A Drinfeld type presentation of affine \imathquantum groups II: split BCFG type
aa r X i v : . [ m a t h . R T ] F e b A DRINFELD TYPE PRESENTATION OF AFFINE ı QUANTUMGROUPS II: SPLIT BCFG TYPE
WEINAN ZHANG
Abstract.
Recently, Lu and Wang formulated a Drinfeld type presentation for ı quantumgroup e U ı arising from quantum symmetric pairs of split affine ADE type. In this paper, wegeneralize their results by establishing a current presentation for e U ı of arbitrary split affinetype. Contents
1. Introduction 12. Preliminaries 43. A Drinfeld type presentation for affine ı quantum groups 84. Verification of the relation (3.5) 145. Verification of Serre relations 19References 241. Introduction
Background.
The affine quantum group, denoted by U , admits two presentations: theSerre presentation introduced by Drinfeld-Jimbo, and the current presentation, also knownas the Drinfeld presentation [Dr88]. The isomorphism between these two presentations werestated by Drinfeld, and a detailed proof were supplied by Beck [Be94] and Damiani [Da12][Da15]. This current presentation has been shown to be crucial in the representation theoryof affine quantum groups; see the survey paper [CH10] for partial references.Quantum symmetric pairs ( U , U ı ς ) were introduced by Letzter in [Let99] for finite typeand generalized to Kac-Moody type by Kolb [Ko14]. The ı quantum group U ı ς arising fromquantum symmetric pairs is a coideal subalgebra of U associated with an involution on theunderlying root system. The universal ı quantum group e U ı [LW19a] is a coideal subalgebra ofDrinfeld double quantum group e U , and U ı ς can be recovered from e U ı by a central reduction.The version e U ı naturally arises in an ı Hall algebra realization of ı quantum groups [LW19a],and a braid group action on e U ı is realized via the reflection functors in this approach [LW19b].According to [BW18], various algebraic, geometric, categorical results for quantum groupsare expected to have corresponding analogues for ı quantum groups. In particular, it isnatural to ask whether there is a current presentation for U ı ς , analogous to the current Mathematics Subject Classification.
Primary 17B37, 17B67.
Key words and phrases.
Affine quantum groups, Drinfeld presentation, ı Quantum groups, Quantum sym-metric pairs . presentation for U . This question is answered positively in the recent work by Lu and Wang[LW20b], who formulated a Drinfeld type presentation for e U ı of arbitrary split affine ADEtype. In the rank one case, Lu-Wang’s Drinfeld type presentation was built on the earlierconstruction of root vectors and relations by Baseilhac and Kolb [BK20]. The braid groupaction on e U ı plays an essential role in Lu and Wang’s construction, just as for affine quantumgroups [Da93, Be94].1.2. Main results.
The goal of this paper is to generalize results in [LW20b] to arbitrarysplit (untwisted) affine type, namely, to provide a Drinfeld type presentation for the split ı quantum group e U ı = e U ı ( b g ) where g is any simple Lie algebra and b g is the correspondinguntwisted affine Lie algebra.Let us explain our approach in details. Let ( c ij ) i,j ∈ I denote the Cartan matrix for g .We start by recalling the Serre presentation of e U ı in Definition 2.2, following [LW19a] and[LW20a]. Such a presentation is obtained by modifying the Serre presentation for U ı ς givenin [CLW18, Theorem 3.1] (built on earlier work by Kolb and Letzter for U ı ς ). The braidgroup action on e U ı (see Lemma 2.7) arises from ı Hall algebras [LW19b, CLW20, LW21], andit recovers the braid group action established in [KP11, BK20] for U ı ς (who worked withspecific parameters).We shall use similar definitions and notations (3.1)-(3.3) for root vectors in e U ı (i.e. gen-erators in the Drinfeld type presentation) as in [LW20b, (3.28)-(3.30)]; that is, we shall use B i,k for the real root vectors, ´Θ i,m for the imaginary root vectors constructed in [BK20] inthe rank one case and adapted to our general case, and Θ i,m for the alternative imaginaryroot vectors originated from ı Hall algebra approach [LRW20]. We shall focus on Θ i,m insteadof ´Θ i,m throughout this paper.Let us explain how we formulate the relations (3.4)-(3.11) for B i,k , Θ i,m which are used ina current presentation for e U ı . Relations (3.4)-(3.8) are natural generalizations of [LW20b,(3.33)-(3.37)] to the case when c ij is arbitrary.The remaining task is to obtain a suitable formulation of the Serre relations (in the currentpresentation). For c ij = −
1, there are two (equivalent) formulations of the Serre relationsavailable: one is the general version formulated in [LW20b, (3.32),(3.38)], and the other is theequal-index version (3.9) (which is the special cases of the general version); the latter is muchsimpler than the general version and is obtained by applying degree shift automorphismsto the corresponding finite type Serre relation (2.15). As for their generalizations to c ij < −
1, the general versions of Serre relations are going to be extremely complicated as Luand Wang’s formulation suggests. However, the equal-index versions (3.10)-(3.11) can beobtained relatively easily. Hence, we choose to use relations (3.9)-(3.11) in our currentpresentation.There are two supporting examples for the use of this equal-index version of Serre relations.For affine quantum groups, Damiani in [Da12, Theorem 11.18] showed that the Drinfeldpresentation Dr U is equivalent to a reduced current presentation Dr U red where Serre relationsare replaced by the corresponding equal-index version. (See Proposition 2.1.) Moreover, Luand Wang showed in [LW20b, § e U ı ofsplit ADE type, if we replace the Serre relation in the current presentation formulated in DRINFELD TYPE PRESENTATION OF AFFINE ı QUANTUM GROUPS, II 3 [LW20b, Definition 3.10] by the equal-index version (3.9) of itself, we obtain an equivalentpresentation.Generalizing these phenomenons, we define an algebra Dr e U ıred in Definition 3.1 with defin-ing relations (3.4)-(3.11). We shall show that Dr e U ıred is isomorphic to e U ı in Theorem 3.2.We call Dr e U ıred a reduced Drinfeld type presentation.In the proof of this isomorphism, most defining relations of Dr e U ıred are verified in e U ı in asimilar way as [LW20b, § i = j , whose originalproof for c ij = − § c ij < −
1, since it requiresthe help of the general Serre relation. In this paper, we provide a new inductive proof of therelation (3.5) for i = j , which is based on a recursive formula (4.2) uniformly established forarbitrary c ij in § c ij in § c ij = −
1, the base case is verified using the finitetype Serre relation. For c ij = −
2, the base case is derived from formulas for the braid groupaction. For c ij = −
3, it turns out we need both of the finite type Serre relation and formulasof the braid group action to prove the base case.Two techniques are widely used in our proof of (3.5) and later in the proof of Serrerelations, analogous to [Da12]. One is the q -brackets, which allow us to write Serre relationsand formulas of the braid group action in compact forms (e.g. (4.10), (4.13) etc.) and thendeal with them efficiently. The other one is the degree shift automorphism T ω i , coming fromthe braid group action, which sends B i,k to B i,k − and fixes B j,l , for all j = i . These degreeshift automorphisms allow us to recover a general relation from a more basic version andthus minimize the required amount of work.It is still desirable to have general Serre relations, which we shall provide when c ij = − c ij = −
1, the general Serre relation (3.26) is first formulated in [LW20b, (3.38),(5.6)]. We offer a more direct proof in terms of generating functions in § § c ij < − c ij = −
2, we generalize this method and formulate a general Serre relation (3.27) interms of generating functions. Details for the proof of (3.27) is included in § c ij = − e U ı of split affine type BCF is formulated and proved in Theorem3.4.For c ij = −
3, while it is still possible to formulate a version of the general Serre relation,computation becomes much more involved and we will skip it. For practical purposes suchas developing the representation theory of e U ı , we do not need this. WEINAN ZHANG
Organization.
This paper is organized as follows. In Section 2, we set up notationsand review the basic theory of affine quantum groups and affine ı quantum groups. In Section3.1, we formulate a current presentation for e U ı of arbitrary split affine type in Definition3.1 and Theorem 3.2. In Section 3.2, we establish a Drinfeld type presentation in terms ofgenerating functions for e U ı in Theorem 3.4.In Section 4, we verify the relation (3.5) in the current presentation using an induction.We establish a recursive formula for the induction in Section 4.1 and check base cases inSection 4.2. In Section 5, we verify general Serre relations: the one for the c ij = − c ij = − Acknowledgement.
The author would like to thank Ming Lu and his advisor WeiqiangWang for sharing their work earlier and for many helpful discussions and advices. This workis partially supported by the GRA fellowship of Wang’s NSF grant DMS-2001351.2.
Preliminaries
Affine Weyl groups.
Set I = { , . . . , n } . Let g be a simple Lie algebra with Car-tan matrix ( c ij ) i,j ∈ I . Let d i be relatively prime positive integers such that ( d i c ij ) i,j ∈ I is asymmetric matrix.Let R denote the root system of g . Fix a set of simple roots { α i | i ∈ I } for R anddenote the corresponding positive system by R +0 . Let Q = L i ∈ I Z α i be the root latticeof g and P be the dual lattice of Q . The bilinear pairing between P, Q is denoted by h· , ·i : P × Q → Z . The lattice P is known as the weight lattice of g and P = L i ∈ I Z ω i ,where ω i are fundamental weights of g given by h ω i , α j i = δ i,j . We identify Q as a sublatticeof P via h α i , α j i = d i c ij . Let θ be the highest root for g .Set I = I ∪ { } . Let b g be the untwisted affine Lie algebra associated to g with the affineCartan matrix ( c ij ) i,j ∈ I . Extend Q to the affine root lattice e Q := Q ⊕ Z α . It is known thatthe element δ = α + θ ∈ e Q satisfies h α i , δ i = 0 , ∀ i ∈ I . The root system R and the set ofpositive roots R + for b g are defined to be R = {± ( β + kδ ) | β ∈ R +0 , k ∈ Z } ∪ { mδ | m ∈ Z \{ }} , (2.1) R + = { kδ + β | β ∈ R +0 , k ≥ } ∪ { kδ − β | β ∈ R +0 , k > } ∪ { mδ | m ≥ } . (2.2)Let s i be the reflection acting on e Q by s i ( x ) = x − h x, α i i α i for i ∈ I . The Weyl group W of g and the affine Weyl group W of b g are subgroups of Aut( e Q ) generated by s i , i ∈ I andby s i , i ∈ I , respectively.The extended affine Weyl group f W is the semi-direct product W ⋉ P . It is known that W ∼ = W ⋉ Q and thus W is identified with a subgroup of f W . For ω ∈ P , write ω for theelement (1 , ω ) ∈ f W .
For s ∈ W , write s for the element ( s, ∈ f W .There is a f W -action on e Q extending the W -action on e Q such that ω ( α i ) = α i − h ω, α i i δ for ω ∈ P, i ∈ I . We identify P/Q with a finite group Ω of Dynkin diagram automorphism,and thus f W ∼ = Ω ⋉ W . The length function l on W extends to f W by setting l ( τ w ) = l ( w )for τ ∈ Ω , w ∈ W . DRINFELD TYPE PRESENTATION OF AFFINE ı QUANTUM GROUPS, II 5
Drinfeld presentation for affine quantum groups.
Let v be the quantum param-eter and v i = v d i . Define, for n ∈ Z , a ∈ Q ( v )[ n ] a = a n − a − n a − a − , [ n ] a ! = [ n ] a [ n − a · · · [1] a , (cid:20) ns (cid:21) a = [ n ] a ![ s ] a ![ n − s ] a ! . Write [
A, B ] = AB − BA and [ A, B ] a = AB − aBA .Let U be the Drinfled-Jimbo quantum group associated to b g with Chevalley generators { E i , F i , K ± i | i ∈ I } . Let U − be the subalgebra of U generated by F i , i ∈ I .It was formulated in [Dr88] that U is isomorphic to Dr U , where Dr U is the Q ( v )-algebragenerated by x ± ik , h il , K ± i , C ± , for i ∈ I , k ∈ Z , and l ∈ Z \{ } , subject to the followingrelations: C , C − are central,(2.3) [ K i , K j ] = [ K i , h jl ] = 0 , K i K − i = C C − = 1 , (2.4) [ h ik , h jl ] = δ k, − l [ kc ij ] v i k C k − C − k v j − v − j , (2.5) K i x ± jk K − i = v ± c ij i x ± jk , (2.6) [ h ik , x ± jl ] = ± [ kc ij ] v i k C ∓ | k | x ± j,k + l , (2.7) [ x + ik , x − jl ] = δ ij ( C k − l K i ψ i,k + l − C l − k K − i ϕ i,k + l ) , (2.8) x ± i,k +1 x ± j,l − v ± c ij i x ± j,l x ± i,k +1 = v ± c ij i x ± i,k x ± j,l +1 − x ± j,l +1 x ± i,k , (2.9) Sym k ,...,k r r X t =0 ( − t (cid:20) rt (cid:21) v i x ± i,k · · · x ± i,k t x ± j,l x ± i,k t +1 · · · x ± i,k n = 0 , for r = 1 − c ij ( i = j ) , (2.10)where Sym k ,...,k r denotes the symmetrization with respect to the indices k , . . . , k r , ψ i,k and ϕ i,k are defined by the following functional equations:1 + X m ≥ ( v i − v − i ) ψ i,m u m = exp (cid:16) ( v i − v − i ) X m ≥ h i,m u m (cid:17) , X m ≥ ( v i − v − i ) ϕ i, − m u − m = exp (cid:16) ( v i − v − i ) X m ≥ h i, − m u − m (cid:17) . We refer to [Be94] for a proof of the isomorphism φ : Dr U → U .In [Da12, § k = k = · · · = k − c ij . Let Dr U red ( red stands for the reduced presentation) denote the Q ( v )-algebra generated by x ± ik , h il , K ± i , C ± subject to relations (2.3) − (2.9) and(2.11) r X t =0 ( − t (cid:20) rt (cid:21) v i (cid:0) x ± i,k (cid:1) r − t x ± j,l (cid:0) x ± i,k (cid:1) t = 0 , for r = 1 − c ij , k, l ∈ Z ( i = j ) . Proposition 2.1 ([Da12, Theorem 11.18]) . Dr U is isomorphic to Dr U red by sending gener-ators x ± ik , h il , K ± i , C ± to those with same names. WEINAN ZHANG
Note Damiani’s original result is stronger than the one stated in Proposition 2.1, since italso involves a reduction of relations (2.5) (2.7), but this version is sufficient for our purpose.Compose the isomorphism in Proposition 2.1 with φ , we have an isomorphism(2.12) φ red : Dr U red −→ U . Universal ı quantum groups of split affine type. We recall the definition of theuniversal ı quantum group of split affine type via its Serre presentation following [LW20b, § Definition 2.2.
The universal (split) affine ı quantum group e U ı := e U ı ( b g ) associated to b g isthe Q ( v ) -algebra generated by B i , K ± i , i ∈ I , subject to K i K − i = K − i K i = 1 , K i is central , (2.13) B i B j − B j B i = 0 , if c ij = 0 , (2.14) B i B j − [2] v i B i B j B i + B j B i = − v − i B j K i , if c ij = − , (2.15) X r =0 ( − r (cid:20) r (cid:21) v i B − ri B j B ri = − v − i [2] v i ( B i B j − B j B i ) K i , if c ij = − , (2.16) X s =0 ( − s (cid:20) s (cid:21) v i B − si B j B si = − v − i (1 + [3] v i )( B j B i + B i B j ) K i (2.17) + v − i [4] v i (1 + [2] v i ) B i B j B i K i − v − i [3] v i B j K i , if c ij = − . Remark . For any ς = ( ς i ) i ∈ I ∈ ( Q × ) I , an affine ı quantum group U ı ς with parametersis introduced in [Ko14], generalizing G.Letzter’s work for finite type. U ı ς admits a Serrepresentation formulated in [Ko14, Theorem 7.1] and also in [CLW18, Theorem 3.1].The presentation for e U ı in Definition 2.2 can be obtained by replacing the parameter − v i ς i in the Serre presentation of U ı ς formulated in [CLW18, Theorem 3.1] by a central element K i for i ∈ I . (set τ = id there for split type) Hence, U ı ς is related to e U ı by a central reduction U ı ς := e U ı / ( K i + v i ς i | i ∈ I ). Remark . A Serre presentation for e U ı is also formulated with generators B i , e k i , i ∈ I in[LW19a, Proposition 6.4] for finite ADE type and in [LW20a, Theorem 4.2] for symmetricKac-Moody type. The central element K i is related to e k i by K i = − v i e k i . We are followingnotations in [LW20b] in this paper. Remark . e U ı has a ZI -grading by settingwt( B i ) = α i , wt( K i ) = 2 α i , i ∈ I . We say that B i has weight α i . Remark . e U ı has a natural filtered algebra structure by setting e U ı,m = Q ( v )-span { B i B i · · · B i r K µ | µ ∈ ZI , r ≤ m, i k ∈ I } . According to [Let02][Ko14], the associated graded algebra with respect to this filtration is(2.18) gr e U ı ∼ = U − ⊗ Q ( v )[ K ± i | i ∈ I ] , B i F i , K i K i ( i ∈ I ) . DRINFELD TYPE PRESENTATION OF AFFINE ı QUANTUM GROUPS, II 7
The following formulas for the braid group action on e U ı of finite ADE type were obtainedin [LW19b]; its generalization to Kac-Moody types is conjectured in [CLW20, Conjecture6.5] and will be proved in [LW21]. Lemma 2.7 ( [LW19b, Lemma 5.1], [CLW20, Conjecture 6.5],[LW21]) . For i ∈ I , there ex-ists an automorphism T i of the Q ( v ) -algebra e U ı such that T i ( K µ ) = K s i µ , and T i ( B j ) = B i K − i , if j = i,B j , if c ij = 0 ,B j B i − v i B i B j , if c ij = − , [2] − v i (cid:0) B j B i − v i [2] v i B i B j B i + v B i B j (cid:1) + B j K i , if c ij = − , [3] − v i [2] − v i (cid:0) B j B i − v i [3] v i B i B j B i + v [3] v i B i B j B i − v i B i B j + v − i [ B j , B i ] v i K i (cid:1) + [ B j , B i ] v i K i , if c ij = − . for µ ∈ ZI and j ∈ I . Moreover, T i ( i ∈ I ) satisfy the braid relations, i.e., T i T j = T j T i if c ij = 0 , and T i T j T i = T j T i T j if c ij c ji = 1 , and T i T j T i T j = T j T i T j T i if c ij c ji = 2 , and T i T j T i T j T i T j = T j T i T j T i T j T i if c ij c ji = 3 . Its inverse T − i is explicitly given by T − i ( K µ ) = K s i µ , andT − i ( B j ) = B i K − i , if j = i,B j , if c ij = 0 ,B i B j − v i B j B i , if c ij = − , [2] − v i (cid:0) B i B j − v i [2] v i B i B j B i + v i B j B i (cid:1) + B j K i , if c ij = − , [3] − v i [2] − v i (cid:0) B i B j − v i [3] v i B i B j B i + v [3] v i B i B j B i − v i B j B i + v − i [ B i , B j ] v i K i (cid:1) + [ B i , B j ] v i K i , if c ij = − . Remark . For specific parameters ς = ( ς i ) i ∈ I , ς i = − v − i , i ∈ I , a braid group action on U ı ς of split finite type is constructed in [KP11, Theorem 3.3]. By taking the central reductionin Remark 2.3, T i descends to an automorphism of U ı ς , which recovers Kolb and Pellegrini’sbraid group action.However, for general parameters ς , T i fails to become an automorphism of U ı ς via thecentral reduction. (A quick way to see this: since T i ( K i ) = K − i , if T i reduces to anautomorphism on U ı ς , then the image of K i , as a scalar in U ı ς , must be ± ς i = ± v − i .)A natural generalization of Kolb and Pellegrini’s braid group action to the split affine rankone case is formulated in [BK20, §
2] for equal parameters ς = ( ς , ς ) , ς = ς .For w ∈ f W with a reduced expression w = σs i . . . s i r , σ ∈ Ω, we define T w = σ T i . . . T i r ,where σ acts on e U ı by σ ( B i ) = B σi , σ ( K i ) = K σi , for all i ∈ I . By Lemma 2.7, T w isindependent of the choice of reduced expressions for w .The first property for this braid group action can be obtained by adapting [Lus89, § Lemma 2.9.
Let x ∈ P , i, j ∈ I . (a) If s i x = xs i , then T i T x = T x T i . WEINAN ZHANG (b) If s i xs i = α − i x = Q k ∈ I ω a k k , then we have T − i T x T − i = Q k ∈ I T a k ω k , in particular, T − i T ω i T − i = T − ω i Q k = i T − c ik ω k . (c) T ω i T ω j = T ω j T ω i . For i ∈ I , just as in [Be94, § ω ′ i = ω i s i and e U ı [ i ] be the subalgebra of e U ı generatedby B i , T ω ′ i ( B i ) , K i , K δ − α i . Since l ( ω ′ i ) = l ( ω i ) −
1, we have(2.19) T ω i = T ω ′ i T i . The following properties for this braid group action on e U ı are natural generalizations ofcorresponding results formulated in [LW20b, § Lemma 2.10 ( [LW20b, Lemma 3.5-3.6 and Proposition 3.9] ) . Let i ∈ I . (a) We have T w ( B i ) = B wi , for any w ∈ W such that wi ∈ I . (b) We have T ω j ( x ) = x , for any j = i and x ∈ e U ı [ i ] . (c) There exists a Q ( v ) -algebra isomorphism ℵ i : e U ı ( b sl ) → e U ı [ i ] , which sends B B i , B T ω ′ i ( B i ) , K K i , K K δ K − i . A Drinfeld type presentation for affine ı quantum groups A current presentation for e U ı of split affine type. New generators B i,k , Θ i,m areintroduced in [LW20b, (3.28)-(3.30)] for e U ı of split affine ADE type. We define elements B i,k , Θ i,m basically in the same way for e U ı of arbitrary split affine type.Define a sign function o ( · ) : I −→ {± } , such that o ( i ) o ( j ) = − c ij < B i,k , ´Θ i,m , Θ i,m in e U ı for i ∈ I , k ∈ Z and m ≥ B i,k = o ( i ) k T − kω i ( B i ) , (3.1) ´Θ i,m = o ( i ) m (cid:16) − B i,m − T ω ′ i ( B i ) + v i T ω ′ i ( B i ) B i,m − (3.2) + ( v i − m − X p =0 B i,p B i,m − p − K − i K δ (cid:17) , Θ i,m = ´Θ i,m − ⌊ m − ⌋ X a =1 ( v i − v − ai ´Θ i,m − a K aδ − δ m,ev v − mi K m δ . (3.3)In particular, B i, = B i . B i,k , Θ i,l are homogeneous with respect to ZI -grading on e U ı withweights wt( B i,k ) = α i + kδ, wt(Θ i,l ) = lδ. Set Θ i, = ( v i − v − i ) − , and Θ i,m = 0 , for m < . With root vectors defined above, a Drinfeld type presentation for the affine ı quantumgroup of split ADE type is introduced in [LW20b, § v by v i and adding theequal-index version of Serre relations, a current presentation for e U ı of arbitrary split affinetype is given in Definition 3.1. We call it a reduced Drinfeld type presentation since it is an ı analogue of reduced Drinfeld presentation U red . DRINFELD TYPE PRESENTATION OF AFFINE ı QUANTUM GROUPS, II 9
Definition 3.1.
Let Dr e U ıred be the Q ( v ) -algebra generated by K ± i , C ± , H i,m and B i,l , where i ∈ I , m ∈ Z ≥ , l ∈ Z , subject to the following relations, for m, n ∈ Z ≥ and k, l ∈ Z : K i , C are central, [ H i,m , H j,n ] = 0 , K i K − i = 1 , CC − = 1 , (3.4) [ H i,m , B j,l ] = [ mc ij ] v i m B j,l + m − [ mc ij ] v i m B j,l − m C m , (3.5) [ B i,k , B j,l +1 ] v − ciji − v − c ij i [ B i,k +1 , B j,l ] v ciji = 0 , if i = j, (3.6) [ B i,k , B i,l +1 ] v − i − v − i [ B i,k +1 , B i,l ] v i = v − i Θ i,l − k +1 C k K i − v − i Θ i,l − k − C k +1 K i (3.7) + v − i Θ i,k − l +1 C l K i − v − i Θ i,k − l − C l +1 K i , [ B i,k , B j,l ] = 0 , if c ij = 0 , (3.8) X s =0 ( − s (cid:20) s (cid:21) v i B − si,k B j,l B si,k = − v − i B j,l K i C k , if c ij = − , (3.9) X s =0 ( − s (cid:20) s (cid:21) v i B − si,k B j,l B si,k = − v − i [2] v i ( B i,k B j,l − B j,l B i,k ) K i C k , if c ij = − , (3.10) X s =0 ( − s (cid:20) s (cid:21) v i B − si,k B j,l B si,k = − v − i (1 + [3] v i )( B j,l B i,k + B i,k B j,l ) K i C k (3.11) + v − i [4] v i (1 + [2] v i ) B i,k B j,l B i,k K i C k − v − i [3] v i B j,l K i C k , if c ij = − , where Θ i,m are related to H i,m by the following functional equation: X m ≥ ( v i − v − i )Θ i,m u m = exp (cid:16) ( v i − v − i ) X m ≥ H i,m u m (cid:17) . (3.12) Theorem 3.2.
There is a Q ( v ) -algebra isomorphism Φ red : Dr e U ıred → e U ı , which sends B i,k B i,k , H i,k H i,k , Θ i,k Θ i,k , K i K i , C K δ , for i ∈ I , k ∈ Z , m ≥ .Proof. Most defining relations for Dr e U ıred are verified in e U ı in similar ways as [LW20b], exceptthe relation (3.5) for i = j . We postpone details of the proof of this relation to Section 4.Relations (3.4)-(3.5) for i = j and the relation (3.7) follow from Lemma 2.10(c) and therank one computation offered by Lu and Wang; see [LW20b, Theorem 2.16] for a summary.For the relation (3.6), observe that its LHS satisfies[ B i,k , B j,l +1 ] v − ciji − v − c ij i [ B i,k +1 , B j,l ] v ciji = [ B j,l , B i,k +1 ] v − cjij − v − c ji j [ B j,l +1 , B i,k ] v cjij . Hence, it suffices to prove it for c ij = − ,
0. For these two cases, it is verified in the sameway as [LW20b, § i = j is given in Section 4, using other definingrelations (3.6)-(3.7); note that proofs for these two relations, as provided above, do not needthe relation (3.5), and hence we did not run into a circular. The relation (3.4) for i = j is verified using (3.5) and (3.7) in the similar way as [LW20b, § − kω i T − lω j to finite type Serre relations(2.14)-(2.17) respectively. Hence, Φ red is a well defined homomorphism.The surjectivity and injectivity of Φ red follows by similar arguments in [LW20b, proof ofTheorem 3.13]. (For surjectivity, one need to replace all Dr e U ı and Dr U in their argumentsby Dr e U ıred and Dr U red respectively, and follow similar arguments there.) (cid:3) Define generating functions B i ( z ) = P k ∈ Z B i,k z k , Θ i ( z ) = 1 + P m> ( v i − v − i )Θ i,m z m , H i ( u ) = P m ≥ H i,m u m , ∆ ( z ) = P k ∈ Z C k z k . (3.13)Then (3.5) can be written in terms of generating functions as( v i − v − i )[ ∂∂z H i ( z ) , B j ( w )](3.14) = − v c ij i zw − − − v − c ij i zw − − − v c ij i zw + 11 − v − c ij i zw ! B j ( w ) , and (3.12) can be written as(3.15) Θ i ( z ) = exp(( v i − v − i ) H i ( z )) . Conjugate (3.14) by Θ i ( z ). Since ( v i − v − i ) Θ i ( z ) ∂∂z H i ( z ) = ∂∂z Θ i ( z ), we have ∂∂z (cid:0) Θ i ( z ) B j ( w ) Θ − i ( z ) (cid:1) = − v c ij i zw − − − v − c ij i zw − − − v c ij i zwC + 11 − v − c ij i zwC ! Θ i ( z ) B j ( w ) Θ i ( z ) − . By integrating it with respect to z , we obtain the following equivalent formulation of (3.14)(3.16) Θ i ( z ) B j ( w ) = − v − c ij i zw − − v c ij i zw − · − v c ij i zwC − v − c ij i zwC ! B j ( w ) Θ i ( z ) . Hence, (3.14) is equivalent to (3.16). Relation (3.16) can be written component-wisely asthe following relation,(3.17) [Θ i,k , B j,l ] + [Θ i,k − , B j,l ] C = v c ij i [Θ i,k − , B j,l +1 ] v − ciji + v − c ij i [Θ i,k − , B j,l − ] v ciji C. Thus, (3.5) is equivalent to (3.17). See also [LW20b, Proposition 2.8] for the rank one case,and [LW20b, Proposition 3.12] for the general case.
Corollary 3.3.
There exists a Q ( v ) -algebra antiautomorphism Ψ : Dr e U ıred → Dr e U ıred givenby B i,k B i, − k , H i,l C − l H i,l , Θ i,r C − r Θ i,r ,C C − , K i K i , DRINFELD TYPE PRESENTATION OF AFFINE ı QUANTUM GROUPS, II 11 for k ∈ Z , r, l > , i ∈ I .Proof. Note that Ψ = 1. It is straightforward to check that Ψ preserves defining relations(3.5)-(3.11) of Dr e U ıred . Hence, Ψ is a well-defined antiautomorphism of Dr e U ıred . (cid:3) By computing weights in the sense of Remark 2.5, we can regard Ψ as the reflection α + kδ α − kδ, α ∈ R , k ∈ Z on the affine root system R . The effect of Ψ can be writtenin the generating function format byΨ( B i ( w )) = B i ( w − ) , Ψ (cid:0) ∆ ( wz ) Θ i ( z ) (cid:1) = ∆ (cid:0) ( zw ) − (cid:1) Θ i ( w − ) . Let Θ i ( s, r ) be a short notation for the RHS of (3.7), i.e.,Θ i ( s, r ) := − Θ i,s − r +1 C r K i + v − i Θ i,s − r − C r +1 K i − Θ i,r − s +1 C s K i + v − i Θ i,r − s − C s +1 K i . Note that Θ i ( s, r ) = Θ i ( r, s ). By a direct computation, we haveΨ(Θ i ( s, r )) = Θ i ( − r − , − s − . (3.18)3.2. A Drinfeld type presentation for e U ı of split affine BCF type. In this section, weadd the general version of Serre relations to the current presentation given in Definition 3.1;then we provide a Drinfeld type presentation in terms of generating functions in Theorem3.4 for e U ı ( b g ), where g is allowed to be any finite type except G .Recall generating functions defined in (3.13). Define S ( w , w , w | z ; i, j ) to be the followingexpression Sym w ,w ,w ( X r =0 ( − − r (cid:20) r (cid:21) v i B i ( w ) · · · B i ( w r ) B j ( z ) B i ( w r +1 ) · · · B i ( w ) ) , (3.19)and similarly define S ( w , w | z ; i, j ) to be the following expressionSym w ,w ( X r =0 ( − r (cid:20) r (cid:21) v i B i ( w ) · · · B i ( w r ) B j ( z ) B i ( w r +1 ) · · · B i ( w ) ) . (3.20)Denote φ i ( w , w , w ) = v − i w w − − w w w − + w w − + w w − + w w − w − + w w w − − ([3] v i − w w − ,ψ i ( w , w , w ) = − − (1 − v − i ) w w − + v − i w w − − w w − + v − i w w w − w w − + w w − + w w − + w w − w − + w w w − − ([3] v i − w w − . Details for the proof of the following general version of Serre relations (3.26) and (3.27) aregiven in Section 5.
Theorem 3.4.
The universal affine ı quantum group e U ı is isomorphic to the Q ( v ) -algebra Dr e U ı which is defined by generators K ± i , C ± , Θ i,m , B i,k ( i ∈ I , m ≥ , k ∈ Z ) , andfollowing defining relations, for i, j ∈ I : K i , C are central, Θ i ( z ) Θ j ( w ) = Θ j ( w ) Θ i ( z ) , (3.21) B j ( w ) Θ i ( z ) = − v c ij i zw − − v − c ij i zw − · − v − c ij i zwC − v c ij i zwC ! Θ i ( z ) B j ( w ) , (3.22) ( v c ij i z − w ) B i ( z ) B j ( w ) + ( v c ij i w − z ) B j ( w ) B i ( z ) = 0 , if i = j, (3.23) ( v i z − w ) B i ( z ) B i ( w ) + ( v i w − z ) B i ( w ) B i ( z )(3.24) = v − i ∆ ( zw ) v i − v − i (cid:0) ( v i z − w ) Θ i ( w ) + ( v i w − z ) Θ i ( z ) (cid:1) K i , B i ( w ) B j ( z ) − B j ( z ) B i ( w ) = 0 , if c ij = 0 , (3.25) S ( w , w | z ; i, j )(3.26) = − Sym w ,w ∆ ( w w ) v i − v − i [2] v i zw − − v i w w − [ Θ i ( w ) , B j ( z )] v − i K i − Sym w ,w ∆ ( w w ) v i − v − i w w − − v i w w − [ B j ( z ) , Θ i ( w )] v − i K i , if c ij = − , S ( w , w , w | z ; i, j )(3.27) = v i [2] v i [3] v i z − Sym w ,w ,w ∆ ( w w ) v i − v − i φ i ( w , w , w ) (cid:2) B i ( w ) , [ B j ( z ) , Θ i ( w )] v − i (cid:3) K i − [3] v i z − Sym w ,w ,w ∆ ( w w ) v i − v − i φ i ( w , w , w ) (cid:2) [ B j ( z ) , B i ( w )] v − i , Θ i ( w ) (cid:3) K i − v i [2] v i Sym w ,w ,w ∆ ( w w ) v i − v − i ψ i ( w , w , w ) (cid:2) [ Θ i ( w ) , B j ( z )] v − i , B i ( w ) (cid:3) K i + Sym w ,w ,w ∆ ( w w ) v i − v − i ψ i ( w , w , w ) (cid:2) Θ i ( w ) , [ B i ( w ) , B j ( z )] v − i (cid:3) K i , if c ij = − . where φ i ( w , w , w ) , ψ i ( w , w , w ) are defined above.Proof. By Theorem 3.2, it suffices to show that Dr e U ıred is isomorphic to Dr e U ı . The compo-nentwise version of relations (3.23)-(3.25) are the same as relations (3.6)-(3.8), and the com-ponentwise version of the relation (3.22) is the relation (3.17). One can find a proof for thisin [LW20b, Theorem 5.1]. By a direct computation, (3.9) is the w k w k z l component of (3.26),and (3.10) is the w k w k w k z l component of (3.27). Hence, the map Φ red : Dr e U ıred −→ Dr e U ı bysending generators K ± i , C ± , Θ i,m , B i,k to those with same names is a well-defined homo-morphism.We will show in Section 5 that relations (3.26) and (3.27) can be derived from definingrelations of Dr e U ıred . Thus, the inverse of Φ red constructed in the obvious way is well-defined,which implies Φ red is an isomorphism. (cid:3) Remark . As pointed out in the proof, when g is of ADE type, this presentation is identicalto the one in [LW20b, § Remark . As originally formulated in [LW20b, (3.32)(3.38)(5.6)], (3.26) can be writtencomponent-wisely as(3.28) S ( k , k | l ; i, j ) = R ( k , k | l ; i, j ) , DRINFELD TYPE PRESENTATION OF AFFINE ı QUANTUM GROUPS, II 13 where S ( k , k | l ; i, j ) = Sym k ,k (cid:0) B i,k B i,k B j,l − [2] v i B i,k B j,l B i,k + B j,l B i,k B i,k (cid:1) , (3.29) R ( k , k | l ; i, j ) = Sym k ,k K i C k (cid:16) − X p ≥ v pi [2] v i [Θ i,k − k − p − , B j,l − ] v − i C p +1 (3.30) − X p ≥ v p − i [2] v i [ B j,l , Θ i,k − k − p ] v − i C p − [ B j,l , Θ i,k − k ] v − i (cid:17) . Remark . One can obtain the componentwise formulas of (3.27) by expanding the de-nominators of φ i ( w , w , w ) and ψ i ( w , w , w ). Note that, after rewriting w − as w C using∆( w w ), denominators of φ i ( w , w , w ) , ψ i ( w , w , w ) have the form 1 + A such that w and nonpositive powers of w do not appear in A . Hence, once we expand the denominators,each component of the RHS will be a finite sum.The constant component of (3.27) is the same as (2.16). The general componentwiseformula of (3.27) is, however, too complicated to write down. Remark . Relations (3.21)-(3.25) are homogeneous by a direct observation on their com-ponentwise formulas. Relations (3.26) and (3.27) are also homogeneous, since they can bederived from relations (3.21)-(3.25) in Section 5.
Remark . Recall the filtration and e U ı,m in Remark 2.5. For any β = P i ∈ I n i α i ∈ R + ,define its height to be ht + ( β ) = X i ∈ I n i . Let d = ht( δ ). By similar arguments in [BK20, Proposition 4.4], B i,k ∈ e U ı, kd \ e U ı,kd , Θ i,l ∈ e U ı,ld \ e U ı,ld − , H i,l ∈ e U ı,ld \ e U ı,ld − , and the images of B i,k , Θ i,l , H i,l in gr U ı are, up to a Q ( v )[ K ± i ] multiple, Drinfeld generators x − i, − k , ϕ i, − l , h i, − l of U − respectively for k ≥ , l >
0. Since B i, − k = − vi [ H i,k , B i ] C − + B i,k C − for k >
0, we have B i, − k ∈ e U ı, kd . We claim the w k w k w k z l component of (3.27) for k , k , k , l ≥ e U ı ∼ = U − ⊗ Q ( v )[ K ± i | i ∈ I ]. Observe that this component has the formSym k ,k ,k X t =0 ( − t (cid:20) t (cid:21) v i B i,k · · · B i,k t B j,l B i,k t +1 · · · B i,k n (3.31)= Sym k ,k ,k (cid:18) X ( ∗ )Θ i,k − s B j,l ′ B i,k + t + X ( ∗ )Θ i,k − s B i,k + t B j,l ′ + X ( ∗ ) B j,l ′ Θ i,k − s B i,k + t + X ( ∗ ) B i,k + t Θ i,k − s B j,l ′ + X ( ∗ ) B j,l ′ B i,k + t Θ i,k − s + X ( ∗ ) B i,k + t B j,l ′ Θ i,k − s (cid:19) . where coefficients ( ∗ ) lie in Q ( v )[ K ± i | i ∈ I ] and each sum ranges in 0 ≤ s ≤ k , − s ≤ t ≤ s, l ′ ∈ { l, l + 1 } . By a direct computation of heights, the RHS lies in e U ı, k + k + k + l ) d , whilethe LHS lies in e U ı, k + k + k + l ) d \ e U ı, k + k + k + l ) d . Hence, the RHS of (3.31) disappears in gr e U ı , and thus the componentwise version of (3.27) reduces to the Serre relation (2.10) inthe original Drinfeld presentation.4. Verification of the relation (3.5)In this section, we establish the relation (3.5) for i = j in U ı and complete the proof ofTheorem 3.2.Recall that (3.5) is equivalent to (3.17). Hence, it suffices to show that (3.17) for i = j holds in U ı . Fix i = j ∈ I and denote(4.1) Y k,l = [Θ i,k , B j,l ] + [Θ i,k − , B j,l ] C − v c ij i [Θ i,k − , B j,l +1 ] v − ciji − v − c ij i [Θ i,k − , B j,l − ] v ciji C. Since Θ i, = v i − v − i and Θ i,k = 0 , ∀ k < Y k,l = 0 if k ≤ Y k,l = 0.We will show that Y k,l = 0 for k > , l ∈ Z in e U ı in this section, in order to verify therelation (3.17). Other two defining relations (3.6) and (3.7) of Dr e U ıred are allowed to be usedin this section, since their proof does not need (3.5).Recall some basic properties for q -brackets, which shall be used heavily in various compu-tations in this section as well as remaining sections. Lemma 4.1 ([Da12, Remark 4.17], also [Ji98, Introduction]) . Let a, b, c ∈ e U ı , u, v, w ∈ C ( q ) \{ } . We have(1) [ a, b ] u = − u [ b, a ] u − ,(2) (cid:2) [ a, b ] u , b (cid:3) v = (cid:2) [ a, b ] v , b (cid:3) u ,(3) (cid:2) [ a, b ] u , c (cid:3) v = (cid:2) a, [ b, c ] v/w (cid:3) uw − u (cid:2) b, [ a, c ] w (cid:3) v/uw . An induction on k . Since the index l of Y k,l can be shifted using T ω j , it suffices tofocus on k . We first establish an inductive formula on k , which relates Y k,l and Y k +2 ,l . Such aninduction is partially inspired by Damiani’s reduction for the relation (2.7) affine quantumgroup [Da12, Proposition 7.15].
Proposition 4.2.
Let k > or k = 0 . We have for l ∈ Z , (4.2) Y k +2 ,l = v − i Y k,l C. We also have Y ,l = (1 − v − i ) Y ,l C for l ∈ Z .Proof. Write Y k +2 ,l K i − v − i Y k,l C K i = Σ + Σ + Σ + Σ for k > k = 0 where eachsummand Σ i is defined and rewritten as follows:Σ := (cid:2) Θ i,k +2 − v − i Θ i,k C, B j,l (cid:3) K i (4.3) = − (cid:2) [ B i,k +2 , B i ] v i , B j,l (cid:3) − (cid:2) [ B i, , B i,k +1 ] v i , B j,l (cid:3) , Σ :=[Θ i,k − v − i Θ i,k − C, , B j,l ] K i C (4.4) = − (cid:2) [ B i,k +1 , B i, ] v i , B j,l (cid:3) − (cid:2) [ B i, , B i,k ] v i , B j,l (cid:3) ,v − c ij i Σ := − [Θ i,k +1 − v − i Θ i,k − C, B j,l +1 ] v − ciji K i (4.5) = (cid:2) [ B i,k +1 , B i ] v i , B j,l +1 (cid:3) v − ciji + (cid:2) [ B i, , B i,k ] v i , B j,l +1 (cid:3) v − ciji DRINFELD TYPE PRESENTATION OF AFFINE ı QUANTUM GROUPS, II 15 = (cid:2) B i,k +1 , [ B i , B j,l +1 ] v − ciji (cid:3) v − ciji − v i (cid:2) B i , [ B i,k +1 , B j,l +1 ] v − ciji (cid:3) v − − ciji + (cid:2) B i, , [ B i,k , B j,l +1 ] v − ciji (cid:3) v − ciji − v i (cid:2) B i,k , [ B i, , B j,l +1 ] v − ciji (cid:3) v − − ciji = − (cid:2) B i,k +1 , [ B j,l , B i, ] v − ciji (cid:3) v − ciji + v i (cid:2) B i , [ B j,l , B i,k +2 ] v − ciji (cid:3) v − − ciji − (cid:2) B i, , [ B j,l , B i,k +1 ] v − ciji (cid:3) v − ciji + v i (cid:2) B i,k , [ B j,l , B i, ] v − ciji (cid:3) v − − ciji ,v c ij i Σ := − [Θ i,k +1 − v − i Θ i,k − C, B j,l − ] v ciji C K i (4.6) = (cid:2) [ B i,k +2 , B i, ] v i , B j,l − (cid:3) v ciji + (cid:2) [ B i, , B i,k +1 ] v i , B j,l − (cid:3) v ciji = (cid:2) B i,k +2 , [ B i, , B j,l − ] v ciji (cid:3) v ciji − v i (cid:2) B i, , [ B i,k +2 , B j,l − ] v ciji (cid:3) v − ciji + (cid:2) B i, , [ B i,k +1 , B j,l − ] v ciji (cid:3) v ciji − v i (cid:2) B i,k +1 , [ B i, , B j,l − ] v ciji (cid:3) v − ciji = − (cid:2) B i,k +2 , [ B j,l , B i ] v ciji (cid:3) v ciji + v i (cid:2) B i, , [ B j,l , B i,k +1 ] v ciji (cid:3) v − ciji − (cid:2) B i, , [ B j,l , B i,k +1 ] v ciji (cid:3) v ciji + v i (cid:2) B i,k +1 , [ B j,l , B i, ] v ciji (cid:3) v − ciji , where relation (3.7) is used in the first equality in each of (4.3)-(4.6), and relation (3.6) isused in the last equality of (4.5)(4.6). Now, adding (4.3)-(4.6) together, we have Y k +2 ,l K i − v − i Y k,l C K i = Σ + Σ + Σ + Σ = − (cid:2) [ B i,k +2 , B i ] v i , B j,l (cid:3) − (cid:2) [ B i, , B i,k +1 ] v i , B j,l (cid:3) − (cid:2) [ B i,k +1 , B i, ] v i , B j,l (cid:3) − (cid:2) [ B i, , B i,k ] v i , B j,l (cid:3) − v c ij i (cid:2) B i,k +1 , [ B j,l , B i, ] v − ciji (cid:3) v − ciji + v c ij i (cid:2) B i , [ B j,l , B i,k +2 ] v − ciji (cid:3) v − − ciji − v c ij i (cid:2) B i, , [ B j,l , B i,k +1 ] v − ciji (cid:3) v − ciji + v c ij i (cid:2) B i,k , [ B j,l , B i, ] v − ciji (cid:3) v − − ciji − v − c ij i (cid:2) B i,k +2 , [ B j,l , B i ] v ciji (cid:3) v ciji + v − c ij i (cid:2) B i, , [ B j,l , B i,k +1 ] v ciji (cid:3) v − ciji − v − c ij i (cid:2) B i, , [ B j,l , B i,k +1 ] v ciji (cid:3) v ciji + v − c ij i (cid:2) B i,k +1 , [ B j,l , B i, ] v ciji (cid:3) v − ciji =0 , where the last step follows by a direct computation using Lemma 4.1. Hence, Y k +2 ,l = v − i Y k,l C for k > k = 0. For k = 1, using a similar method, we have Y ,l = (1 − v − i ) Y ,l C . (cid:3) Base cases.
By Proposition 4.2, Y m,l is a scalar multiple of Y ,l and Y m − ,l is a scalarmultiple of Y ,l for m > , l ∈ Z . Since Y ,l = 0 as discussed in the beginning of Section 4, itremains to show that Y ,l = 0 for l ∈ Z .We explain the underlying idea for proving the base case Y ,l = 0, since details in the proofare quite technical. By the definition (4.1), we have(4.7) Y ,l = [Θ i, , B j,l ] − [ c ij ] v i B j,l +1 + [ c ij ] v i B j,l − C. Since Θ i, = − [ B i, , B i ] v i by (3.7), we replace Θ i, in (4.7) by the q -brackets of real rootvectors and we obtain(4.8) Y ,l = − (cid:2) [ B i, , B i ] v i , B j,l (cid:3) − [ c ij ] v i B j,l +1 + [ c ij ] v i B j,l − C. We prove that the RHS of (4.8) equal 0 in separate cases depending on c ij , i, j ∈ I . For c ij = − , we use the finite type Serre relation (2.14). For c ij = −
2, we use the formulas ofT i , T − i in Lemma 2.7. For c ij = −
3, we use both of the finite type Serre relation (2.16) andthe formulas of T i , T − i .We also recall that, by Lemma 2.10(b) and the construction of real root vectors, T ω j fixes B i,k for any j = i, k ∈ Z while T ω i sends B i,k to o ( i ) B i,k − .We now start to prove Y ,l = 0 case by case.(1) c ij = c ji = 0. In this case, since both B i , B i, commute with B j,l , Θ i, commutes with B j,l for l ∈ Z . Hence, Y ,l = 0.(2) c ij = c ji = −
1. We rewrite the finite type Serre relation (2.15) in terms of q -bracketsas(4.9) (cid:2) B i , [ B i , B j ] v i (cid:3) v − i = − v − i B j K i , (cid:2) [ B j , B i ] v i , B i (cid:3) v − i = − v − i B j K i . i.e. each of these two relations is equivalent to (2.15).Applying o ( j ) l T − lω j T − kω i to them, for k, l ∈ Z , we have(4.10) (cid:2) B i,k , [ B i,k , B j,l ] v i (cid:3) v − i = − v − i B j,l K i C k , (cid:2) [ B j,l , B i,k ] v i , B i,k (cid:3) v − i = − v − i B j,l K i C k . We now compute[Θ i, , B j ] K i (3.7) = − (cid:2) [ B i, , B i ] v i , B j (cid:3) = − (cid:2) B i, , [ B i , B j ] v i (cid:3) v i + v i (cid:2) B i , [ B i, , B j ] v − i (cid:3) v − i (3.6) = (cid:2) B i, , [ B j, − , B i, ] v i (cid:3) v i − v i (cid:2) B i , [ B j, , B i ] v − i (cid:3) v − i (4.10) = B j, − K i C − B j, K i . Hence, Y , = 0, and by applying T − lω j , we get Y ,l = 0.(3) c ij = − , c ji = −
1. We first write T i ( B j ) defined in Lemma 2.7 in terms of q -bracketsas [2] v i T i ( B j ) = (cid:2) [ B j , B i ] v i , B i (cid:3) + [2] v i B j K i , (4.11) [2] v i T − i ( B j ) = (cid:2) B i , [ B i , B j ] v i (cid:3) + [2] v i B j K i . (4.12)Since T i , T ω j commute by Lemma 2.9(a), applying o ( j ) l T − lω j to these equalities, we have for l ∈ Z , [2] v i T i ( B j,l ) = (cid:2) [ B j,l , B i ] v i , B i (cid:3) + [2] v i B j,l K i , (4.13) [2] v i T − i ( B j,l ) = (cid:2) B i , [ B i , B j,l ] v i (cid:3) + [2] v i B j,l K i . (4.14)Apply T − ω i to (4.13), we have(4.15) [2] v i T − ω i T i ( B j,l ) = (cid:2) [ B j,l , B i, ] v i , B i, (cid:3) + [2] v i B j,l K i C. We now compute[Θ i, , B j ] K i (3.7) = − (cid:2) [ B i, , B i ] v i , B j (cid:3) = − (cid:2) B i, , [ B i , B j ] v i (cid:3) + v i (cid:2) B i , [ B i, , B j ] v − i (cid:3) DRINFELD TYPE PRESENTATION OF AFFINE ı QUANTUM GROUPS, II 17 (3.6) = (cid:2) B i, , [ B j, − , B i, ] v i (cid:3) − v i (cid:2) B i , [ B j, , B i ] v − i (cid:3) (4.14) = (cid:2) B i, , [ B j, − , B i, ] v i (cid:3) + [2] v i T − i ( B j, ) − [2] v i B j, K i (4.15) = − [2] v i T − ω i T i ( B j, − ) + [2] v i B j, − K i C + [2] v i T − i ( B j, ) − [2] v i B j, K i = [2] v i B j, − C K i − [2] v i B j, K i − [2] v i (cid:0) T − ω i T i ( B j, − ) − T − i ( B j, ) (cid:1) = [2] v i B j, − C K i − [2] v i B j, K i , where the last step follows from T i T − ω i T i = T ω i T − ω j Q k = i,j T c ik ω k , which is given by Lemma2.9(b). Hence, Y , = 0, and by applying T − lω j , we get Y ,l = 0.(4) c ij = − , c ji = −
1. Without loss of generality, assume o ( i ) = 1. In this case, we rewriteT i ( B j ) in Lemma 2.7 asT i ( B j ) = 1[3] v i ! (cid:20)(cid:2) [ B j , B i ] v i , B i (cid:3) v i , B i (cid:21) v − i + 1[3] v i ! v − i [ B j , B i ] v i K i + [ B j , B i ] v i K i , (4.16) T − i ( B j ) = 1[3] v i ! (cid:20) B i , (cid:2) B i , [ B i , B j ] v i (cid:3) v i (cid:21) v − i + 1[3] v i ! v − i [ B i , B j ] v i K i + [ B i , B j ] v i K i . (4.17)Since T i , T ω j commute, applying o ( j ) l T − lω j for l ∈ Z to these equalities, we haveT i ( B j,l ) = 1[3] v i ! (cid:20)(cid:2) [ B j,l , B i ] v i , B i (cid:3) v i , B i (cid:21) v − i + 1[3] v i ! v − i [ B j,l , B i ] v i K i + [ B j,l , B i ] v i K i , (4.18) T − i ( B j,l ) = 1[3] v i ! (cid:20) B i , (cid:2) B i , [ B i , B j,l ] v i (cid:3) v i (cid:21) v − i + 1[3] v i ! v − i [ B i , B j,l ] v i K i + [ B i , B j,l ] v i K i . (4.19)In particular, for l = − i ( B j, − ) , B i ] v − i = 1[3] v i ! (cid:20)(cid:20)(cid:2) [ B j, − , B i ] v i , B i (cid:3) v i , B i (cid:21) v − i , B i (cid:21) v − i (4.20) + 1[3] v i ! v − i (cid:2) [ B j, − , B i ] v i , B i (cid:3) v − i K i + (cid:2) [ B j, − , B i ] v i , B i (cid:3) v − i K i , [ B i , T − i ( B j, )] v − i = 1[3] v i ! (cid:20) B i , (cid:20) B i , (cid:2) B i , [ B i , B j, ] v i (cid:3) v i (cid:21) v − i (cid:21) v − i (4.21) + 1[3] v i ! v − i (cid:2) B i , [ B i , B j, ] v i (cid:3) v − i K i + (cid:2) B i , [ B i , B j, ] v i (cid:3) K i . Apply T − ω i to (4.20), since T − ω i T i ( B i ) = T − ω i ( B i K − i ) = B i, K − i C − , we haveT − ω i T i (cid:0) [ B j, − , B i ] v − i (cid:1) K i C = 1[3] v i ! (cid:20)(cid:20)(cid:2) [ B j, − , B i, ] v i , B i, ] (cid:3) v i , B i, (cid:21) v − i , B i, (cid:21) v − i (4.22) + 1[3] v i ! v − i (cid:2) [ B j, − , B i, ] v i , B i, (cid:3) v − i K i C + (cid:2) [ B j, − , B i, ] v i , B i, (cid:3) v − i K i C. On the other hand, we also rewrite the finite type Serre relation (2.17) as (cid:20)(cid:20)(cid:2) [ B j , B i ] v i , B i (cid:3) v i , B i (cid:21) v − i , B i (cid:21) v − i = − v − i (1 + [3] v i )( B j B i + B i B j ) K i (4.23) + v − i [4] v i (1 + [2] v i ) B i B j B i K i − v − i [3] v i B j K i . Apply o ( j )T − ω j and we obtain (cid:20)(cid:20)(cid:2) [ B j, , B i ] v i , B i (cid:3) v i , B i (cid:21) v − i , B i (cid:21) v − i = − v − i (1 + [3] v i )( B j, B i + B i B j, ) K i (4.24) + v − i [4] v i (1 + [2] v i ) B i B j, B i K i − v − i [3] v i B j, K i . Note the leading term (of degree 5) on the RHS of (4.21) coincides with the LHS of (4.24).We substitute it using (4.24) and simplify as[3] v i [2] v i [ B i , T − i ( B j, )] v − i = [3] v i (cid:2) B i , [ B j, , B i ] v − i (cid:3) v i K i − v − i [3] v i B j, K i . (4.25)Similarly, we apply T ω j T − ω i to (4.23), by Lemma 4.1(a), (cid:20) B i, , (cid:20) B i, , (cid:2) B i, , [ B i, , B j, − ] v i (cid:3) v i (cid:21) v − i (cid:21) v − i = − v − i (1 + [3] v i )( B j, − B i, + B i, B j, − ) K i + v − i [4] v i (1 + [2] v i ) B i, B j, − B i, K i C − v − i [3] v i B j, − K i C . (4.26)and then we substitute the leading term of RHS of (4.22) using (4.26). We obtain[3] v i [2] v i T − ω i T i [ B j, − , B i ] v − i K i C = [3] v i (cid:2) [ B i, , B j, − ] v − i , B i, (cid:3) v i K i C − v − i [3] v i B j, − K i C , which can be simplified as[2] v i T − ω i T i [ B j, − , B i ] v − i = (cid:2) [ B i, , B j, − ] v − i , B i, (cid:3) v i − v − i [3] v i B j, − K i C. (4.27)We now compute Y , in this case.[Θ i, , B j ] K i (3.7) = − (cid:2) [ B i, , B i ] v i , B j (cid:3) = − (cid:2) B i, , [ B i , B j ] v i (cid:3) v − i + v i (cid:2) B i , [ B i, , B j ] v − i (cid:3) v i (3.6) = (cid:2) B i, , [ B j, − , B i, ] v i (cid:3) v − i − v i (cid:2) B i , [ B j, , B i ] v − i (cid:3) v i = v i (cid:2) [ B i, , B j, − ] v − i , B i, (cid:3) v i − v i (cid:2) B i , [ B j, , B i ] v − i (cid:3) v i ( ∗ ) = [3] v i B j, − K i C − [3] v i B j, K i + v i [2] v i (cid:18) T − ω i T i [ B j, − , B i ] v − i − [ B i , T − i ( B j, )] v − i K − i (cid:19) = [3] v i B j, − K i C − [3] v i B j, K i + v i [2] v i T − i (cid:18) T i T − ω i T i [ B j, − , B i ] v − i − [ B i , B j, ] v − i (cid:19) ( ∗∗ ) = [3] v i B j, − K i C − [3] v i B j, K i + v i [2] v i T − i (cid:18) − [ B j, , B i, − ] v − i − [ B i , B j, ] v − i (cid:19) , (3.6) = [3] v i B j, − K i C − [3] v i B j, K i DRINFELD TYPE PRESENTATION OF AFFINE ı QUANTUM GROUPS, II 19 where step (*) follows by applying (4.27) to the first term and applying (4.25) to the secondterm, and step (**) follows from T i T − ω i T i = T ω i T − ω j given in Lemma 2.9(b). (also o ( j ) = − Y , = 0 and by applying T − lω j we have Y ,l = 0 for l ∈ Z . Verification of Serre relations
The goal of this section is to establish general Serre relations (3.26)-(3.27) in Dr e U ıred . Wefirst recover the general Serre relation (3.26) formulated in [LW20b] for c ij = −
1, using amore straightforward approach compared with the original one. We generalize this approachand offer several formulations for the Serre relation for c ij = −
2. We obtain two symmetricformulations in § § Serre relation for c ij = − . Let c ij = −
1. Recall the notation S ( k , k | l ; i, j ) intro-duced in (3.29) and denote it by S ( k , k | l ) for short. The Serre relation (3.9), together withthe relation (3.5), is verified in [LW20b, § Lemma 5.1 ([LW20b, Lemma 4.13]) . For k , k , l ∈ Z , we have S ( k , k + 1 | l ) + S ( k + 1 , k | l ) − [2] v i S ( k , k | l + 1)= Sym k ,k (cid:16) − [Θ i,k − k +1 , B jl ] v − i C k + v − i [Θ i,k − k − , B jl ] v − i C k +1 (cid:17) K i . Lemma 5.2 ([LW20b, Lemma 4.9]) . For k , k , l ∈ Z , we have S ( k , k + 1 | l ) + S ( k + 1 , k | l ) − [2] v i S ( k + 1 , k + 1 | l − k ,k (cid:16) − [ B jl , Θ i,k − k +1 ] v − i C k + v − i [ B jl , Θ i,k − k − ] v − i C k +1 (cid:17) K i . Denote S ( w , w | z ) = Sym w ,w (cid:8) B i ( w ) B i ( w ) B j ( z ) − [2] v i B i ( w ) B j ( z ) B i ( w ) + B j ( z ) B i ( w ) B i ( w ) (cid:9) . Lemma 5.1 and 5.2 can be written in terms of generating functions respectively as( w − + w − − [2] v i z − ) S ( w , w | z )(5.1) = Sym w ,w ∆ ( w w ) v i − v − i ( v − i w − − w − )[ Θ i ( w ) , B j ( z )] v − i K i , and ( w + w − [2] v i z ) S ( w , w | z )(5.2) = Sym w ,w ∆ ( w w ) v i − v − i ( v − i w − w )[ B j ( z ) , Θ i ( w )] v − i K i . Then we calculate (5.1) × [2] z + (5.2) × ( w − + w − ) and we obtain( w − v i w )( w − − v − i w − ) S ( w , w | z ) =[2] z Sym w ,w ∆ ( w w ) v i − v − i ( v − i w − − w − )[ Θ i ( w ) , B j ( z )] v − i K i (5.3) + Sym w ,w ∆ ( w w ) v i − v − i ( v − i w − w )( w − + w − )[ B j ( z ) , Θ i ( w )] v − i K i . We also calculate (5.1) × ( w + w ) + (5.2) × [2] z − and we obtain( w − v i w )( w − − v − i w − ) S ( w , w | z )= Sym w ,w ∆ ( w w ) v i − v − i ( v − i w − − w − )( w + w )[ Θ i ( w ) , B j ( z )] v − i K i (5.4) + [2] z − Sym w ,w ∆ ( w w ) v i − v − i ( v − i w − w )[ B j ( z ) , Θ i ( w )] v − i K i . Simplify (5.3) as S ( w , w | z ) = − Sym w ,w ∆ ( w w ) v i − v − i [2] zw − v i w [ Θ i ( w ) , B j ( z )] v − i K i (5.5) − Sym w ,w ∆ ( w w ) v i − v − i w + w w − v i w [ B j ( z ) , Θ i ( w )] v − i K i , which is exactly (3.26).5.2. Symmetric formulation.
We now forward to the case c ij = −
2. Two symmetricformulations (5.6) and (5.7), generalizing Lemma 5.2 and 5.1, are formulated and verified inthis section. Denote S ( k , k , k | l ) = Sym k ,k ,k X s =0 ( − s (cid:20) s (cid:21) v i B i,k · · · B i,k s B j,l B i,k s +1 · · · B i,k . Note that S ( k , k , k | l ) is symmetric with respect to the first three components. Proposition 5.3.
We have, for any k , k , k , S ( k , k , k + 1 | l ) + S ( k + 1 , k , k | l ) + S ( k , k + 1 , k | l ) − [3] v i S ( k , k , k | l + 1)= − v − i [2] v i (cid:2) [Θ i ( k , k ) , B j,l ] v − i , B i,k (cid:3) v i + (cid:2) Θ i ( k , k ) , [ B i,k , B j,l ] v i (cid:3) v − i (5.6) − v − i [2] v i (cid:2) [Θ i ( k , k ) , B j,l ] v − i , B i,k (cid:3) v i + (cid:2) Θ i ( k , k ) , [ B i,k , B j,l ] v i (cid:3) v − i − v − i [2] v i (cid:2) [Θ i ( k , k ) , B j,l ] v − i , B i,k (cid:3) v i + (cid:2) Θ i ( k , k ) , [ B i,k , B j,l ] v i (cid:3) v − i . The following relation can be obtained from (5.6) by applying Ψ. S ( k − , k , k | l ) + S ( k , k − , k | l ) + S ( k , k , k − | l ) − [3] v i S ( k , k , k | l − v − i [2] v i (cid:2) B i,k , [ B j,l , Θ i ( k − , k − v − i (cid:3) v i − (cid:2) [ B j,l , B i,k ] v i , Θ i ( k − , k − (cid:3) v − i (5.7) v − i [2] v i (cid:2) B i,k , [ B j,l , Θ i ( k − , k − v − i (cid:3) v i − (cid:2) [ B j,l , B i,k ] v i , Θ i ( k − , k − (cid:3) v − i v − i [2] v i (cid:2) B i,k , [ B j,l , Θ i ( k − , k − v − i (cid:3) v i − (cid:2) [ B j,l , B i,k ] v i , Θ i ( k − , k − (cid:3) v − i . DRINFELD TYPE PRESENTATION OF AFFINE ı QUANTUM GROUPS, II 21 proof of Proposition 5.3.
Denote R ( k , k , k | l ) = Sym k ,k (cid:18) B i,k B i,k [ B i,k , B j,l ] v i − v − i [2] v i B i,k [ B i,k , B j,l ] v i B i,k + v − i [ B i,k , B j,l ] v i B i,k B i,k (cid:19) . (5.8)Note that R ( k , k , k | l ) is only symmetric w.r.t its first two components. In fact, R ( k , k , k | l )plays the role of breaking the symmetry of S ( k , k , k | l ) as it satisfies(5.9) S ( k , k , k | l ) = R ( k , k , k | l ) + R ( k , k , k | l ) + R ( k , k , k | l ) . We compute S ( k , k , k + 1 | l ) − [3] v i R ( k , k , k | l + 1)= (cid:26) (1 + v i ) B i,k [ B i,k +1 , B i,k ] v i B j,l + [ B i,k +1 , B i,k ] v i B i,k B j,l (5.10) − [3] v i [ B i,k +1 , B i,k ] v i B j,l B i,k − v − i [3] v i B i,k B j,l [ B i,k +1 , B i,k ] v i + v − i B j,l B i,k [ B i,k +1 , B i,k ] v i + ( v − i + v − i ) B j,l [ B i,k +1 , B i,k ] v i B i,k (cid:27) + { k ↔ k } . Rewrite (5.10) using the symmetrizer as S ( k , k , k + 1 | l ) − [3] v i R ( k , k , k | l + 1)= (cid:26) (1 + v i ) B i,k [ B i,k +1 , B i,k ] v i B j,l + [ B i,k +1 , B i,k ] v i B i,k B j,l (5.11) − [3] v i [ B i,k +1 , B i,k ] v i B j,l B i,k − v − i [3] v i B i,k B j,l [ B i,k +1 , B i,k ] v i + v − i B j,l B i,k [ B i,k +1 , B i,k ] v i + ( v − i + v − i ) B j,l [ B i,k +1 , B i,k ] v i B i,k (cid:27) + { k ↔ k } . On the other hand, the relation (3.7) implies that[ B i,k +1 , B i,k ] v i = Θ i ( k , k ) − [ B i,k +1 , B i,k ] v i , (5.12) [ B i,k +1 , B i,k ] v i = Θ i ( k , k ) − [ B i,k +1 , B i,k ] v i . (5.13)Substitute (5.12) and (5.13) into (5.11), and we have S ( k , k , k + 1 | l ) − [3] v i R ( k , k , k | l + 1)= − (cid:26) (1 + v i ) B i,k [ B i,k +1 , B i,k ] v i B j,l + [ B i,k +1 , B i,k ] v i B i,k B j,l (5.14) − [3] v i [ B i,k +1 , B i,k ] v i B j,l B i,k − v − i [3] v i B i,k B j,l [ B i,k +1 , B i,k ] v i + v − i B j,l B i,k [ B i,k +1 , B i,k ] v i + ( v − i + v − i ) B j,l [ B i,k +1 , B i,k ] v i B i,k (cid:27) − { k ↔ k } + Q , , where Q , denotes all terms involving the imaginary root vectors Q , = − v − i [2] v i (cid:2) [Θ i ( k , k ) , B j,l ] v − i , B i,k (cid:3) v i + (cid:2) Θ i ( k , k ) , [ B i,k , B j,l ] v i (cid:3) v − i − v − i [2] v i (cid:2) [Θ i ( k , k ) , B j,l ] v − i , B i,k (cid:3) v i + (cid:2) Θ i ( k , k ) , [ B i,k , B j,l ] v i (cid:3) v − i . We recognize that the first (cid:26) (cid:27) -term on the RHS of (5.14) is the same as the (cid:26) (cid:27) -term onthe RHS of (5.11), up to a swap of indices k ↔ k . Hence, we replace the one in (5.14) by(5.11). We do the same thing for the term { k ↔ k } in (5.14). S ( k , k , k + 1 | l ) − [3] v i R ( k , k , k | l + 1)= − (cid:18) S ( k , k , k + 1 | l ) − [3] v i R ( k , k , k | l + 1) (cid:19) + (cid:26) (1 + v i ) B i,k [ B i,k +1 , B i,k ] v i B j,l + [ B i,k +1 , B i,k ] v i B i,k B j,l − [3] v i [ B i,k +1 , B i,k ] v i B j,l B i,k − v − i [3] v i B i,k B j,l [ B i,k +1 , B i,k ] v i (5.15) + v − i B j,l B i,k [ B i,k +1 , B i,k ] v i + ( v − i + v − i ) B j,l [ B i,k +1 , B i,k ] v i B i,k (cid:27) − (cid:18) S ( k , k , k + 1 | l ) − [3] v i R ( k , k , k | l + 1) (cid:19) + (cid:26) k ↔ k (cid:27) + Q , . By (3.7), we have the following relation(5.16) [ B i,k +1 , B i,k ] v i + [ B i,k +1 , B i,k ] v i = Θ i ( k , k ) . Now we can apply the above relation to those two (cid:26) (cid:27) -terms in (5.15) and we obtain S ( k , k , k + 1 | l ) + S ( k + 1 , k , k | l ) + S ( k , k + 1 , k | l ) − [3] v i (cid:0) R ( k , k , k | l + 1) + R ( k , k , k | l + 1) + R ( k , k , k | l + 1) (cid:1) = − v − i [2] v i (cid:2) [Θ i ( k , k ) , B j,l ] v − i , B i,k (cid:3) v i + (cid:2) Θ i ( k , k ) , [ B i,k , B j,l ] v i (cid:3) v − i (5.17) − v − i [2] v i (cid:2) [Θ i ( k , k ) , B j,l ] v − i , B i,k (cid:3) v i + (cid:2) Θ i ( k , k ) , [ B i,k , B j,l ] v i (cid:3) v − i − v − i [2] v i (cid:2) [Θ i ( k , k ) , B j,l ] v − i , B i,k (cid:3) v i + (cid:2) Θ i ( k , k ) , [ B i,k , B j,l ] v i (cid:3) v − i . Finally, by (5.9), we obtain (5.6) from (5.17) as desired. (cid:3)
Generating function formulation.
By taking suitable linear combination of twosymmetric formulations, we derive the Serre relation (3.27) and thus finish the proof ofTheorem 3.4.Fix i, j ∈ I such that c ij = −
2. Recall the notation S ( w , w , w | z ; i, j ) from (3.19) anddenote it by S ( w , w , w | z ) for short. DRINFELD TYPE PRESENTATION OF AFFINE ı QUANTUM GROUPS, II 23
We can rewrite (5.6) in terms of generating functions as( w − + w − + w − − [3] v i z − ) S ( w , w , w | z )= − v i [2] v i Sym w ,w ,w ∆ ( w w ) v i − v − i ( v − i w − − w − ) (cid:2) [ Θ i ( w ) , B j ( z )] v − i , B i ( w ) (cid:3) K i (5.18) + Sym w ,w ,w ∆ ( w w ) v i − v − i ( v − i w − − w − ) (cid:2) Θ i ( w ) , [ B i ( w ) , B j ( z )] v − i (cid:3) K i , and rewrite (5.7) in terms of generating function as( w + w + w − [3] v i z ) S ( w , w , w | z )= v i [2] v i Sym w ,w ,w ∆ ( w w ) v i − v − i ( v − i w − w ) (cid:2) B i ( w ) , [ B j ( z ) , Θ i ( w )] v − i (cid:3) K i (5.19) − Sym w ,w ,w ∆ ( w w ) v i − v − i ( v − i w − w ) (cid:2) [ B j ( z ) , B i ( w )] v − i , Θ i ( w ) (cid:3) K i . We calculate (5.19) × [3] v i z − + (5.18) × ( w + w + w ) and obtain (cid:0) ( w + w + w )( w − + w − + w − ) − [3] v i (cid:1) S ( w , w , w | z ) K − i (5.20) =[3] v i z − Sym w ,w ,w ∆ ( w w ) v i − v − i ( v − i w − w ) × (cid:18) v i [2] v i (cid:2) B i ( w ) , [ B j ( z ) , Θ i ( w )] v − i (cid:3) − (cid:2) [ B j ( z ) , B i ( w )] v − i , Θ i ( w ) (cid:3)(cid:19) + Sym w ,w ,w ∆ ( w w ) v i − v − i ( v − i w − − w − )( w + w + w ) × (cid:18) − v i [2] v i (cid:2) [ Θ i ( w ) , B j ( z )] v − i , B i ( w ) (cid:3) + (cid:2) Θ i ( w ) , [ B i ( w ) , B j ( z )] v − i (cid:3)(cid:19) . Dividing both sides of (5.20) by the coefficient of S ( w , w , w | z ), we obtain the definingrelation (3.27) of Dr e U ı . Remark . Our formulation (3.27) of the Serre relation for c ij = − × ( w − + w − + w − ) + (5.18) × [3] v i z and obtain a variant of (5.20) as (cid:0) ( w + w + w )( w − + w − + w − ) − [3] v i (cid:1) S ( w , w , w | z ) K − i (5.21) = Sym w ,w ,w ∆ ( w w ) v i − v − i ( v − i w − w )( w − + w − + w − ) × (cid:18) v i [2] v i (cid:2) B i ( w ) , [ B j ( z ) , Θ i ( w )] v − i (cid:3) − (cid:2) [ B j ( z ) , B i ( w )] v − i , Θ i ( w ) (cid:3)(cid:19) + [3] v i z Sym w ,w ,w ∆ ( w w ) v i − v − i ( v − i w − − w − ) × (cid:18) − v i [2] v i (cid:2) [ Θ i ( w ) , B j ( z )] v − i , B i ( w ) (cid:3) + (cid:2) Θ i ( w ) , [ B i ( w ) , B j ( z )] v − i (cid:3)(cid:19) . Dividing both sides of (5.21) by the coefficient of S ( w , w , w | z ), we get an alternativeformulation for the Serre relation, which looks different from (3.27). In fact, (5.21) canbe obtained from (5.20) by applying Ψ, and thus the alternative formulation of the Serrerelation can also be obtained from (3.27) by applying Ψ. References [Be94] J. Beck,
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