 represeted dagrammatcally by x, τ, η, ε. (1.10) These satsfy further relatos whch we wll recall more detal later the troducto. It s evdet that all of the defg relatos of U(g) recorded above are satsfed the Khovaov-Lauda 2-category. Hece there s a strct k-lear 2-fuctor from U(g) to the Khovaov-Lauda 2-category whch s the detty o obects ad 1-morphsms, ad maps the geeratg 2-morphsms x, τ, η ad ε to the correspodg 2-morphsms from [KL, CL]. To see that ths fuctor s a somorphsm, we costruct a two-sded verse. I order to do ths, we eed to detfy approprate 2-morphsms x, τ, η ad ε U(g) that wll be the mages of the addtoal geerators (1.10) uder the verse fuctor. The deftos of η ad ε that follow are essetally the same as Rouquer s caddates for secod aducto from [R, 4.1.4], except that we have reormalzed by the sg (1) h, +1 order to be cosstet wth the covetos of [KL, CL]. We wll also defe a leftward crossg σ : F E 1 E F 1, (1.11) whch we have chose to ormalze dfferetly from the leftward crossg [CL]. Defto 1.2. Defe the dowward dots ad crossgs x ad τ to be the rght mates of x ad τ (up to the factor t 1 the latter case): x :, (1.12) τ : t 1 (1.6) t 1. (1.13) The to defe η, ε ad σ, we assume tally that h, > 0. Thkg of (1.8) as a colum vector of morphsms, ts verse s a row vector. We defe the

4 4 J. BRUNDAN 2-morphsms σ ad η so that σ s the leftmost etry of ths row vector ad η s ts rghtmost etry: ( : h, 1 ) 1. (1.14) Istead, f h, < 0, the morphsm (1.9) s a row vector ad ts verse s a colum vector. We defe σ ad ε so that σ s the top etry of ths colum vector ad ε s ts bottom oe: ( : h, 1 ) 1. (1.15) To complete the deftos of σ, η ad ε, t remas to set ( : ) 1 f h, 0, (1.16) : h, f h, 0, (1.17) : ( : h, f h, 0, (1.18) ) 1 f. (1.19) Now to prove the Ma Theorem we must show that all of the defg relatos for the Khovaov-Lauda 2-category from [CL] are satsfed by the 2-morphsms U(g) ust troduced. Frst we wll show that the fte Grassmaa relato holds U(g). Ths relato was formulated orgally by Lauda [L]. It asserts that, as well as the dotted bubble 2-morphsms r, s Ed(1 ) already defed for r, s 0, there are uque dotted bubble 2-morphsms for r, s < 0 such that r 0 f r < h, 1, r 1 1 f r h, 1, (1.20) s 0 f s < h, 1, s 1 1 f s h, 1, (1.21) r,s Z r+st2 r s 0 for all t > 0. (1.22)

5 KAC-MOODY 2-CATEGORIES 5 Usg the fte Grassmaa relato, we deduce that the verses of the 2- morphsms from (1.8) ad (1.9) are ad h, 1 0 h, 1 0 r 0 r 0 r (1.23) r2 r2, (1.24) r respectvely. Several other of the Khovaov-Lauda relatos follow from ths asserto; see (3.16) (3.18) below. After that, we wll show that the 2-morphsms η ad ε defe a ut ad a cout makg the rght dual F 1 +α of E 1 also to ts left dual:,. (1.25) Cosequetly, U(g) s rgd,.e. all of ts 1-morphsms admt both a left ad a rght dual. Fally we wll show that the other geeratg 2-morphsms are cyclc (up to scalars):, t 1. (1.26) Ackowledgemets. The dagrams ths paper were created usg Tll Tatau s TkZ package. 2. Chevalley voluto I ths secto we troduce a useful symmetry. Frst though we record some of the most basc addtoal relatos that hold the 2-category U(g). Lemma 2.1. The followg relatos hold:,, (2.1), f, 0 otherwse,. (2.2) (2.3)

6 6 J. BRUNDAN k k r+sd 1 r t + s k s pq 0 p<d 0 q<d r+sp1 r q f k, s k 0 otherwse. (2.4) Proof. The frst two relatos follow from the defto (1.12) of the dowward dot usg the aducto relatos (1.5). The secod two follow smlarly from the defto of the rghtward crossg (1.6). For (2.3), attach a rghtward cap to the top rght strad ad a rghtward cup to the bottom left strad of (1.2), the use (2.1) ad the defto (1.6). Fally for (2.4), attach a rghtward cap to the top rght strad ad a rghtward cup to the bottom left strad (1.4), the use (2.2) ad the defto of the rghtward crossg. Takg otato from [CL], we defe ew parameters from t : t 1, The ext lemma explas the sgfcace of these scalars. s pq : t 1 t1 sqp. (2.5) Lemma 2.2. The followg relatos hold: f, 0 otherwse, k k 0 f, t f d 0, d t + t r+sd 1 d + s pq 0 p<d 0 q<d r s + t k p 0 p<d 0 q<d r+sp1 q s pq k r s otherwse, (2.6) (2.7) q f k, 0 otherwse. (2.8) Proof. Put rghtward caps o the top ad rghtward cups o the bottom of the relatos (1.2) (1.4), the use (1.5), the deftos (1.6), (1.12), (1.13), ad (2.1) (2.2). For ay strct k-lear 2-category C, we wrte C opp for the 2-category wth the same obects as C but wth morphsm categores defed from Hom C opp(, µ) : Hom C (, µ) opp, so the vertcal composto C opp s the opposte of the oe C, whle the horzotal composto C opp s the same as C.

7 KAC-MOODY 2-CATEGORIES 7 Theorem 2.3. Let U(g) be the Kac-Moody 2-category defed as Defto 1.1 but usg the prmed parameters from (2.5) place of t ad s pq. The there s a somorphsm of strct k-lear 2-categores T : U(g) U(g) opp defed o obects by T() :, o geeratg 1-morphsms by T(E 1 ) : F 1 ad T(F 1 ) : E 1, ad o geeratg 2-morphsms by,,,. The effect of T o the other amed 2-morphsms U(g) s as follows:,,,, t 1,. t Moreover we have that T T d T T where T : U(g) U(g) opp s the aalog of T wth U(g) replaced by U(g) ad U(g) replaced by U(g) U(g). Proof. To see that T s well defed we eed to verfy that the mages uder T of the relatos (1.2) (1.5) ad (1.7) (1.9) wth prmed parameters hold U(g) opp. For the frst three, ths follows from Lemma 2.2, whle (1.5) s clear. For the remag oes, we frst ote that example for (1.8), we must show for h, 0 that by the defto (1.6). The for h, 1 0 s vertble U(g). Ths follows by composg (1.9) wth dag(1, 1,..., 1), usg also (2.1). The rest of the theorem s a route check from the deftos (1.12) (1.19). We wll ofte appeal to Theorem 2.3 to establsh mrror mages of relatos a horzotal axs. For example, applyg t to the aalog of the relato (2.2) U(g), we obta the followg relato (whch could also be deduced drectly from the defto of the dowward crossg): Corollary 2.4. The followg relatos hold U(g): t, t. (2.9)

8 8 J. BRUNDAN 3. The fte Grassmaa relato The goal ths secto s to show that the fte Grassmaa relato holds U(g). We beg by troducg otato for the other etres of the row vector that s the two-sded verse of (1.8) ad of the colum vector that s the two-sded verse of (1.9): let h, 1 0 h, 1 0 ( : ( : h, 1 0 h, 1 0 ) 1 f h, 0, ) 1 f h, 0. (3.1) (3.2) We wll gve a more explct descrpto of these 2-morphsms (3.12) (3.13) below. Note rght away comparg the preset deftos wth (1.14) (1.15) that h, 1 f h, > 0, h, 1 Also for all admssble values of we have that ( ) ( T, T f h, < 0. (3.3) ), (3.4) as follows o applyg the somorphsm T from Theorem 2.3 to the deftos (3.1) (3.2) ad usg (2.1). Lemma 3.1. The followg relatos hold: h, 1 0, h, 1 0, (3.5) 0, 0, δ, h, 11 1 all assumg 0 < h,, (3.6) 0, 0, Proof. Ths follows from (3.1) (3.3). δ, h, 11 1 all assumg 0 < h,. r (3.7) The dotted bubbles, s defe edomorphsms of 1 for r, s 0. We also gve meag to egatvely dotted bubbles by makg the followg deftos

9 KAC-MOODY 2-CATEGORIES 9 for r, s < 0: r : s : r1 h, f r > h, 1, 1 1 f r h, 1, 0 f r < h, 1, h, s1 Note by Theorem 2.3, (2.1) ad (3.4) that ( ) ( r T r, T f s > h, 1, 1 1 f s h, 1, 0 f s < h, 1. r ) r (3.8) (3.9), (3.10) for all r, s Z. Gve (3.6) (3.9), the followg theorem mples the fte Grassmaa relato as formulated formulae (1.20) (1.22) the troducto. Theorem 3.2. The followg holds for all t > 0: r,s Z r+st2 r s 0. (3.11) Proof. We prove ths uder the assumpto that h, 0; the result whe h, 0 the follows usg Theorem 2.3 ad (3.10). We have that r r,s Z r+st2 (3.9) s (3.8) (3.9) h, 0 h, +t1 h, (3.5) (1.17) t1 t1 +t1 h, r 1,s 0 r+st2 r 1,s 0 r+st2 +t1 r r 1 + r 1,s 0 r+st2 h, s s (2.3) + r r 1,s 0 r+st2 + t1 1 r s s t1. It ust remas to show that the fal expresso here s zero. Whe h, > 0 ths follows by (3.6) ad (3.8). Fally f h, 0 the + t1 1 t1 (1.18) (3.8) t1 + t1 (3.5) 0.

10 10 J. BRUNDAN Corollary 3.3. The followg relatos hold: r 0 r 0 r f 0 < h,, (3.12) r2 r2 r f 0 < h,. (3.13) Proof. We expla the proof of (3.12); the proof of (3.13) s etrely smlar. Rememberg the defto (3.1), t suffces to show that the vertcal composto cosstg of (1.8) o top of (1.23) s equal to the detty. Usg (3.5) (3.6), ths reduces to checkg that r 0 m+r r 0 r r2 0 f 0 < h,, (3.14) r2 δ m, 1 1 f 0 m, < h,. (3.15) For (3.14), each term the summato s zero: f r h, the couterclockwse dotted bubble s zero by (3.9); f 0 r < h, we have that r (1.2) r s,t 0 s+tr1 t s (3.6) 0. To prove (3.15), ote by (3.6) ad (3.9) that order for m+r r2 be o-zero we must have m+r h, 1 ad r 2 h, 1. Addg these equaltes mples that m. Moreover f m the oly o-zero term the summato s the term wth r h, 1 m, whch equals 1 1 by (3.6) ad (3.9) aga. Fally f m > the left had sde of (3.15) ca be rewrtte as whch s zero by (3.11). r,s Z r+sm2 r s, O substtutg (3.12) (3.13) to the deftos (3.1) (3.2), ths establshes the assertos about the 2-morphsms (1.23) (1.24) made the troducto. Corollary 3.4. The followg relatos hold: to h, 1 r 0 r 0 r2, (3.16)

11 KAC-MOODY 2-CATEGORIES 11 h, 1 0 r 0 r2 r. (3.17) Proof. Substtute (3.12) (3.13) to (3.5). Corollary 3.5. The followg relatos hold: h, 0 1, h, 0 1. (3.18) Proof. I vew of Theorem 2.3, (2.1) ad (3.10), t suffces to prove the left had relato. We are doe already by (3.7) f h, < 0. If h, 0 the: (1.17) h, (3.16) h, h, 1 0 r 0 h, +r r2 h, (3.6) (3.8) (3.11) h, + h, 1 0 r,s Z r+s h, 2 h, 1 0 r s + h 1 (3.9), 0 h, 1 0 h, Left aducto relatos I ths secto we show that the leftward cup ad cap satsfy aducto relatos. Lemma 4.1. The followg relatos hold: δ, h, 2 f 0 h, 2, (4.1) δ, h, f 0 h,. (4.2) Proof. Proceed by ducto o. For the base case, covert the upward crossgs to rghtward oes usg (1.5) (1.6), apply (3.18) ad (3.8) (3.9), the voke (1.5). For the ducto step, pull a dot past the crossg usg (2.1) ad (1.2), the use (3.6) (3.7) ad the ducto hypothess.

12 12 J. BRUNDAN Lemma 4.2. The followg relatos hold: f h, < 0, (4.3) f h, > 2. (4.4) Proof. Let h : h, for short. Frst we prove (4.3), so h < 0. We clam that δh,1. (4.5) To establsh the clam, we vertcally compose o the bottom wth the somorphsm h1 arsg from (1.9) to reduce to showg equvaletly that 0 δ h,1, (4.6) δ h,1 for 0 h 1. (4.7) Here s the verfcato of (4.6): (2.4) h+2 (3.18) 0 1 (2.2) h (1.3) (3.9) δ h,1 (1.3) δ h,1 (1.2) δ h,1 (3.7) (2.2) δ h,1 δ h,1. For (4.7), we ote by (3.7) ad (1.5) that the rght had sde s equal to f h 1 > 0, ad t s zero otherwse. Now we smplfy the left had sde: (2.2) (1.2) + r+s1 r s (1.3) (2.1) r+s1 r. s Ths s obvously zero f 0. Assumg > 0, we apply (4.1) to see that t s zero uless h 1, whe the term wth r h 2, s 0 cotrbutes. Ths completes the proof of the clam. Now to establsh the relato (4.3), we vertcally

13 KAC-MOODY 2-CATEGORIES 13 compose (4.5) o the bottom wth to obta the desred relato: (4.1) (4.5) h3 0 δ h,1 r 0 r2 (3.6) r (3.17) (2.1) The proof of (4.4) follows by a very smlar argumet; oe frst checks that. δ h,1 whe h > 2 the vertcally composes o the top wth. Theorem 4.3. The followg relatos hold:,. (4.8) Proof. It suffces to prove the left had relato; the rght had oe the follows usg Theorem 2.3. Let h : h, for short. If h 0 the (1.17) h (4.4) h (2.1) h (4.2). If h 2 the (1.18) h2 (4.3) h2 (2.1) h2 (4.1). Fally f h 1 the (3.7) (3.16) (3.9) + (2.2) (4.3) + (1.3) (1.5). Ths completes the proof.

14 14 J. BRUNDAN 5. Cyclcty relatos At ths pot, the proof of the Ma Theorem s reduced to checkg the cyclcty relatos. We proceed to do ths. Lemma 5.1. The followg relato holds: f h, 1. (5.1) Proof. The 2-morphsms o both sdes of the desred detty go from 1 to E F 1. To show that they are equal, we vertcally compose them both o the top wth the somorphsm (1.8) to reduce to provg stead that h, h, 1 0. I these two colum vectors of 2-morphsms, the etres volvg dotted crcles are equal thaks to (2.1). It ust remas to observe that (2.3) (3.6) (3.6) Lemma 5.2. Assumg that, the followg relatos hold: (2.3) t f h, < d, (5.2). t f h, d. (5.3) Proof. Let h : h, for short. Frst we prove (5.2) assumg that h 0. By (1.7) ad (1.9), the followg 2-morphsm s vertble: h1 0. Vertcally composg wth ths o the bottom, we deduce that the relato we are tryg to prove s equvalet to the followg relatos: t, t for 0 < h. (5.4) To establsh the frst of these, we pull the -strg past the -crossg: (2.4) + t r+sd 1 s r + s pq 0 p<d 0 q<d r+sp1 q s r.

15 KAC-MOODY 2-CATEGORIES 15 If h < 0 the all the terms o the rght had sde vash thaks to (3.7). If h 0 ad d > 0 everythg except for the r d 1 term from the frst sum vashes, ad we get t. Fally f h d 0, we oly have the frst term o the rght had sde, whch cotrbutes t ad (1.3). Ths s what we wat because: aga thaks to (3.18), (3.9), (2.2) (1.19) (3.18) (3.9) δ h,0. We are ust left wth the rght had relatos from (5.4) volvg bubbles: (2.2) (1.2) (1.3) t +d + a lear comb. of p q wth p < + d (3.7) δ,h1 t (3.7) t t. (1.19) The relato (5.3) follows by a very smlar argumet to the oe explaed the prevous paragraph; the frst step s to vertcally compose o the top wth the somorphsm hd 1 0. Fally we must prove (5.2) f 0 < h < d. We vertcally compose o the bottom wth the somorphsm to reduce to provg that t. (5.5) To see ths we apply (3.17) to trasform the left had sde to (2.4) t r+sd 1 s r s pq 0 p<d 0 q<d r+sp1 r q. The frst term o the rght had sde here vashes by (3.6). Also the terms the summatos are zero uless r d h 1 ad s h by (3.6) (3.7), hece we are s

16 16 J. BRUNDAN left ust wth the r d h 1, s h term, whch equals t h (4.8) t h (4.4) t h Ths s equal to the rght had sde of (5.5) thaks to (1.19). (2.1) (4.2) t Theorem 5.3. The followg relatos hold for all, I ad P :,., (5.6) t, t, (5.7),. (5.8) Proof. For (5.6), we already proved the left had relato whe h, 1 (5.1). Now take ths relato wth replaced by + α, attach leftward caps to the top left ad top rght strads, the apply (4.8) to prove the rght had relato whe h, 1. Fally apply Theorem 2.3 to the cases establshed so far to get the rght had relato whe h, 1 ad the left had relato whe h, 1. The proofs of (5.7) (5.8) follow by a smlar strategy to the prevous paragraph, startg from (4.3) (4.4) ad (5.2) (5.3). The fal set of relatos (1.26) eeded to complete the proof of the Ma Theorem follow easly from (5.6) (5.8) usg also (4.8). Refereces [CL] S. Cauts ad A. Lauda, Implct structure 2-represetatos of quatum groups, Selecta Math. 21 (2015), [KL] M. Khovaov ad A. Lauda, A dagrammatc approach to categorfcato of quatum groups III, Quatum Top. 1 (2010), [L] A. Lauda, A categorfcato of quatum sl(2), Adv. Math. 225 (2010), [R] R. Rouquer, 2-Kac-Moody algebras; arxv: Departmet of Mathematcs, Uversty of Orego, Eugee, OR 97403, USA E-mal address: bruda@uorego.edu