A Drinfeld type presentation of affine ı quantum groups I: split ADE type
aa r X i v : . [ m a t h . R T ] S e p A DRINFELD TYPE PRESENTATION OF AFFINE ı QUANTUMGROUPS I: SPLIT ADE TYPE
MING LU AND WEIQIANG WANG
Abstract.
We establish a Drinfeld type new presentation for the ı quantum groups arisingfrom quantum symmetric pairs of split affine ADE type, which includes the q -Onsageralgebra as the rank 1 case. This presentation takes a form which can be viewed as adeformation of half an affine quantum group in the Drinfeld presentation. Contents
1. Introduction 12. The q -Onsager algebra and its Drinfeld type presentation 53. A Drinfeld type presentation of affine ı quantum groups 144. Verification of Drinfeld type new relations 265. Variants of Drinfeld type presentations 34References 371. Introduction
Background.
Affine Lie algebras distinguish themselves among Kac-Moody Lie alge-bras, largely due to 2 distinct constructions: a Serre presentation (as definition) and a loopalgebra realization. Such a dual nature of affine Lie algebras has led to numerous applicationsto string theory, modular forms, algebraic combinatorics, and so on.Remarkably, these constructions repeat themselves at the quantum level for Drinfeld-Jimbo quantum groups: besides the Serre presentation (as definition) [Dr87, Jim85], anaffine quantum group admits a current realization, also known as Drinfeld’s new presentation[Dr88]. An explicit isomorphism between these 2 presentations with proof was suppliedby Beck [Be94] in the untwisted affine type. Damiani [Da12, Da15] consolidated part ofBeck’s arguments and established interconnections among different relations in the currentrealization.Drinfeld’s current presentation of affine quantum groups has played a fundamental rolein numerous subsequent algebraic, geometric, and categorical developments. It has been in-strumental in the active area of (finite-dimensional) representation theory of affine quantumgroups U , in algebraic or combinatorial approach, by numerous authors including Chari,Kashiwara, and their collaborators and others; see the survey paper [CH10] for partial ref-erences; there have been connections with cluster algebras and monoidal categorification Mathematics Subject Classification.
Primary 17B37, 17B67.
Key words and phrases.
Affine quantum groups, Drinfeld presentation, ı Quantum groups, Quantum sym-metric pairs, q -Onsager algebra. [HL15]. A powerful geometric approach to representation theory of U was developed byGinzburg, Vasserot, and Nakajima (see [V98, Nak00]). Moreover, the Drinfeld presentationarises categorically in Hall algebras of coherent sheaves over (weighted) projective lines asinitiated by Kapranov; see [Ka97, BKa01, Sch04, DJX12].According to the ı program as outlined in [BW18], various algebraic, geometric, and cate-gorical constructions of quantum groups should be generalizable to ı quantum groups arisingfrom quantum symmetric pairs. As an ı -analogue of Drinfeld double quantum groups e U , theuniversal ı quantum groups e U ı was introduced by the authors in [LW19a] (also see [LW20])as they arise naturally from the ı Hall algebra constructions of ı quivers; moreover, there is abraid group action on e U ı which is realized by reflection functors in ı Hall algebras [LW19b]. Acentral reduction of e U ı recovers the ı quantum groups U ı = U ı ς with parameters ς ∈ Q ( v ) × , I introduced earlier by G. Letzter and generalized by Kolb [Let99, Let02, Ko14].We view ı quantum groups as a vast generalization of quantum groups, as quantum groupscan be regarded as ı quantum groups of diagonal type. The ı quantum groups of finite type areclassified by Satake diagrams (or the real forms of complex simple Lie algebras) of roughly2 dozens types, and there is an even richer family of affine ı quantum groups. The split andmore generally quasi-split ı quantum groups make sense in Kac-Moody generality and forma distinguished class of ı quantum groups. For some recent progress in representation theoryof ı quantum groups, see [Wat19].1.2. Goal.
The goal of this paper is to initiate a current realization of affine ı quantumgroups. We shall restrict ourselves to the split affine ADE type in this paper, where mostof the new features are already present; the Drinfeld type presentation for affine ı quantumgroup of rank one (known as q -Onsager algebra) is already new. We expect that the currentrealizations of affine ı quantum groups will open up algebraic, categorical, and geometricdevelopments as did Drinfeld’s presentation for the affine quantum group U = U ( b g ).Throughout the paper, we shall work with the universal ı quantum group e U ı , which con-tains central generators K i , for i ∈ I . The corresponding results for U ı (with arbitraryparameters) can be obtained from e U ı readily by a central reduction which specializes K i toscalars.1.3. Rank one.
The split affine ı quantum group U ı = U ı ς ( b sl ), with parameters ς = { ς , ς } ∈ ( Q ( v ) × ) , is known as q -Onsager algebra in the literature. There has been some at-tempts with mixed success by Baseilhac and collaborators toward a Dinfeld type presentationfor U ı , cf. [BS10, BK20] and references therein.Making an ansatz with the constructions for affine quantum group U ( b sl ) by Damiani[Da93], Baseilhac and Kolb [BK20] defined the q -root vectors in the q -Onsager algebra U ı (where both parameters were set to be equal: ς = ς , for technical reasons), and establisheda PBW basis for U ı . Along the way, an affine braid group action on U ı (when ς = ς ) isgiven. Various relations among the q -root vectors were computed, but clearly they do notresemble the relations in Drinfeld’s current presentation for U ( b sl ).In this paper, we first upgrade the main results of [BK20] for U ı to e U ı , such as theconstructions of v -root vectors (denoted by B ,k , ´Θ m , for k ∈ Z , m ∈ Z ≥ ) and their relations,with the help of a braid group action coming from the ı Hall algebra realization of e U ı . By a DRINFELD TYPE PRESENTATION OF AFFINE ı QUANTUM GROUPS, I 3 central reduction this in turn allows us to obtain the v -root vectors and PBW basis for U ı with 2 arbitrary parameters { ς , ς } , somewhat improving [BK20].As a point of departure, we introduce new v -imaginary root vectors Θ m and especially H m , for m ≥
1, and work with generating functions. This allows us to greatly simplify someof the BK relations to be Drinfeld type relations. The normalization from ´Θ m to Θ m inrank 1 is inspired by the realization of q -Onsager algebra via ı Hall algebra of the projectiveline in a forthcoming work [LRW20]. We also obtain new Drinfeld type relations among real v -root vectors. In this way, we formulate a Drinfeld type presentation for e U ı as an algebraisomorphism Dr e U ı ∼ = e U ı ; see Definition 2.15 and Theorem 2.16.Via generating functions in a variable z , Θ ( z ) = 1 + ( v − v − ) X m ≥ Θ m z m , B ( z ) = X r ∈ Z B ,r z r , ∆ ( z ) = X k ∈ Z C k z k , where C = K δ , the Drinfeld presentation for the q -Onsager algebra e U ı is reformulated as: Θ ( z ) Θ ( w ) = Θ ( w ) Θ ( z ) , (1.1) Θ ( z ) B ( w ) = (1 − v − zw − )(1 − v zwC )(1 − v zw − )(1 − v − zwC ) B ( w ) Θ ( z ) , (1.2) ( v z − w ) B ( z ) B ( w ) + ( v w − z ) B ( w ) B ( z )(1.3) = v − K v − v − ∆ ( zw ) (cid:0) ( v z − w ) Θ ( w ) + ( v w − z ) Θ ( z ) (cid:1) . Higher rank.
Now let e U ı = e U ı ς ( b g ) be the universal affine ı quantum group of splitADE type, where g is the Lie algebra of type ADE with root datum I . (The corresponding ı quntum groups U ı were introduced in [BB10] and referred to as generalized q -Onsageralgebras; cf. [Ko14].) By definition, e U ı is generated by B i , K i , K − i , for i ∈ I = I ∪ { } ,where K i are central and the B i satisfy some inhomogenous Serre relations; see (3.11)–(3.12).The algebra e U ı can be realized via an ı Hall algebra construction associated to ı quivers[LW19b, LW20]; moreover, there are automorphisms T i ( i ∈ I ) of e U ı which are realized asreflection functors on ı Hall algebras, which gives rise to an affine braid group action on e U ı .It is worth pointing out that such a braid group action is not a restriction of the braid groupaction of Lusztig on quantum groups [Lus90, Lus94].In a way similar to [Be94], with the help of the braid group action and the rank oneconstruction above, we construct real v -root vectors B i,k and imaginary v -root vectors ´Θ i,m (or Θ i,m , H i,m ), for i ∈ I , k ∈ Z , m ∈ Z ≥ . Then the formulation of relations among theseelements are used to define a new Drinfeld type algebra Dr e U ı ; see Definition 3.10. Our mainresult (see Theorem 3.13) asserts that Dr e U ı ∼ = e U ı , providing a new presentation for the affine ı quantum group e U ı .The defining relations (3.33)–(3.38) for Dr e U ı turn out to be strikingly neat and similar toDrinfeld’s current relations for U ( b g ). Actually, one can regard half the Drinfeld’s realizationof U ( b g ) as the associated graded algebra with respect to a filtration of Dr e U ı over its centralsubalgebra. In other words, the relations (3.34), (3.37) and (3.38) for Dr e U ı look like Drinfeld’srelations for U ( b g ) plus lower terms (involving powers of C = K δ ). Yet another view of the MING LU AND WEIQIANG WANG relations (3.34), (3.37) and (3.38) for Dr e U ı is that they exhibit a hybrid phenomenon mixingrelations in the current negative half with relations between current positive and negativegenerators for U ( b g ).One new relation which was not present in the rank one case is the Serre type relation(3.38). As the Serre relations among B i = B i, (3.11) are inhomogeneous, it is understandablethat the general Serre type relations (3.38) among B i,k for Dr e U ı are much more challengingto formulate than its counterpart for U ( b g ), where the RHS of (3.38) is simply set to 0 as ina standard Serre relation. What is perhaps surprising to us is that such a relation can beformulated concretely after all.Damiani [Da12] made a careful analysis of the relations in Drinfeld’s current realizationof U ( b g ), and showed that they can be derived from a few distinguished relations. To thatend, the triangular decomposition of U ( b g ) was very helpful. In contrast, e U ı does not admita triangular decomposition. Because of this, the verifications of the new relations for e U ı ,in particular (3.37)–(3.38), require a very different strategy from [Be94, Da12], though ouroverall plan is somewhat similar by showing that all the new relations for e U ı can be derivedfrom a few simpler ones.By verifying all the new relations in e U ı we obtain a homomorphism Φ : Dr e U ı → e U ı ,and it remains to show that Φ is an isomorphism. Our argument of the surjectivity of Φis adapted from the proof of Damiani [Da12, Theorem 12.11]. The injectivity of Φ followsby applying some filtered algebra argument to reduce to the corresponding injectivity in theaffine quantum group setting.We obtain several natural variants of the current presentation of e U ı , including one in thegenerating function form similar to (1.1)–(1.3) for q -Onsager algebra; see Theorem 5.1. Asurprising bonus of working with e U ı (instead of U ı ) is that the canonical central element C in affine quantum group naturally appear as K δ = Q i ∈ I K a i i associated to the basic imag-inary root δ = P i ∈ I a i α i . This is especially clear in a symmetrized variant of the currentpresentation of e U ı (see Definition 5.2 and Proposition 5.3). This phenomenon is even moreremarkable, as the classical ( v
1) limit of e U ı or U ı does not contain the canonical centralelement of the affine Lie algebra b g ; see § Applications.
While the ı Hall algebras are not used in this paper in any explicit man-ner, they have played a fundamental role in guiding our work. Since we have realized the uni-versal ı quantum group in its Serre presentation via ı Hall algebra of ı quivers [LW19a, LW20],it is natural to expect that ı Hall algebras of coherent sheaves over (weighted) projectivelines should provide a realization of e U ı in a new current presentation (keeping in mind theclassic works [Ka97, BKa01, Sch04, DJX12] relating (weighted) projective lines to the cur-rent realization of affine quantum groups). Preliminary computations at earlier stages of[LRW20, LR20] on ı Hall algebras of (weighted) projective lines have been very helpful inpinning down some new relations for e U ı . The current presentation of e U ı in this paper willplay an essential role in the ı Hall algebra realization therein.It is our hope that this work can be of interest to people with diverse algebraic, categorical,geometric backgrounds. With this in mind, we have tried to make the presentation in thispaper to be self-contained and come up with proofs independent of ı Hall algebras. (As aresult, a multiple of proofs for various relations in U ı are known to us.) DRINFELD TYPE PRESENTATION OF AFFINE ı QUANTUM GROUPS, I 5
In sequels to this paper, we shall further generalize this work to obtain Drinfeld type pre-sentations for affine ı quantum groups beyond split ADE type. The new current presentationshould be very helpful in developing an algebraic approach toward the representation theoryof affine ı quantum groups (cf., e.g., [CH10]), to which we hope to return elsewhere.In yet another direction, this work makes it possible to develop the geometric realizationof affine ı quantum groups via equivariant K-theory [SuW20], building on the works of Y. Liand collaborators [BKLW18, Li19] and generalizing earlier works of Vasserot and Nakajima[V98, Nak00].1.6. Organization.
In Section 2, we define the v -root vectors and give a Drinfeld typepresentation for the q -Onsager algebra e U ı .In Section 3, we define the v -root vectors for e U ı of split affine ADE type, and formulatea Drinfeld type presentation for e U ı in the form of isomorphism Φ : Dr e U ı → e U ı . We show Φis a bijection under the assumption that Φ is an algebra homomorphism, while postponingthe verification of the new relations in e U ı to Section 4. We verify the new relations in e U ı one-by-one in Section 4, with the most challenging ones being (3.38) and (3.34) for c ij = − e U ı , one in generating func-tion format (Theorem 5.1), and in a symmetrized form which resembles Drinfeld’s realizationfor U better (Definition 5.2 and Proposition 5.3), and yet another one in terms of different v -imaginary root vectors (Theorem 5.4). We also formulate a Drinfeld type presentation for U ı ς (Theorem 5.5). Acknowledgement.
We thank Shiquan Ruan for his collaboration on related Hall algebraprojects, and thank Weinan Zhang who inspired us to simplify much our earlier proofs in § The q -Onsager algebra and its Drinfeld type presentation In this section, we derive Drinfeld type new relations among the generators of the universal q -Onsager algebra, and recast them in the form of generating functions. This is built on areformulation and enhancement of the results in [BK20] for q -Onsager algebra. We introducenew imaginary root vectors in the universal q -Onsager algebra (with motivation coming fromthe ı Hall algebra of the projective line), and establish a Drinfeld type presentation.2.1.
Root vectors.
For n ∈ Z , r ∈ N , denote by[ n ] = v n − v − n v − v − , (cid:20) nr (cid:21) = [ n ][ n − . . . [ n − r + 1][ r ]! . For
A, B in a Q ( v )-algebra, we shall denote [ A, B ] v a = AB − v a BA , and [ A, B ] = AB − BA . Definition 2.1.
The (universal) q -Onsager algebra e U ı = e U ı ( b sl ) is the Q ( v ) -algebra gener-ated by B , B , K , K , subject to the following relations: K , K are central, and X r =0 ( − r (cid:20) r (cid:21) B − ri B j B ri = − v − [2] ( B i B j − B j B i ) K i , for i = j. (2.1) (This algebra is also known as the universal ı quantum group of split type A (1)1 .) MING LU AND WEIQIANG WANG
Remark . The generator K i here is related to k i used in [LW19a] by K i = − v k i ; seeRemark 3.2. The q -Onsager algebra U ı ς , for ς = ( ς , ς ) ∈ ( Q ( v ) × ) , is obtained from e U ı by a central reduction U ı ς = e U ı / ( K i + v ς i | i = 0 , − ) denotes an ideal. The1-parameter specialization U ı ς by taking ς = ς = − c recovers the q -Onsager algebra B c studied in [BK20].Let { α , α } be the simple roots of the affine Lie algebra b sl , and δ = α + α is the basicimaginary root. The root system for b sl is R = {± ( α + kδ ) , mδ | k, m ∈ Z , m = 0 } . For µ, ν ∈ Z α ⊕ Z α and i = 0 ,
1, set K α i = K i , K − α i = K − i , K δ = K K , K µ + ν = K µ K ν . (2.2)Let † be the involution of the Q ( v )-algebra e U ı such that † : B ↔ B , K ↔ K . (2.3)We have the following two automorphisms T , T , which has an interpretation in ı Hall al-gebras, see [LW19b] and a forthcoming sequel in the Kac-Moody setting (This shows someconceptual advantage of e U ı over U ı ς [BK20], where the parameters ς i are set to be equal):T ( K ) = K − , T ( K ) = K δ K , (2.4) T ( B ) = K − B , (2.5) T ( B ) = [2] − (cid:0) B B − v [2] B B B + v B B (cid:1) + B K , (2.6) T − ( B ) = [2] − (cid:0) B B − v [2] B B B + v B B (cid:1) + B K . (2.7)The action of T is obtained from the above formulas by switching indices 0 ,
1, that is,T = † ◦ T ◦ † . (2.8)For n ∈ Z , following [BK20], we define the real v -root vectors B ,n = ( † T ) − n ( B ) . (2.9)Slightly modifying [BK20], we further define, for m ≥ m = − B ,m − B + v B B ,m − + ( v − m − X p =0 B ,p B ,m − p − K . (2.10)Note that B , = B by definition. (Our − v − ´Θ m corresponds to B mδ in [BK20, (3.11)].) Inparticular, we have ´Θ = − B B + v B B . We also set´Θ m := ( v − v − if m = 0 , m < . (2.11)From (2.9), we have B , − = ( † T )( B ) = B K − , B = B , − K . DRINFELD TYPE PRESENTATION OF AFFINE ı QUANTUM GROUPS, I 7
So (2.10) can be rewritten as´Θ m = (cid:0) − B ,m − B , − + v B , − B ,m − + ( v − m − X p =0 B ,p B ,m − p − (cid:1) K (2.12) = − m − X p =0 [ B ,p , B ,m − − p ] v K . We note that [BK20, Corollary 5.12] in our setting of e U ı reads † T ( ´Θ m ) = ´Θ m . (2.13)Applying ( † T ) − to (2.12), we have ´Θ m = − P mp =1 [ B ,p , B ,m − p ] v K − . Relations `a la Baseilhac-Kolb.
The following relations in e U ı are the counterpartsof the main relations for B c established in [BK20]; see Remark 2.2. They are obtained byliterally repeating the arguments loc. cit. , and hence we shall skip the proofs altogether inthis subsection. Our formulation in turn strengthens somewhat the results in [BK20], asthe central reduction in Remark 2.2 provides relations and then a presentation for U ı with arbitrary ς i ( i = 0 , Proposition 2.3 ([BK20, Corollary 5.11]) . We have [ ´Θ n , ´Θ m ] = 0 holds in e U ı , for n, m ≥ . For m ∈ N , define a mp := (cid:26) v p − (1 + v ) , if p = 1 , , . . . , ⌊ m − ⌋ ,v m − , if 2 | m and p = m . (2.14) Proposition 2.4 ([BK20, Proposition 5.5, Corollary 5.13]) . The following relation holds in e U ı , for m ∈ N and r ∈ Z : [ B ,r + m +1 ,B ,r ] v = − ´Θ m +1 K rδ + α − ( v − ⌊ m ⌋ X p =1 v p − ´Θ m − p +1 K ( p + r ) δ + α (2.15) + ( v − ⌊ m +12 ⌋ X p =1 a m +1 p B ,r + p B ,m + r − p +1 . Proposition 2.5 ([BK20, Proposition 5.8, Corollary 5.13]) . The following relation holds in e U ı , for m ≥ and r ∈ Z : [ ´Θ m , B ,r ](2.16) = [2] (cid:16) v m − B ,r + m − ( v − v − ) m − X h =1 v m − h ) B ,r + m − h K hδ − v − m ) B ,r − m K mδ (cid:17) + ( v − v − ) × m − X a =1 (cid:16) v a − B ,r + a − ( v − v − ) a − X h =1 v a − h ) B ,r + a − h K hδ − v − a ) B ,r − a K aδ (cid:17) ´Θ m − a . MING LU AND WEIQIANG WANG
Drinfeld type relations in rank 1.
We shall introduce a new imagnary v -root vectors´ H m and formulate several Drinfeld type relations in e U ı among the v -root vectors.2.3.1. We start with the relations among real v -root vectors B ,k . Proposition 2.6.
The following relation holds in e U ı , for r, s ∈ Z : [ B ,r , B ,s +1 ] v − − v − [ B ,r +1 , B ,s ] v = v − ´Θ r − s +1 K sδ + α − v − ´Θ r − s − K ( s +1) δ + α (2.17) + v − ´Θ s − r +1 K rδ + α − v − ´Θ s − r − K ( r +1) δ + α . Proof.
Note that − v − [ B ,r +1 , B ,s ] v = [ B ,s , B ,r +1 ] v − , and hence (2.17) is invariant under r ↔ s . So we can assume m = s − r ≥ B ,r + m +1 , B ,r ] v and [ B ,r + m , B ,r +1 ] v for m ≥ B ,r + m +1 , B ,r ] v − v [ B ,r + m , B ,r +1 ] v (2.18) = − ´Θ m +1 K rδ + α + ´Θ m − K ( r +1) δ + α + ( v − B ,r +1 B ,m + r . One then rewrites (2.18) equivalently as (2.17), for m ≥ = v − v − , one observes that (2.17) for m = s − r = 1 is equivalent to therelation [ B ,r , B ,r +2 ] v − = v − ´Θ K rδ + α +( v − − B ,r +1 , and then equivalent to the followingrelation in (2.15): [ B ,r +2 , B ,r ] v = − ´Θ K rδ + α + ( v − B ,r +1 . The relation (2.17) for m = s − r = 0 is equivalent to [ B ,r , B ,r +1 ] v − = v − ´Θ K rδ + α ,another relation in (2.15). The proof of (2.17) is completed. (cid:3) Remark . The relation (2.17) was derived from (2.15) above; they are actually equivalent,as the converse can be shown by induction on m .2.3.2. Define elements ´ H m in e U ı , for m ≥
1, by the following equation:1 + X m ≥ ( v − v − ) ´Θ m z m = exp (cid:16) ( v − v − ) X m ≥ ´ H m z m (cid:17) . (2.19)Introduce the following generating functions in a variable z :´ H ( z ) = X m ≥ ´ H m z m , B ( z ) = X r ∈ Z B ,r z r , ´ Θ ( z ) = 1 + ( v − v − ) X m ≥ ´Θ m z m . Then we have ´ Θ ( z ) = exp (cid:0) ( v − v − ) ´ H ( z ) (cid:1) . (2.20)The relations between imaginary and real v -root vectors can now be formulated as follows;here the imaginary v -root vectors refer to ´ H m . DRINFELD TYPE PRESENTATION OF AFFINE ı QUANTUM GROUPS, I 9
Proposition 2.8.
The following identities hold, for m ≥ , l ∈ Z : [ ´ H m , B ,l ] = [2 m ] m B ,l + m − [2 m ] m B ,l − m K mδ , (2.21) ´ Θ ( z ) B ( w ) = (1 − v − zw − )(1 − v zw K δ )(1 − v zw − )(1 − v − zw K δ ) B ( w ) ´ Θ ( z ) , (2.22) [ ´Θ m , B ,l ] + [ ´Θ m − , B ,l ] K δ = v [ ´Θ m − , B ,l +1 ] v − + v − [ ´Θ m − , B ,l − ] v K δ . (2.23) Indeed, the identities (2.16) , (2.21) , (2.22) and (2.23) are all equivalent. The proof of Proposition 2.8 will be given in § m , B ,l ](2.24) = [2] (cid:16) v − m ) B ,l + m + ( v − v − ) m − X h =1 v h − m ) B ,l + m − h K hδ − v m − B ,l − m K mδ (cid:17) + ( v − v − ) × m − X a =1 ´Θ m − a (cid:16) v − a ) B ,l + a + ( v − v − ) a − X h =1 v h − a ) B ,l + a − h K hδ − v a − B ,l − a K aδ (cid:17) . Proof of Proposition 2.8.
We shall establish the equivalences among identities (2.16),(2.21), (2.22) and (2.23).2.4.1.
Proof of equivalences of (2.16) , (2.21) and (2.22) . The identity (2.21) can be equiva-lently reformulated via generating functions as( v − v − )[ ´ H ( z ) , B ( w )] = X k ≥ ,m ∈ Z B ,m + k w m + k (cid:18) ( v zw − ) k k − ( v − zw − ) k k (cid:19) − X k ≥ ,m ∈ Z B ,m − k w m − k (cid:18) ( v zw K δ ) k k − ( v − zw K δ ) k k (cid:19) = ln (cid:18) − v − zw − − v zw − · − v zw K δ − v − zw K δ (cid:19) B ( w ) . Via integration this is then equivalent to e ( v − v − ) ´ H ( z ) B ( w ) e − ( v − v − ) ´ H ( z ) = B ( w ) (1 − v − zw − )(1 − v zw K δ )(1 − v zw − )(1 − v − zw K δ ) , (2.25)which can be reformulated as (2.22). We have the following identities:1 − v − zw − − v zw − = 1 + ( v − v − ) X h ≥ v h − z h w − h , − v zw K δ − v − zw K δ = 1 − ( v − v − ) X h ≥ v − h ) z h w h K hδ , and hence(1 − v − zw − )(1 − v zw K δ )(1 − v zw − )(1 − v − zw K δ )(2.26)= 1 + X a ≥ z a ( v − v − ) (cid:16) v a − w − a − ( v − v − ) a − X h =1 v a − h ) w h − a K hδ − v − a ) w a K aδ (cid:17) . It follows by (2.26) that the identity (2.22) is equivalent to the following identity: (cid:16) X m ≥ ( v − v − ) ´Θ m z m (cid:17) B ( w ) = B ( w ) (cid:16) X m ≥ ( v − v − ) ´Θ m z m (cid:17) + X a ≥ z a ( v − v − ) v a − w − a − ( v − v − ) a − X h =1 v a − h ) w h − a K hδ − v − a ) w a K aδ ! × B ( w ) (cid:16) X n ≥ ( v − v − ) ´Θ n z n (cid:17) . Equating the coefficients of z m w l on both sides, for m ≥
1, we obtain( v − v − )[ ´Θ m , B ,l ]= ( v − v − ) × (cid:16) v m − B ,l + m − ( v − v − ) m − X h =1 v m − h ) B ,l + m − h K hδ − v − m ) B ,l − m K mδ (cid:17) + ( v − v − )( v − v − ) × m − X a =1 (cid:16) v a − B ,l + a − ( v − v − ) a − X h =1 v a − h ) B ,l + a − h K hδ − v − a ) B ,l − a K aδ (cid:17) ´Θ m − a , which is equivalent to the identity (2.16).2.4.2. Equivalence of (2.22) and (2.23) . The identity (2.22) can be rephrased as(1 − v zw − )(1 − v − zw K δ ) ´ Θ ( z ) B ( w ) = (1 − v − zw − )(1 − v zw K δ ) B ( w ) ´ Θ ( z ) . DRINFELD TYPE PRESENTATION OF AFFINE ı QUANTUM GROUPS, I 11
This can be rewritten as(1 − v zw − )(1 − v − zw K δ ) (cid:0) X m ≥ ( v − v − ) ´Θ m z m (cid:1) X r ∈ Z B ,r w r (2.27) =(1 − v − zw − )(1 − v zw K δ ) X r ∈ Z B ,r w r (cid:0) X m ≥ ( v − v − ) ´Θ m z m (cid:1) . Equating the coefficients of z m w l on both sides of (2.27), for m ≥
1, we obtain the following.If m ≥
3, then´Θ m B ,l − v ´Θ m − B ,l +1 − v − ´Θ m − B ,l − K δ + ´Θ m − B ,l K δ = B ,l ´Θ m − v − B ,l +1 ´Θ m − − v B ,l − ´Θ m − K δ + B ,l ´Θ m − K δ , which can be transformed into (2.23).If m = 2, then´Θ B ,l − v ´Θ B ,l +1 − v − ´Θ B ,l − K δ = B ,l ´Θ − v − B ,l +1 ´Θ − v B ,l − ´Θ K δ . If m = 1, then ( v − v − ) ´Θ B ,l − v B ,l +1 − v − B ,l − K δ =( v − v − ) B ,l ´Θ − v − B ,l +1 − v B ,l − K δ . The above identities in both cases for m = 1 , New imaginary root vectors.
With motivation from and application to ı Hall algebraof the projective line [LRW20] in mind, we shall introduce a somewhat normalized versionsof the elements ´Θ m and ´ H m , denoted by Θ m and H m , respectively.For m ≥
1, defineΘ m = ´Θ m − ⌊ m − ⌋ X a =1 ( v − v − a ´Θ m − a K aδ − δ m,ev v − m K m δ , (2.28)where δ m,ev = ( , for m even , , for m odd . Note that Θ = ´Θ . We also set Θ = v − v − , and Θ m = 0 for m <
0. Let Θ ( z ) = 1 + ( v − v − ) X m ≥ Θ m z m . (2.29)We define the new imaginary v -root vectors H m by letting1 + X m ≥ ( v − v − )Θ m u m = exp (cid:16) ( v − v − ) X m ≥ H m u m (cid:17) . (2.30) Lemma 2.9.
The identity (2.28) can be reformulated as a generating function identity: Θ ( z ) = 1 − K δ z − v − K δ z ´ Θ ( z ) . (2.31) Proof.
By (2.28), we have (with a change of variables k = m − a in the double summandbelow) Θ ( z ) = 1 + ( v − v − ) X m ≥ ´Θ m z m − ( v − v − ) X m ≥ X m/ >a ≥ ( v − v − a ´Θ m − a K aδ z m − ( v − v − ) X n ≥ v − n K nδ z n = ´ Θ ( z ) − ( v − (cid:16) X a ≥ v − a K aδ z a (cid:17) · (cid:16) X k ≥ ( v − v − ) ´Θ k z k (cid:17) − ( v − v − ) v − K δ z − v − K δ z = ´ Θ ( z ) − ( v − v − K δ z − v − K δ z (cid:0) ´ Θ ( z ) − (cid:1) − ( v − v − ) v − K δ z − v − K δ z = 1 − K δ z − v − K δ z ´ Θ ( z ) . The lemma is proved. (cid:3)
Lemma 2.10.
We have, for m ∈ Z , Θ m +1 − v − Θ m − K δ = ´Θ m +1 − ´Θ m − K δ . (2.32) Proof.
By (2.31), we have (1 − v − K δ z ) Θ ( z ) = (1 − K δ z ) ´ Θ ( z ) . The lemma follows bycomparing the coefficients of z m +1 on both sides of this identity. (cid:3) Proposition 2.11.
We have [ H m , H n ] = 0 = [Θ m , Θ n ] , for m, n ≥ .Proof. Follows from Proposition 2.3 by using (2.31) and noting K δ is central. (cid:3) The following is a counterpart via H m and Θ m of the identities (2.21) and (2.23), and theylook formally the same. Proposition 2.12.
The following identity holds in e U ı , for m ≥ and r ∈ Z : [ H m , B ,l ] = [2 m ] m B ,l + m − [2 m ] m B ,l − m K mδ , (2.33) [Θ m , B ,r ] + [Θ m − , B ,r ] K δ = v [Θ m − , B ,r +1 ] v − + v − [Θ m − , B ,r − ] v K δ . (2.34) Proof.
By (2.22), (2.31) and noting K δ is central, we have Θ ( z ) B ( w ) = (1 − v − zw − )(1 − v zw K δ )(1 − v zw − )(1 − v − zw K δ ) B ( w ) Θ ( z ) , which takes the same form as (2.22). Now (2.34) (which takes the same form as (2.23))follows by exactly the same argument for the equivalence between (2.22) and (2.23) in § (cid:3) Proposition 2.13.
We have, for r, s ∈ Z , [ B ,r ,B ,s +1 ] v − − v − [ B ,r +1 , B ,s ] v (2.35) = v − Θ ( s − r +1) δ K rδ + α − v − Θ ( s − r − δ K ( r +1) δ + α + v − Θ ( r − s +1) δ K sδ + α − v − Θ ( r − s − δ K ( s +1) δ + α . Proof.
Follows from (2.17) by using (2.32). (cid:3)
DRINFELD TYPE PRESENTATION OF AFFINE ı QUANTUM GROUPS, I 13
Grading and filtration.
We shall relabel the v -root vectors by a set of roots { α + kδ | k ∈ Z } ∪ { mδ | m ≥ } :(2.36) B α + kδ = B ,k , B mδ = H m , for k ∈ Z , m ≥ . Denote by e U ı the subalgebra of e U ı generated by Θ m , for m ≥
1. By Definition 2.1, thealgebra e U ı is ZI ( ≡ Z α ⊕ Z α )-graded by weights, withwt( B i ) = α i , wt( K i ) = 2 α i , for i ∈ { , } . (2.37)It follows that wt( B ,k ) = α + kδ, wt( H m ) = mδ , whence the notation (2.36).For any fixed positive integer h (with a standard choice being h = 1), the algebra e U ı isendowed with a filtered algebra structure | · | h by setting | B | h = 1 , | B | h = h , | K i | h = 0 , for i ∈ { , } . (2.38)Then there is an algebra isomorphism relating the associated graded gr e U ı to half a quantumaffine sl , U − = h F , F i in the setting of [LW20], which goes back to [Let02, Ko14]:gr e U ı ∼ = U − ⊗ Q ( v )[ K ± , K ± δ ] , B i F i ( i = 0 , . (2.39) Proposition 2.14 (cf. [BK20]) . The following holds in e U ı :(1) The subalgebra e U ı is a polynomial algebra in Θ m , for m ≥ ; it is also a polynomialalgebra in H m , for m ≥ . (2) Fix any total order < on the roots { α + kδ | k ∈ Z } ∪ { mδ | m ≥ } . Then e U ı admits a basis (cid:8) K r K sδ B a γ B a γ . . . B a N γ N | r, s ∈ Z , a , a , . . . , a N ∈ N , N ∈ N , γ < γ < . . . < γ N (cid:9) . Proof.
Part (2) follows as a variant of [BK20, Theorem 4.5] on a PBW basis for U ı , whichis proved by using the filtration (2.38)–(2.39) and comparing with the v -root vectors in U given in [Da93]. (A mild difference is that e U ı admits the central elements K β , for β ∈ R ,and B mδ for U ı used loc. cit. is understood as a version of ´Θ m .)Clearly, the algebraic independence among { ´Θ m | m ≥ } implies the algebraic indepen-dence of { Θ m | m ≥ } as well as of { H m | m ≥ } . Part (1) follows. (cid:3) A Drinfeld type presentation in rank 1.Definition 2.15.
Let Dr e U ı = Dr e U ı ( b sl ) be the Q ( v ) -algebra generated by K ± , C ± , H m and B ,r , where m ≥ , r ∈ Z , subject to the following defining relations, for m, n ≥ and r, s ∈ Z : K K − = 1 , CC − = 1 , K , C are central , (2.40) [ H m , H n ] = 0 , (2.41) [ H m , B ,r ] = [2 m ] m B ,r + m − [2 m ] m B ,r − m C m , (2.42) [ B ,r , B ,s +1 ] v − − v − [ B ,r +1 , B ,s ] v = v − Θ s − r +1 C r K − v − Θ s − r − C r +1 K (2.43) + v − Θ r − s +1 C s K − v − Θ r − s − C s +1 K , where P m ≥ ( v − v − )Θ m z m = exp (cid:0) ( v − v − ) P m ≥ H m z m (cid:1) . Theorem 2.16.
There is a Q ( v ) -algebra isomorphism Φ : Dr e U ı → e U ı , which sends B ,r B ,r , Θ m Θ m , K K , C K δ , for m ≥ , r ∈ Z . The inverse Φ − : e U ı → Dr e U ı sends K K , K C K − , B B , , B B , − C K − . Proof.
The relations (2.41), (2.42) and (2.43) in Dr e U hold for the images of the generators of Dr e U ı under Φ, thanks to Propositions 2.11, 2.12, and 2.13, respectively. (The relation (2.40)under Φ holds trivially.) Thus Φ is a homomorphism.By the defining relations of Dr e U ı , one shows that Dr e U ı has a spanning set { K r C s B a γ B a γ . . . B a N γ N | r, s ∈ Z , a , a , . . . , a N ∈ N , N ∈ N , γ < γ < . . . < γ N } . Since this set is mapped to a basis of e U ı according to Proposition 2.14, it must be a basisfor Dr e U ı as well. Therefore, Φ is an isomorphism. (cid:3) Remark . There is another Drinfeld type presentation of the Q ( v )-algebra e U ı with thesame set of generators as in Theorem 2.16, subject to the relations (2.40)–(2.42) and (2.44)below (in place of (2.43)):[ B ,r , B ,s +1 ] v − − v − [ B ,r +1 , B ,s ] v = v − Θ s − r +1 C r K − v − Θ s − r − C r +1 K (2.44) + v − Θ r − s +1 C s K − v − Θ r − s − C s +1 K . One should (secretly) regard the Θ m in this presentation as ´Θ m defined earlier, and then therelation (2.44) is simply (2.17). In this way, the equivalence between (2.17) and (2.35) (or(2.43)) has been established earlier. Hence the equivalence between this presentation andthe presentation Dr e U ı in Definition 2.15 follows.3. A Drinfeld type presentation of affine ı quantum groups In this section, we study in depth the ı quantum groups e U ı of split affine ADE type.We introduce a new set of generators for e U ı and use them to formulate a Drinfeld typepresentation for e U ı , generalizing the one for q -Onsager algebra in § Affine Weyl and braid groups.
Let ( c ij ) i,j ∈ I be the Cartan matrix of the simple Liealgebra g of type ADE. Let R be the set of roots for g , and fix a set R +0 of positive rootswith simple roots α i ( i ∈ I ). Denote by θ the highest root of g .Let b g be the (untwisted) affine Lie algebra with affine Cartan matrix denoted by ( c ij ) i,j ∈ I ,where I = { } ∪ I with the affine node 0. Let α i ( i ∈ I ) be the simple roots of b g , and α = δ − θ , where δ denotes the basic imaginary root. The root system R for b g and itspositive system R + are defined to be R = {± ( β + kδ ) | β ∈ R +0 , k ∈ Z } ∪ { mδ | m ∈ Z \{ }} , (3.1) R + = { kδ + β | β ∈ R +0 , k ≥ } ∪ { kδ − β | β ∈ R +0 , k > } ∪ { mδ | m ≥ } . (3.2)For γ = P i ∈ I n i α i ∈ NI , the height ht( γ ) is defined as ht( γ ) = P i ∈ I n i .Let P and Q denote the weight and root lattices of g , respectively. Let ω i ∈ P ( i ∈ I )be the fundamental weights of g . Note α i = P j ∈ I c ij ω j . We define a bilinear pairing h· , ·i : P × Q → Z such that h ω i , α j i = δ i,j , for i, j ∈ I , and thus h α i , α j i = c ij . DRINFELD TYPE PRESENTATION OF AFFINE ı QUANTUM GROUPS, I 15
The Weyl group W of g is generated by the simple reflection s i , for i ∈ I . It acts on P by s i ( x ) = x − h x, α i i α i for x ∈ P . The extended affine Weyl group f W is the semi-directproduct W ⋉ P , which contains the affine Weyl group W := W ⋉ Q = h s i | i ∈ I i as asubgroup; we denote t ω = (1 , ω ) ∈ f W , for ω ∈ P. We identify
P/Q with a finite group Ω of Dynkin diagram automorphism, and so f W = Ω .W .There is a length function ℓ ( · ) on f W such that ℓ ( s i ) = 1, for i ∈ I , and each element in Ωhas length 0.For i ∈ I , as in [Be94], we have(3.3) ℓ ( t ω i ) = ℓ ( ω ′ i ) + 1 , where ω ′ i := t ω i s i ∈ W. Let U = U ( b g ) denote the Drinfeld-Jimbo affine quantum group, a Q ( v )-algebra generatedby E i , F i , K ± i , for i ∈ I . Following Lusztig [Lus90, Lus94], we have the braid group actionof T w on e U , for w ∈ f W ; for example T i , for i ∈ I , acts on U by, for i = j ∈ I , µ ∈ ZI (our K i corresponds to e K i in [Lus94]): T i ( E i ) = − F i K i , T i ( F i ) = − K − i E i , T i ( K µ ) = K s i ( µ ) ,T i ( E j ) = X r + s = − c ij ( − r v − ri E ( s ) i E j E ( r ) i ,T i ( F j ) = X r + s = − c ij ( − r v ri F ( r ) i F j F ( s ) i . (3.4)The formulas above are written in terms of v i (dependent on the length of α i ) for theconvenience of future references. In the current ADE setting, we always have v i = v . Similarremarks apply elsewhere, e.g., [ n ] i = [ n ] v i .The following was known to Bernstein and Lusztig. Lemma 3.1 ([Lus89, § . Let x ∈ P , i, j ∈ I . (a) If s i x = xs i , then T i T x = T x T i . (b) If s i xs i = t − α i x = Q k ∈ I t a k ω k , then we have T − i T x T i = Q k T a k ω k ; in particular, we have 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 . We record some useful formulas: T i (cid:0) T ω ′ i ( F i ) (cid:1) = T ω ′ i ( F i ) F (2) i − v i F i T ω ′ i ( F i ) F i + v i F (2) i T ω ′ i ( F i ) , (3.5) T − i (cid:0) T ω ′ i ( F i ) (cid:1) = F (2) i T ω ′ i ( F i ) − v i F i T ω ′ i ( F i ) F i + v i T ω ′ i ( F i ) F (2) i , (3.6) T i (cid:0) [ T ω ′ i ( F i ) , F i ] v − i (cid:1) = [ F i , T ω ′ i ( F i )] v − i . (3.7)The formula (3.5) follows by applying the Chevalley involution to its E -version, which canbe found in [Be94]. The formula (3.6) can be derived (and is equivalent to) (3.5). Formula(3.7) can also be derived directly from (3.5). Drinfeld presentation of an affine quantum group.
Let C = ( c ij ) i,j ∈ I be theCartan matrix of untwisted affine type. Let Dr U be the Q ( v )-algebra generated by x ± ik , h il , K ± i , C ± , for i ∈ I , k ∈ Z , and l ∈ Z \{ } , subject to the following relations: C , C − arecentral such that [ K i , K j ] = [ K i , h jl ] = 0 , K i K − i = C C − = 1 , [ h ik , h jl ] = δ k, − l [ kc ij ] i k C k − C − k v j − v − j ,K i x ± jk K − i = v ± c ij x ± jk , [ h ik , x ± jl ] = ± [ kc ij ] i k C ∓ | k | x ± j,k + l , [ x + ik , x − jl ] = δ ij ( C k − l K i ψ i,k + l − C l − k K − i ϕ i,k + l ) ,x ± i,k +1 x ± j,l − v ± c ij x ± j,l x ± i,k +1 = v ± c ij x ± i,k x ± j,l +1 − x ± j,l +1 x ± i,k , Sym k ,...,k r r X t =0 ( − t (cid:20) rt (cid:21) 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 ) , 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 omit a degree operator D in the version of Dr U above.)It was stated by Drinfeld [Dr88] (and proved by Beck [Be94] and Damiani [Da12, Da15])that there exists a Q ( v )-algebra isomorphism φ : Dr U −→ U . (3.8)We omit the explicit formulas for the images under φ of generators of Dr U ; see [Be94, BCP99].3.3. Affine ı quantum groups of split ADE type. Recall the Cartan matrix ( c ij ) i,j ∈ I ofaffine ADE type, for I = I ∪ { } with the affine node 0. The notion of (quasi-split) universal ı quantum groups e U ı was formulated in [LW19a]. The universal affine ı quantum group ofsplit ADE type is the Q ( v )-algebra e U ı = e U ı ( b g ) with generators B i , K ± i ( i ∈ I ), subject tothe following relations, for i, j ∈ I : K i K − i = K − i K i = 1 , K i is central , (3.9) B i B j − B j B i = 0 , if c ij = 0 , (3.10) B i B j − [2] B i B j B i + B j B i = − v − B j K i , if c ij = − , (3.11) X r =0 ( − r (cid:20) r (cid:21) B − ri B j B ri = − v − [2] ( B i B j − B j B i ) K i , if c ij = − . (3.12) DRINFELD TYPE PRESENTATION OF AFFINE ı QUANTUM GROUPS, I 17
The universal split ı quantum group of rank 1 is also known as the q -Onsager algebra, andthis is the only case where the relation (3.12) is needed; see Definition 2.1.For µ = µ ′ + µ ′′ ∈ ZI := ⊕ i ∈ I Z α i , define K µ such that K α i = K i , K − α i = K − i , K µ = K µ ′ K µ ′′ , K δ = K K θ . (3.13)The algebra e U ı is endowed with a filtered algebra structure e U ı, ⊂ e U ı, ⊂ · · · ⊂ e U ı,m ⊂ · · · (3.14)by setting e U ı,m = Q ( v )-span { B i B i . . . B i r K µ | µ ∈ NI , i , . . . , i r ∈ I , r ≤ m } . (3.15)Note that e U ı, = M µ ∈ NI Q ( v ) K µ , (3.16)is the Q ( v )-subalgebra generated by K i for i ∈ I . Then, according to a basic result of Letzterand Kolb on quantum symmetric pairs adapted to our setting of e U ı (cf. [Let02, Ko14]), theassociated graded gr e U ı with respect to (3.14)–(3.15) can be identified withgr e U ı ∼ = U − ⊗ Q ( v )[ K ± i | i ∈ I ] , B i F i , K i K i ( i ∈ I ) . (3.17) Remark . The generator K i here, which corresponds to a (generalized) simple module inthe ı Hall algebra, is related to the generator ˜ k i used in [LW19a, LW20] (which is naturalfrom the viewpoint of Drinfeld doubles) by a rescaling: K i = − v ˜ k i . The precise relationbetween the algebra e U ı and the ı quantum group U ı arising from quantum symmetric pairs[Ko14] is explained loc. cit. ; also see § Remark . The Q ( v )-algebra e U ı is ZI -graded by lettingwt( B i ) = α i , wt( K i ) = 2 α i , for i ∈ I . (3.18)A variant of e U ı , in which K i is not assumed to be invertible, is NI -graded by (3.18). Lemma 3.4 (also cf. [KP11, BK20]) . For i ∈ I , there exists 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 ) = K − i B i , if j = i,B j , if c ij = 0 ,B j B i − vB i B j , if c ij = − , [2] − (cid:0) B j B i − v [2] B i B j B i + v B i B j (cid:1) + B j K i , if c ij = − , for µ ∈ ZI and j ∈ I . Moreover, T i ( i ∈ I ) satisfy the braid group 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 = − .Proof. The formulas for T i when c ij = 0 , − ı Hall algebra approach and the braid relations are verified therein; the formulas for T i when c ij = − ı quantum groups of Kac-Moodytype in a forthcoming sequel to [LW19b]); also see [KP11] and [BK20] for closely relatedversions on U ı via computer packages. It follows from the reflection functor constructionthat these formulas define an automorphism T i of e U ı . (cid:3) We also have T − i ( K µ ) = K s i µ , andT − i ( B j ) = K − i B i , if j = i,B j , if c ij = 0 ,B i B j − vB j B i , if c ij = − , [2] − (cid:0) B i B j − v [2] B i B j B i + v B j B i (cid:1) + B j K i , if c ij = − . Just as the quantum group setting [Lus94], T − i is related to T i byT − i = † ◦ T i ◦ † , (3.19)where † denotes the anti-involution of the Q ( v )-algebra e U ı , which fixes the generators B i , K i .For w ∈ f W with a reduced expression w = σs i . . . s i r and σ ∈ Ω, we define T w = σ T i . . . T i r , where σ acts on e U ı by permuting the indices of generators, σ ( B i ) = B σi , σ ( K i ) = K σi , for all i ∈ I . Hence, T w is well defined by Lemma 3.4. In particular, the standard resultssuch as Lemma 3.1 on braid groups associated to f W apply to T w . Lemma 3.5.
We have T w ( B i ) = B wi , for i ∈ I and w ∈ W such that wi ∈ I .Proof. This statement is well known for quantum groups [Lus94], and the proof here isadapted from the proof in [J95, Lemma 8.20] in the usual quantum group setting.We first prove the lemma in the rank 2 setting. Assume that w ∈ h s i , s j i , for some j ∈ I .The case when s i s j = s j s i is trivial as we must have wi = i . Otherwise, c ij = − i ∈ I suchthat wi ∈ I only happen when w = s i s j , and in this case, a direct computation shows wi = j and T i T j ( B i ) = B j .In general, we prove by induction on ℓ ( w ). We shall assume ℓ ( w ) > j ∈ I such that wα j < j = i ). Then, as in the proof of [J95, Lemma 8.20], we havea decomposition w = w ′ w ′′ such that w ′′ ∈ h s i , s j i , ℓ ( w ) = ℓ ( w ′ ) + ℓ ( w ′′ ), wα i > , and wα j <
0. Thus, w ′′ α i > , w ′′ α j <
0. Following the proof loc. cit. , wi ∈ I implies w ′′ i ∈ I .The inductive assumption applied to w ′ (and w ′′ i ∈ I ) yields T w ′ ( B w ′′ i ) = B w ′ w ′′ i = B wi .The rank 2 result in a previous paragraph shows B w ′′ i = T w ′′ B i . Therefore, we obtainT w ( B i ) = T w ′ T w ′′ ( B i ) = T w ′ ( B w ′′ i ) = B wi . (cid:3) For i ∈ I , let e U ı [ i ] be the subalgebra of e U ı generated by B i , T ω ′ i ( B i ) , K i , K δ − α i . Lemma 3.6.
Let i, j ∈ I be such that i = j . Then T ω i ( x ) = x , for all x ∈ e U ı [ j ] .Proof. Clearly, T ω i ( K j ) = K j and T ω i ( K δ − α j ) = K δ − α j . By Lemma 3.5, T ω i ( B j ) = B j .By Lemma 3.1(b) and noting T − j ( B i ) = T i ( B j ), we haveT − ω i T ω j ( B j ) =T − i T ω i T − i ( B j )(3.20) =T − i T ω j T − j ( B i ) = T ω j T − i T − j ( B i ) = T ω j ( B j ) . On the other hand, we haveT ω j ( B j ) = T ω ′ j T j ( B j ) = T ω ′ j ( B j K − j ) = T ω ′ j ( B j ) K δ − α j . (3.21)If follows by (3.20)–(3.21) that T − ω i T ω ′ j ( B j ) = T ω ′ j ( B j ) . As T ω i fixes all generators of e U ı [ j ] ,the lemma follows. (cid:3) DRINFELD TYPE PRESENTATION OF AFFINE ı QUANTUM GROUPS, I 19
Lemma 3.7.
We have T ω ′ i ( B i ) = T − ω ′ i ( B i ) , for i ∈ I .Proof. Note that Lemma 3.1 remains valid when the T ’s therein are replaced by T’s. Thenwe have T ω ′ i = T − ω ′ i Q j = i T − c ij ω j , thanks to T ω i = T ω ′ i T i . Therefore, it follows by Lemma 3.6that T ω ′ i ( B i ) = T − ω ′ i Q j = i T − c ij ω j ( B i ) = T − ω ′ i ( B i ) . (cid:3) Here are some additional useful formulas:T i (cid:0) T ω ′ i ( B i ) (cid:1) = T ω ′ i ( B i ) B (2) i − v i B i T ω ′ i ( B i ) B i + v i B (2) i T ω ′ i ( B i ) + T ω ′ i ( B i ) K i , (3.22) T − i (cid:0) T ω ′ i ( B i ) (cid:1) = B (2) i T ω ′ i ( B i ) − v i B i T ω ′ i ( B i ) B i + v i T ω ′ i ( B i ) B (2) i + T ω ′ i ( B i ) K i , (3.23) T i (cid:0) [T ω ′ i ( B i ) , B i ] v − i (cid:1) = [ B i , T ω ′ i ( B i )] v − i . (3.24)In the rank 1 case, formulas (3.22)–(3.23) reduce to (2.6)–(2.7) while (3.24) reduces to (2.13).The formulas (3.22)–(3.24) are obtained by an Ansatz with the corresponding formulas (3.5)–(3.7) in the setting of affine quantum groups. Formulas (3.22) and (3.23) are equivalent toeach other in the presence of (3.24) or in light of (3.19); Equation (3.24) is compatible with(3.19) as well. These formulas shall have interpretations in the ı Hall algebras of ı quivers and ı -weighted projective lines. We skip the details. Lemma 3.8.
For i ∈ I , we have X a =0 ( − a (cid:20) a (cid:21) i B − ai T ω ′ i ( B i ) B ai = − v − i [2] i [ B i , T ω ′ i ( B i )] K i , (3.25) X a =0 ( − a (cid:20) a (cid:21) i (T ω ′ i ( B i )) − a B i (T ω ′ i ( B i )) a = − v − i [2] i [T ω ′ i ( B i ) , B i ] K δ K − i . (3.26) Proof.
Recall K i is central in e U ı . Using (3.23)–(3.24), we compute[2] − i X a =0 ( − a (cid:20) a (cid:21) i B − ai T ω ′ i ( B i ) B ai = [2] − i h B i , (cid:2) B i , [ B i , T ω ′ i ( B i )] v i (cid:3) i v − i = (cid:2) B i , T − i (T ω ′ i ( B i )) − T ω ′ i ( B i ) K i (cid:3) v − i = (cid:2) B i , T − i (T ω ′ i ( B i )) (cid:3) v − i − (cid:2) B i , T ω ′ i ( B i ) K i (cid:3) v − i = K i T − i (cid:0) [ B i , T ω ′ i ( B i )] v − i (cid:1) − (cid:2) B i , T ω ′ i ( B i ) (cid:3) v − i K i = K i [T ω ′ i ( B i ) , B i ] v − i − (cid:2) B i , T ω ′ i ( B i ) (cid:3) v − i K i = − v − i [2] i [ B i , T ω ′ i ( B i )] K i . The first formula (3.25) follows.Applying T ω ′ i to both sides of the first formula (3.25), we obtain the second formula (3.26)by using T ω ′ i ( K i ) = K δ K − i and Lemma 3.7. (cid:3) Let us denote the q -Onsager algebra in Definition 2.1 by e U ı ( b sl ) in the proposition below,in order to distinguish from e U ı of higher rank. Proposition 3.9.
Let i ∈ I . There is 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 . Proof.
It follows by Definition 2.1 and Lemma 3.8 that ℵ i is a surjective homomorphism.It remains to show that ℵ i is injective. Let h be the Coxeter number of g . Recall from(2.38)–(2.39) a filtered algebra structure on e U ı ( b sl ) given by | · | h − with | B | h − = 1 , | B | h = h − , | K | h = | K δ | h = 0. As wt(T ω ′ i ( B i )) = δ − α i , we have | T ω ′ i ( B i ) | = h −
1, cf. (3.17).Hence ℵ i is a homomorphism of filtered algebras, and it induces a homomorphism of theassociated graded: ℵ gr i : gr e U ı ( b sl ) −→ gr e U ı [ i ] , which can be identified with ℵ − i : U − ( b sl ) ⊗ Q ( v )[ K ± , K ± δ ] −→ U − ⊗ Q ( v )[ K ± i , K ± δ ] , sending F F i , F T ω ′ i ( F i ) and K K i , K δ K δ . The homomorphism ℵ − i | U − ( b sl ) : U − ( b sl ) → U − is well known to be injective; cf. [Be94]. It follows that ℵ − i and ℵ gr i areinjective, hence so is ℵ i . (cid:3) As in [Be94], we have (cf. (3.22))T i | e U ı [ i ] = ℵ i ◦ T ◦ ℵ − i , T ω i | e U ı [ i ] = ℵ i ◦ T ω ◦ ℵ − i , for i ∈ I . (3.27)Define a sign function o ( · ) : I −→ {± } such that o ( i ) o ( j ) = − c ij < e U ı , for i ∈ I , k ∈ Z and m ≥ ω = † T ): B i,k = o ( i ) k T − kω i ( B i ) , (3.28) ´Θ i,m = o ( i ) m (cid:16) − B i,m − T ω ′ i ( B i ) + v T ω ′ i ( B i ) B i,m − (3.29) + ( v − 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 − v − a ´Θ i,m − a K aδ − δ m,ev v − m K m δ . (3.30)In particular, B i, = B i .3.4. A Drinfeld type presentation.
Let k , k , l ∈ Z and i, j ∈ I . We introduce short-hand notations: S ( k , k | l ; i, j ) = B i,k B i,k B j,l − [2] B i,k B j,l B i,k + B j,l B i,k B i,k , S ( k , k | l ; i, j ) = S ( k , k | l ; i, j ) + { k ↔ k } . (3.31) DRINFELD TYPE PRESENTATION OF AFFINE ı QUANTUM GROUPS, I 21
Here and below, { k ↔ k } stands for repeating the previous summand with k , k switched,so the sums over k , k are symmetrized. We also denote R ( k , k | l ; i, j ) = K i C k (cid:16) − X p ≥ v p [2][Θ i,k − k − p − , B j,l − ] v − C p +1 − X p ≥ v p − [2][ B j,l , Θ i,k − k − p ] v − C p − [ B j,l , Θ i,k − k ] v − (cid:17) , R ( k , k | l ; i, j ) = R ( k , k | l ; i, j ) + { k ↔ k } . (3.32)Sometimes, it is convenient to rewrite part of the summands in (3.32) as − X p ≥ v p − [2][ B j,l , Θ i,k − k − p ] v − C p − [ B j,l , Θ i,k − k ] v − = − X p ≥ v p − [2][ B j,l , Θ i,k − k − p ] v − C p + v − [ B j,l , Θ i,k − k ] v − . We shall often omit i, j and write S ( k , k | l ) = S ( k , k | l ; i, j ), S ( k , k | l ) = S ( k , k | l ; i, j ),and similarly for R and R , whenever i, j are clear from the context. Definition 3.10.
Let Dr e U ı 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 , (3.33) [ H i,m , B j,l ] = [ mc ij ] m B j,l + m − [ mc ij ] m B j,l − m C m , (3.34) [ B i,k , B j,l ] = 0 , if c ij = 0 , (3.35) [ B i,k , B j,l +1 ] v − cij − v − c ij [ B i,k +1 , B j,l ] v cij = 0 , if i = j, (3.36) [ B i,k , B i,l +1 ] v − − v − [ B i,k +1 , B i,l ] v = v − Θ i,l − k +1 C k K i − v − Θ i,l − k − C k +1 K i (3.37) + v − Θ i,k − l +1 C l K i − v − Θ i,k − l − C l +1 K i , S ( k , k | l ; i, j ) = R ( k , k | l ; i, j ) , if c ij = − . (3.38) Here we set Θ i, = ( v − v − ) − , Θ i,m = 0 , for m < , (3.39) and H i,m are related to Θ i,m by the following equation: X m ≥ ( v − v − )Θ i,m u m = exp (cid:16) ( v − v − ) X m ≥ H i,m u m (cid:17) . (3.40)Let us mention that in spite of its appearance the RHS of (3.37) typically has two nonzeroterms, thanks to the convention (3.39).The Q ( v )-algebra Dr e U ı clearly exhibits certain symmetries as follows. Lemma 3.11.
For each i ∈ I , we have an automorphism ω i of the algebra Dr e U ı given by ω i ( B j,r ) = B j,r − δ i,j , ω i ( H j,m ) = H j,m , ω i ( K j ) = K j C − δ ij , ω i ( C ) = C, (and hence ω i (Θ j,m ) = Θ j,m ), for all j ∈ I , r ∈ Z , m ≥ . Moreover, ω i ω k = ω k ω i for all i, k ∈ I . Proof.
Follows by inspection on the defining relations for Dr e U ı in Definition 3.10. (cid:3) Define the generating functions, for i ∈ I : B i ( z ) = P k ∈ Z B i,k z k , Θ i ( z ) = 1 + P m> ( v − v − )Θ i,m z m , ∆ ( z ) = P k ∈ Z C k z k . (3.41)The following is a generalization of Proposition 2.8, which follows by a similar formalmanipulation of generating functions as in § Proposition 3.12.
The identity (3.34) is equivalent to either of the following identities, for m ≥ , l ∈ Z and i, j ∈ I : Θ i ( z ) B j ( w ) = (1 − v − c ij zw − )(1 − v c ij zwC )(1 − v c ij zw − )(1 − v − c ij zwC ) B j ( w ) Θ i ( z ) , (3.42) [Θ i,m , B j,l ] + [Θ i,m − , B j,l ] C = v c ij [Θ i,m − , B j,l +1 ] v − cij + v − c ij [Θ i,m − , B j,l − ] v cij C. (3.43)Recall the elements B i,k , Θ i,m in e U ı defined in (3.28) and (3.30), while elements in thesame notation are generators for the algebra Dr e U ı . Below is the main result of this paper. Theorem 3.13.
There is a Q ( v ) -algebra isomorphism Φ : Dr e U ı → e U ı , which sends B i,k B i,k , H i,m H i,m , Θ i,m Θ i,m , K i K i , C K δ , (3.44) for i ∈ I , k ∈ Z , and m ≥ . The proof of Theorem 3.13 will be given in § Dr e U ı is ZI -graded by lettingwt( B i,k ) = α i + kδ, wt(Θ i,m ) = mδ, wt( K i ) = 2 α i , wt( C ) = 2 δ, (3.45)for i ∈ I , k ∈ Z , m ≥ . (It follows that wt( K ) = 2 α .) Moreover, Φ preserves the ZI -gradings.3.5. Proof of the main isomorphism.
The details of the proof of Proposition 3.14 belowwill occupy Section 4.
Proposition 3.14.
There is a homomorphism
Φ : Dr e U ı → e U ı as prescribed in (3.44) .Proof. By Propositions 4.1, 4.3, 4.5, 4.6, 4.8 and 4.16 in a later Section 4, all the definingrelations (3.33)–(3.38) for the generators in Dr e U ı are satisfied by their images under Φ in e U ı . We shall specify precisely what relations are used when deriving a given relation, tomake sure our proofs of these propositions do not run circularly. Hence Φ : Dr e U ı → e U ı is ahomomorphism. (cid:3) Assuming Proposition 3.14 (whose proof is much more difficult), we can now complete theproof of Theorem 3.13.
DRINFELD TYPE PRESENTATION OF AFFINE ı QUANTUM GROUPS, I 23
Proof of Theorem 3.13.
We first show that Φ : Dr e U ı → e U ı is surjective. All generators for e U ı except B are clearly in the image of Φ, and so it remains to show that B ∈ Im(Φ). Weshall adapt and modify the arguments in the proof of [Da12, Theorem 12.11].The automorphisms ω i ∈ Aut( Dr e U ı ) (see Lemma 3.11) and T ω i ∈ Aut( e U ı ), for i ∈ I ,satisfy Φ ◦ ω i = T ω i ◦ Φ , for i ∈ I . It follows that Im(Φ) is T ω i -stable, for each i ∈ I .Recall θ is the highest root in R +0 . We can choose and fix i ∈ I such that α + α i ∈ R + , θ i = θ − α i ∈ R +0 , and s θ i ( α i ) = θ. (3.46)Write t ω i = σ i s i . . . s i N with σ i ∈ Ω. Then t ω i ( θ ) = θ − δ ∈ −R + , and so there exists p suchthat s i N . . . s i p +1 ( α i p ) = θ . Hence, y := T − i N . . . T − i p +1 ( B i p ) ∈ Im(Φ); note wt( y ) = θ .Consider T ω i ( y ) = T σ i T i . . . T i p − ( B i p ) ∈ Im(Φ), since Im(Φ) is T ω i -stable. Note that σ i s i = s σ i (see [Be94, Lemma 3.1]), and so T s ω i = T − T ω i . Since wt(T − T ω i ( y )) = α (i.e., s ω i s i N . . . s i p +1 ( α i p ) = α ), it follows by Lemma 3.5 that T − T ω i ( y ) = B , and henceT ω i ( y ) = T ( B ) = K B . Therefore, we have K B ∈ Im(Φ) and thus B ∈ Im(Φ).It remains to show that the map Φ : Dr e U ı → e U ı is injective. Let us explain the simpleunderlying idea of the arguments for injectivity: we shall first show that Φ on the associatedgraded level when restricted to “a positive half” of e U ı (which correspond to “a quarter” inthe affine quantum group U , for which the roots share the same sign in Kac-Moody sense andin the Drinfeld current sense) is injective; this is summarized by the commutative diagram(3.54) below. Then we show the injectivity on “the positive half” implies the injectivity ofΦ : Dr e U ı → e U ı fully via the translation automorphisms ω i .Denote by e U ı> (respectively, Dr e U ı> ) the subalgebra of e U ı (respectively, Dr e U ı ) generatedby B i,m , H i,m , K i , for m ≥
1, and i ∈ I . Then Φ : Dr e U ı −→ e U ı restricts to a surjectivehomomorphism Φ : Dr e U ı> −→ e U ı> .Define a filtration on Dr e U ı> by( Dr e U ı> ) ⊂ ( Dr e U ı> ) ⊂ · · · ⊂ ( Dr e U ı> ) m ⊂ · · · (3.47)by setting( Dr e U ı> ) m = Q ( v )-span (cid:8) x = B i ,m B i ,m . . . B i r ,m r Θ j ,n Θ j ,n . . . Θ j s ,n s K µ (3.48) | µ ∈ NI , i , . . . , i r , j , . . . j s , ∈ I , m , . . . , m r , n , . . . , n s ≥ , ht + ( x ) ≤ m (cid:9) . Here we have denoted ht + ( x ) := r X a =1 ht( m a δ + α i a ) + s X b =1 n b ht( δ ) , (3.49)where ht( β ) denotes the height of a positive root β ; compare with the NI -grading on e U ı by(3.45). Recalling e U ı, from (3.16), we have( Dr e U ı> ) = e U ı, = Q ( v )[ K ± i | i ∈ I ] . The filtration (3.47)–(3.48) on Dr e U ı> defined via a height function is compatible with thefiltration (3.14)–(3.15) on e U ı under Φ, and thus the surjective homomorphism Φ : Dr e U ı> −→ e U ı> induces a surjective homomorphism gr Φ > : gr Dr e U ı> −→ gr e U ı> . (3.50)Recall from (3.8) an isomorphism φ : Dr U → U for the affine quantum group U . Denoteby Dr U − < the Q ( v )-subalgebra of U − generated by x − i, − k , for i ∈ I , k >
0, and denote by U − < = φ ( Dr U − < ). Then φ restricts to an isomorphism φ : Dr U − < ∼ = −→ U − < . (3.51)On the other hand, consider the associated graded gr e U ı with respect to the filtration on e U ı given by (3.14)–(3.15). Recall the algebra isomorphism in (3.17) G : U − ⊗ e U ı, −→ gr e U ı , F i B i , K i K i , where U − denotes half a Drinfeld-Jimbo quantum group generated by F i , for i ∈ I . Thehomomorphism G above restricts to an isomorphism G : U − < ⊗ e U ı, ∼ = −→ gr e U ı> . (3.52)Finally, by definition (3.48)–(3.49) of the filtration on Dr e U ı> , we have a surjective homomor-phism Ξ : Dr U − < ⊗ e U ı, −→ gr Dr e U ı> , (3.53)which sends x − i, − k B i,k , for k > Dr U − < ⊗ e U ı, / / φ, ∼ = (cid:15) (cid:15) gr Dr e U ı> gr Φ > (cid:15) (cid:15) U − < ⊗ e U ı, G , ∼ = / / gr e U ı> (3.54)Since the homomorphisms Ξ and gr Φ > are surjective while φ and G are isomorphisms, weconclude that gr Φ > : gr Dr e U ı> −→ gr e U ı> is injective (and an isomorphism).Now we show that gr Φ : gr Dr e U ı −→ gr e U ı is injective (and hence an isomorphism); asimilar argument was used in [Da15, Proposition 5.2]. Recall ρ = P i ∈ I ω i is half the sumof positive roots in Φ, and thus T ρ = Q i ∈ I T ω i ; the automorphism T ρ on e U ı induces anautomorphism (with the same notation) on gr e U ı . Assume that a finite linear combination X = P ( ∗ ) B i ,r B i ,r . . . B i t ,r t Θ j ,n Θ j ,n . . . Θ j s ,n s K µ (with r a ∈ Z , n b ≥
1, for various a, b )lies in the kernel of gr Φ. Recall the automorphisms ω i of Dr e U ı from Lemma 3.11. Applying anautomorphism Q i ∈ I ω − Ni to X produces an element X N = Q i ∈ I ω − Ni ( X ), which is obtainedfrom X with all indices r a of B ’s in each summand of X shifted to r a + N . Pick and fix an N large enough so that all relevant r a + N are positive, that is, X N ∈ gr Dr e U ı> . Thanks to DRINFELD TYPE PRESENTATION OF AFFINE ı QUANTUM GROUPS, I 25 the following commutative diagramgr Dr e U ı> Q i ∈ I ω Ni / / gr Φ > (cid:15) (cid:15) gr Dr e U ı gr Φ (cid:15) (cid:15) gr e U ı> T Nρ / / gr e U ı We have T Nρ ◦ gr Φ > ( X N ) = gr Φ ◦ ( Q i ∈ I ω Ni )( X N ) = gr Φ( X ) = 0, and hence gr Φ > ( X N ) = 0.Since gr Φ > : gr Dr e U ı> −→ gr e U ı> is injective, we must have X N = 0 and hence X = 0.This proves the injectivity of gr Φ. It follows that Φ : Dr e U ı −→ e U ı is injective. Thiscompletes the proof of Theorem 3.13. (cid:3) Remark . It follows by Lemma 3.5 that B = T θ i T ω ′ i ( B i ) = o ( i ) K T θ i ( B i , − ), where i ∈ I is chosen as in (3.46). The inverse Φ − : e U ı → Dr e U ı to the isomorphism Φ in (3.44) isgiven by K j K j , K C K − θ , B j B j, , B o ( i )T θ i ( B i , − ) C K − θ , for j ∈ I . Another formula for Φ − ( B ) can be read off from the proof of Theorem 3.13. Remark . One can construct all real v -root vectors in e U ı with the help of braid groupaction. Together with the imaginary v -root vectors which have been constructed, one canwrite down a natural PBW basis for e U ı , following [BCP99]. Remark . Definition 3.10 for Dr e U ı formally makes sense for a generalized Cartan matrix(GCM) ( c ij ) i,j ∈ I of a simply-laced Kac-Moody algebra g , and hence can be regarded asa definition of ı quantum loop Kac-Moody algebras for g . A most interesting subclass ofthese new algebras will be ı quantum toroidal algebras when g is of affine type. Once wehave extended Definition 3.10 to a wide class of affine ı quantum groups, similar relaxing ofconditions on GCMs will allow us to define more general ı quantum loop Kac-Moody algebras . Remark . There is a nonstandard comultiplication ∆ n-std (due to Drinfeld) on the affinequantum group Dr U (or its Drinfeld double e U ) via its Drinfeld presentation. It will beinteresting to ask if there is a natural coideal subalgebra of Dr U with respect to ∆ n-std (which, if it exists, could be different from Dr e U ı ).3.6. Classical limit.
Denote the Chevalley generators of the semisimple (or even Kac-Moody) Lie algebra g over Q by e i , f i , h i , for i ∈ I . Denote L g = g ⊗ Q [ t, t − ], and the affineLie algebra (as a vector space) b g = L g ⊕ Q c ; set x k = x ⊗ t k , for x ∈ g , k ∈ Z . Denote by ω the Chevalley involution on g such that ω ( c ) = − c, ω ( e i ) = − f i , ω ( f i ) = − e i , ω ( h i ) = − h i ,for all i . (More generally, one can take ω = ω a , for any fixed nonzero scalar a , such that ω a ( c ) = − c, ω a ( e i ) = − af i , ω a ( f i ) = − a − e i , ω a ( h i ) = − h i .) Then ω induces an involution b ω on b g such that b ω ( x k ) = ω ( x ) − k , for all x ∈ g , k ∈ Z . The algebra e U ı of split affine type ADEspecializes at v = 1 to the enveloping algebra of the b ω -fixed point subalgebra ( L g ) b ω .Let us examine in detail ( L g ) b ω in the case when g = sl with standard basis { e, h, f } . Set b r := f r + e − r , t r := h r − h − r , for r ∈ Z . Note t − r = − t r and t = 0. Then { b r , t m | r ∈ Z , m ≥ } forms a basis for ( L sl ) b ω . They satisfy the relations, for r, s ∈ Z , m ≥ b r , b s ] = t s − r , (3.55) [ t m , t n ] = 0 , (3.56) [ t m , b r ] = − b m + r + 2 b − m + r , (3.57) [ b r , b s +1 ] − [ b r +1 , b s ] = t s − r +1 − t s − r − . (3.58)Clearly, (3.55)–(3.57) are defining relations for ( L sl ) b ω . One checks that (3.55) and (3.58)are equivalent. Hence (3.56)–(3.58) are also defining relations for ( L sl ) b ω , providing a non-standard presentation for the Onsager algebra, which is compatible with our Drinfeld typepresentation up to suitable changes of indices: b r ↔ B , − r and t r ↔ H r ; this index issueoriginates from the identification of the associated graded of the q -Onsager algebra with thepositive (instead of the negative) half of quantum loop sl in [BK20] (which is followed inour Section 2).A Drinfeld type presentation of ( L g ) b ω in ADE case (generalizing (3.56)–(3.58) for g = sl )can be similarly written down, compatible with Definition 3.10.4. Verification of Drinfeld type new relations
In this section, we prove that Φ : Dr e U ı → e U ı defined by (3.44) is a homomorphism. Weshall first establish the relations (3.33), (3.35)–(3.37), and an easier case of (3.34) in e U ı . Wethen establish the more challenging relations (3.38) and (3.34) (when c ij = −
1) in e U ı .4.1. Relation (3.35) .Proposition 4.1.
Assume c ij = 0 , for i, j ∈ I . Then [ B i,k , B j,l ] = 0 , for all k, l ∈ Z .Proof. The identity for k = l = 0, i.e., [ B i , B j ] = 0 , is exactly the defining relation (3.10) for e U ı . The identity for general k, l follows by applying T − kω i T − lω j to the above identity and usingLemma 3.1(c) and Lemma 3.6. (cid:3) Relation (3.36) .Lemma 4.2 (cf. [Be94, Lemma 3.3]) . For i = j ∈ I such that c ij = − , denote X ij := T − j B i = B j B i − vB i B j . Then we have T ω i ( X ji ) = T ω j ( X ij ) . Proof.
This follows by a direct computation using Lemma 3.1 and Lemma 3.6:T ω j ( X ij ) =T ω j T − j ( B i ) = T j T − ω j T ω i ( B i ) = T ω i T j ( B i ) = T ω i ( X ji ) . (cid:3) Now we are ready to establish the relation (3.36).
Proposition 4.3.
We have [ B i,k , B j,l +1 ] v − cij − v − c ij [ B i,k +1 , B j,l ] v cij = 0 , for i = j ∈ I and k, l ∈ Z . DRINFELD TYPE PRESENTATION OF AFFINE ı QUANTUM GROUPS, I 27
Proof. If c ij = 0, then the identity in the proposition follows directly by (3.35).Assume c ij = −
1. Note that v [ B i,k +1 , B j ] v − = − o ( i ) k +1 T − ( k +1) ω i ( X ij ) . Hence we have[ B i,k , B j, ] v = B i,k B j, − vB j, B i,k = o ( i ) k o ( j )T − kω i T − ω j ( X ji ) = − o ( i ) k +1 T − kω i T − ω i ( X ij )= − o ( i ) k +1 T − ( k +1) ω i ( X ij ) = v [ B i,k +1 , B j ] v − . So we have obtained an identity [ B i,k , B j, ] v − v [ B i,k +1 , B j ] v − = 0 . The identity in theproposition follows by applying T − lω j to this identity. (cid:3) Relation (3.34) for c ij = 0 . We shall identify C = K δ below. Lemma 4.4.
For j ∈ I , we have Θ j,n = v − C Θ j,n − + v K − j (cid:0) [ B j, , B j,n ] v − + [ B j,n − , B j, ] v − (cid:1) , if n ≥ , − v − C + v K − j (cid:0) [ B j, , B j, ] v − + [ B j, , B j, ] v − (cid:1) , if n = 2 ,v K − j [ B j, , B j, ] v − , if n = 1 . In particular, for any n ≥ , the element Θ j,n is a Q ( v )[ C ± , K ± j ] -linear combination of and [ B j,k , B j,l +1 ] v − + [ B j,l , B j,k +1 ] v − , for l, k ∈ Z .Proof. The recursion formulas in the lemma are reformulations of (3.37) with k = 0 and l = n −
1. The second statement follows by an induction on n using the recursion formulas.(A precise linear combination can be written down, but will not be needed.) (cid:3) Proposition 4.5.
Assume c ij = 0 , for i, j ∈ I . Then, for m ≥ and r ∈ Z , we have [Θ i,m , B j,r ] = 0 = [ H i,m , B j,r ] . Proof.
We shall prove the first equality only (as H i,n can be expressed in terms of Θ i,m forvarious m ). By Lemma 4.4 (with index j replaced by i ), it suffices to check that [ B i,k , B i,l ] v − commutes with B j,r for all k, l, r . But this clearly follows by the commutativity between B i,k and B j,r (3.35). (cid:3) Relations (3.37) and (3.33) – (3.34) for i = j .Proposition 4.6. Relation (3.37) and relations (3.33) – (3.34) for i = j ∈ I hold in e U ı .Proof. These rank one relations (for a fixed i ∈ I ) follow by transporting the correspondingrelations in q -Onsager algebra (see Theorem 2.16) to e U ı using Proposition 3.9. Note thatfor (3.37), the overall sign o ( i ) k + l +1 originated from (3.28)–(3.30) cancels out. (cid:3) Relation (3.33) for i = j . We shall derive the identity [ H i,m , H j,n ] = 0 in (3.33), for i = j ∈ I , from the relations (3.37) (proved above) and (3.34). The proof of (3.34) in thefollowing subsections will not use the relation (3.33). Lemma 4.7.
For i, j ∈ I , l, k ∈ Z and m ≥ , we have h H i,m , [ B j,k , B j,l +1 ] v − + [ B j,l , B j,k +1 ] v − i = 0 . Proof.
Using (3.34) we have h H i,m , [ B j,k , B j,l +1 ] v − + [ B j,l , B j,k +1 ] v − i =[ H i,m , B j,k ] B j,l +1 + v − B j,l +1 [ B j,k , H i,m ]+ [ H i,m , B j,l ] B j,k +1 + v − B j,k +1 [ B j,l , H i,m ]+ B j,l [ H i,m , B j,k +1 ] + v − [ B j,k +1 , H i,m ] B j,l + B j,k [ H i,m , B j,l +1 ] + v − [ B j,l +1 , H i,m ] B j,k = [ mc ij ] m (cid:16) ( B j,k + m − B j,k − m C m ) B j,l +1 − v − B j,l +1 ( B j,k + m − B j,k − m C m )+ ( B j,l + m − B j,l − m C m ) B j,k +1 − v − B j,k +1 ( B j,l + m − B j,l − m C m )+ B j,l ( B j,k + m +1 − B j,k − m +1 C m ) − v − ( B j,k + m +1 − B j,k − m +1 C m ) B j,l + B j,k ( B j,l + m +1 − B j,l − m +1 C m ) − v − ( B j,l + m +1 − B j,l − m +1 C m ) B j,k (cid:17) . The above equality can be further rewritten as h H i,m , [ B j,k , B j,l +1 ] v − + [ B j,l , B j,k +1 ] v − i = [ mc ij ] m × (cid:16) [ B j,k + m , B j,l +1 ] v − − [ B j,k − m , B j,l +1 ] v − C m + [ B j,l + m , B j,k +1 ] v − − [ B j,l − m , B j,k +1 ] v − C m + [ B j,l , B j,k + m +1 ] v − − [ B j,l , B j,k − m +1 ] v − C m + [ B j,k , B j,l + m +1 ] v − − [ B j,k , B j,l − m +1 ] v − C m (cid:17) . There are 4 terms in each of the above 2 lines, and we add up column by column using(3.37). Note that the sum of column 1 cancels with the sum of column 4 since both are equalto (modulo an opposite sign) v − Θ j,l − k − m +1 C k + m K j − v − Θ j,l − k − m − C k + m +1 K j , if l > k + m,v − Θ j,k + m − l +1 C l K j − v − Θ j,k + m − l − C l +1 K j , if l < k + m, v − Θ j, C l K j , if l = k + m. Similarly, the sum of column 2 cancels with the sum of column 3 since both are equal to(modulo an opposite sign) v − Θ j,k − l − m +1 C l + m K j − v − Θ j,k − l − m − C l + m +1 K j , if k > l + m,v − Θ j,l + m − k +1 C k K j − v − Θ j,l + m − k − C k +1 K j , if k < l + m, v − Θ j, C k K j , if k = l + m. This proves the lemma. (cid:3)
Proposition 4.8.
Relation (3.33) for i = j ∈ I in e U ı follows from the relations (3.34) and (3.37) in e U ı .Proof. It follows by Lemma 4.4 and Lemma 4.7 that [ H i,m , Θ j,a ] = 0, for all m, a ≥ . Since H j,n for any n ≥ j,a , for various a ≥ H i,m , H j,n ] = 0, whence (3.33). (cid:3) DRINFELD TYPE PRESENTATION OF AFFINE ı QUANTUM GROUPS, I 29
Two more relations rephrased.
It remains to establish relations (3.38) and (3.34)for c ij = − e U ı , with the help of the finite type Serre relation (3.11) and (3.36)–(3.37).By Proposition 3.12, the relation (3.34) (for c ij = −
1) is equivalent to the following relation[Θ i,k , B j,r ] + [Θ i,k − , B j,r ] K δ (4.1) = v − [Θ i,k − , B j,r +1 ] v + v [Θ i,k − , B j,r − ] v − K δ , for k ≥ r ∈ Z . Clearly (4.1) also holds for k < e U ı (where we can assume k ≥ k without loss of generality) reads: S ( k , k | l ) = R ( k , k | l ) , for k = k − k ≥ l ∈ Z . (4.2)We shall prove (4.1) and (4.2) simultaneously and inductively on k in § k and (4.2) k , respectively. We then denoteby (4.1) ≤ k (respectively, (4.1) For k , k , l ∈ Z , we have S ( k , k + 1 | l ) + S ( k + 1 , k | l ) − [2] S ( k + 1 , k + 1 | l − (cid:16) − [ B jl , Θ i,k − k +1 ] v − C k + v − [ B jl , Θ i,k − k − ] v − C k +1 (cid:17) K i + { k ↔ k } . (The proof of the lemma uses only the relations (3.36)–(3.37).) Proof. We rewrite (3.31) as S ( k , k | l ) = B i,k [ B i,k , B j,l ] v − − v [ B i,k , B j,l ] v − B i,k + { k ↔ k } . This together with (3.36) implies that S ( k + 1 , k + 1 | l − B i,k +1 [ B i,k +1 , B j,l − ] v − − v [ B i,k +1 , B j,l − ] v − B i,k +1 ) + { k ↔ k } = ( v − B i,k +1 [ B i,k , B j,l ] v − [ B i,k , B j,l ] v B i,k +1 ) + { k ↔ k } . Using (4.3), we compute S ( k , k + 1 | l ) + S ( k + 1 , k | l ) − [2] S ( k + 1 , k + 1 | l − (cid:16) S ( k , k + 1 | l ) − [2] (cid:0) v − B i,k +1 [ B i,k , B j,l ] v − [ B i,k , B j,l ] v B i,k +1 (cid:1)(cid:17) + { k ↔ k } = (cid:16)(cid:0) B i,k [ B i,k +1 , B j,l ] v − v − [ B i,k +1 , B j,l ] v B i,k + B i,k +1 [ B i,k , B j,l ] v − v − [ B i,k , B j,l ] v B i,k +1 (cid:1) − [2] (cid:0) v − B i,k +1 [ B i,k , B j,l ] v − [ B i,k , B j,l ] v B i,k +1 (cid:1)(cid:17) + { k ↔ k } = (cid:0) B jl [ B i,k +1 , B i,k ] v − v − [ B i,k +1 , B i,k ] v B jl (cid:1) + { k ↔ k } , where the last identity is obtained by first combining the third and fifth terms (and respec-tively, the fourth and sixth terms) and then further adding the first and second terms. The relation (3.37) can be rewritten as[ B i,l +1 , B i,k ] v + [ B i,k +1 , B i,l ] v = − (Θ i,l − k +1 C k − v − Θ i,l − k − C k +1 ) K i + { k ↔ l } . (4.5)Using (4.5), we rewrite the RHS of the identity (4.4) as (cid:0) B jl [ B i,k +1 , B i,k ] v − v − [ B i,k +1 , B i,k ] v B jl (cid:1) + { k ↔ k } = B jl (cid:0) [ B i,k +1 , B i,k ] v + [ B i,k +1 , B i,k ] v (cid:1) − v − (cid:0) [ B i,k +1 , B i,k ] v + [ B i,k +1 , B i,k ] v (cid:1) B jl = − B jl (cid:0) Θ i,k − k +1 C k − v − Θ i,k − k − C k +1 (cid:1) K i + v − (cid:0) Θ i,k − k +1 C k − v − Θ i,k − k − C k +1 (cid:1) B jl K i + { k ↔ k } = (cid:16) − [ B jl , Θ i,k − k +1 ] v − C k + v − [ B jl , Θ i,k − k − ] v − C k +1 (cid:17) K i + { k ↔ k } . The lemma is proved. (cid:3) Denote, for n ∈ Z , X n | l = X p ≥ v p +1 [ B j,l +1 , Θ i,n − p − ] v − C p +1 + X p ≥ v p − [2][Θ i,n − p , B j,l ] C p +1 (4.6) − X p ≥ v p +1 [Θ i,n − p − , B j,l − ] v − C p +2 + [Θ i,n , B j,l ] C. Clearly, X n | l = 0, for n ≤ Lemma 4.10. If (4.1) ≤ k holds, then X n | l − = 0 , for all n ≤ k and l ∈ Z . (The converse isalso true.)Proof. Recalling (4.6), we compute X n | l − − v CX n − | l − = v [ B j,l , Θ i,n − ] v − + v [2][Θ i,n − , B j,l − ] C − v [Θ i,n − , B j,l − ] v − C + [Θ i,n , B j,l − ] − v [Θ i,n − , B j,l − ] C = − v − [Θ i,n − , B j,l ] v + [Θ i,n − , B j,l − ] C + [Θ i,n , B j,l − ] − v [Θ i,n − , B j,l − ] v − C. For 0 ≤ n ≤ k , the RHS is 0, which is equivalent to the assumption that (4.1) n holds. Thelemma follows by an induction on n and noting that X − | l − = X | l − = 0. (cid:3) Recall R ( k , k | l ) from (3.32). Let us establish an R -counterpart of Lemma 4.9. Lemma 4.11. For k , k , l ∈ Z with k ≥ k , we have R ( k , k + 1 | l ) + R ( k + 1 , k | l ) − [2] R ( k + 1 , k + 1 | l − − [2] X k − k | l − C k K i + (cid:0) − [ B j,l , Θ i,k − k +1 ] v − C k + v − [ B j,l , Θ i,k − k − ] v − C k +1 (cid:1) K i . Proof. This proof is based on only formal algebraic manipulations, and does not use anynontrivial relations in e U ı .Following the format of (3.32) for R , we write R ( k , k + 1 | l ) = (cid:0) R ′ + R ′ − [ B j,l , Θ i,k − k +1 ] v − (cid:1) C k K i ,R ( k + 1 , k | l ) = (cid:0) R + R − [ B j,l , Θ i,k − k − ] v − C (cid:1) C k K i ,R ( k + 1 , k + 1 | l − 1) = (cid:0) R + R − [ B j,l − , Θ i,k − k ] v − C (cid:1) C k K i . DRINFELD TYPE PRESENTATION OF AFFINE ı QUANTUM GROUPS, I 31 where R ′ and R ′ denote the first and second summands of R ( k , k + 1 | l ) as in (3.32); thenotations in the other two identities are understood similarly.By a direct computation, we have R ′ + R = − X p ≥ v p [2] [ B j,l , Θ i,k − k − p − ] v − C p +1 + v − [2][ B j,l , Θ i,k − k − ] v − C. By first summing up R ′ + R , we also obtain by a direct computation that R ′ + R − [2] R = − X p ≥ v p − [2] [Θ i,k − k − p , B j,l − ] C p +1 − [2][Θ i,k − k , B j,l − ] v − C. Note the main summands in ( R ′ + R ), ( R ′ + R − [2] R ) and − [2] R above are precisely − v − [2] times the three main summands in X k − k | l − defined in (4.6). Hence we have R ( k , k + 1 | l ) + R ( k + 1 , k | l ) − [2] R ( k + 1 , k + 1 | l − R ′ + R ) + ( R ′ + R − [2] R ) − [2] R − [ B j,l , Θ i,k − k +1 ] v − − [ B j,l , Θ i,k − k − ] v − C + [2][ B j,l − , Θ i,k − k ] v − C = ( − v − [2] X k − k | l − + v − [2] [Θ i,k − k , B j,l − ] C )+ v − [2][ B j,l , Θ i,k − k − ] v − C − [2][Θ i,k − k , B j,l − ] v − C − [ B j,l , Θ i,k − k +1 ] v − − [ B j,l , Θ i,k − k − ] v − C + [2][ B j,l − , Θ i,k − k ] v − C = − v − [2] X k − k | l − − [ B j,l , Θ i,k − k +1 ] v − + v − [ B j,l , Θ i,k − k − ] v − C. The lemma is proved. (cid:3) Now we can show that (4.1) Let k ≥ . If (4.1) ≤ k holds, then (4.2) ≤ k +1 holds.Proof. Assume (4.1) ≤ k holds. Then by Lemma 4.10, X k − k | l − = 0. Hence by comparingLemma 4.9 and Lemma 4.11 we obtain, for k = k − k ≥ S ( k , k + 1 | l ) + S ( k + 1 , k | l ) − [2] S ( k + 1 , k + 1 | l − R ( k , k + 1 | l ) + R ( k + 1 , k | l ) − [2] R ( k + 1 , k + 1 | l − . We induct on k . Consider the base case when k = 0. We have S ( r, r | l − 1) = R ( r, r | l − r, l , by applying T − ri T − lj to the finite type Serre relation (3.11). Hence (4.2) ≤ holdsby (4.7) for k = k and noting S ( k , k + 1 | l ) = S ( k + 1 , k | l ) and R ( k , k + 1 | l − 1) = R ( k + 1 , k | l − 1) by the symmetrization definition of S and R .By the inductive assumption, (4.2) ≤ k holds; in particular, we have, for k = k − k > S ( k + 1 , k | l ) = R ( k + 1 , k | l ) , S ( k + 1 , k + 1 | l − 1) = R ( k + 1 , k + 1 | l − . We conclude from this and (4.7) that S ( k , k + 1 | l ) = R ( k , k + 1 | l ), whence (4.2) k +1 . (cid:3) Implication from (4.2) ≤ k to (4.1) ≤ k . We shall fix i, j ∈ I such that c ij = − Lemma 4.13. For k , k , l ∈ Z , we have S ( k , k + 1 | l ) + S ( k + 1 , k | l ) − [2] S ( k , k | l + 1)= (cid:16) − [Θ i,k − k +1 , B jl ] v − C k + v − [Θ i,k − k − , B jl ] v − C k +1 (cid:17) K i + { k ↔ k } . (The proof of the lemma uses only the relations (3.36)–(3.37).) Proof. We rewrite (3.31) as S ( k , k | l ) = B i,k [ B i,k , B j,l ] v − v − [ B i,k , B j,l ] v B i,k + { k ↔ k } . This together with (3.36) implies that S ( k , k | l + 1) = ( B i,k [ B i,k , B j,l +1 ] v − v − [ B i,k , B j,l +1 ] v B i,k ) + { k ↔ k } (4.8) = ( vB i,k [ B i,k +1 , B j,l ] v − − [ B i,k +1 , B j,l ] v − B i,k ) + { k ↔ k } . Using (4.8), we compute S ( k , k + 1 | l ) + S ( k + 1 , k | l ) − [2] S ( k , k | l + 1)(4.9)= (cid:16) S ( k , k + 1 | l ) − [2] (cid:0) vB i,k [ B i,k +1 , B j,l ] v − − [ B i,k +1 , B j,l ] v − B i,k (cid:1)(cid:17) + { k ↔ k } = (cid:16)(cid:0) B i,k [ B i,k +1 , B j,l ] v − v − [ B i,k +1 , B j,l ] v B i,k + B i,k +1 [ B i,k , B j,l ] v − v − [ B i,k , B j,l ] v B i,k +1 (cid:1) − [2] (cid:0) vB i,k [ B i,k +1 , B j,l ] v − − [ B i,k +1 , B j,l ] v − B i,k (cid:1)(cid:17) + { k ↔ k } = (cid:0) [ B i,k +1 , B i,k ] v B jl − v − B jl [ B i,k +1 , B i,k ] v (cid:1) + { k ↔ k } , where the last identity is obtained by first adding the first and fifth terms (and respectively,the second and sixth terms) and then further simplifying when adding with the third andfourth terms.Using (4.5), we rewrite the RHS of the identity (4.9) as (cid:0) [ B i,k +1 , B i,k ] v B jl − v − B jl [ B i,k +1 , B i,k ] v (cid:1) + { k ↔ k } = (cid:0) [ B i,k +1 , B i,k ] v + [ B i,k +1 , B i,k ] v (cid:1) B jl − v − B jl (cid:0) [ B i,k +1 , B i,k ] v + [ B i,k +1 , B i,k ] v (cid:1) = − (cid:0) Θ i,k − k +1 C k − v − Θ i,k − k − C k +1 (cid:1) K i B jl + v − B jl (cid:0) Θ i,k − k +1 C k − v − Θ i,k − k − C k +1 (cid:1) K i + { k ↔ k } = (cid:16) − [Θ i,k − k +1 , B jl ] v − C k + v − [Θ i,k − k − , B jl ] v − C k +1 (cid:17) K i + { k ↔ k } . The lemma is proved. (cid:3) Denote Y k +1 = [ B j,l +1 , Θ i,k ] v − − [Θ i,k , B j,l − ] v − C + v − [Θ i,k +1 , B j,l ] + v − [Θ i,k − , B j,l ] C. (4.10)(The identity (4.1) can be stated equivalently as Y k +1 = 0.) Here is a variant of Lemma 4.11. DRINFELD TYPE PRESENTATION OF AFFINE ı QUANTUM GROUPS, I 33 Lemma 4.14. For k , k , l ∈ Z with k ≥ k , we have R ( k , k + 1 | l ) + R ( k + 1 , k | l ) − [2] R ( k , k | l + 1)= [2] X k − k − | l C k K i + [2] Y k − k +1 C k K i + (cid:0) − [Θ i,k − k +1 , B j,l ] v − C k + v − [Θ i,k − k − , B j,l ] v − C k +1 (cid:1) K i . Proof. This proof is based on only formal algebraic manipulations, and does not use anynontrivial relations in e U ı .Following the format of (3.32) for R , we write R ( k , k + 1 | l ) = (cid:0) R ′ + R ′ − [ B j,l , Θ i,k − k +1 ] v − (cid:1) C k K i ,R ( k + 1 , k | l ) = (cid:0) R + R − [ B j,l , Θ i,k − k − ] v − C (cid:1) C k K i ,R ( k , k | l + 1) = (cid:0) R + R − [ B j,l +1 , Θ i,k − k ] v − (cid:1) C k K i . where R ′ and R ′ denote the first and second summands of R ( k , k + 1 | l ) as in (3.32); thenotations in the other two identities are understood similarly.By a direct computation, we have R ′ + R = − X p ≥ v p +1 [2] [Θ i,k − k − p − , B j,l − ] v − C p +2 − [2][Θ i,k − k , B j,l − ] v − C. By first summing up R ′ + R , we also obtain by a direct computation that R ′ + R − [2] R = − X p ≥ v p − [2] [Θ i,k − k − p − , B j,l ] C p − v [2][ B j,l , Θ i,k − k − ] v − C + [2] [Θ i,k − k − , B j,l ] v − C Note the main summands in ( R ′ + R ), ( R ′ + R − [2] R ) and − [2] R above are precisely[2] times the three main summands (in reversed order) in X k − k defined in (4.6). Hence wehave R ( k , k + 1 | l ) + R ( k + 1 , k | l ) − [2] R ( k , k | l + 1)(4.11) = ( R ′ + R ) + ( R ′ + R − [2] R ) − [2] R − [ B j,l , Θ i,k − k +1 ] v − − [ B j,l , Θ i,k − k − ] v − C + [2][ B j,l +1 , Θ i,k − k ] v − = [2] ( X k − k − | l − [Θ i,k − k − , B j,l ] C ) − [2][Θ i,k − k , B j,l − ] v − C − v [2][ B j,l , Θ i,k − k − ] v − C + [2] [Θ i,k − k − , B j,l ] v − C − [ B j,l , Θ i,k − k +1 ] v − − [ B j,l , Θ i,k − k − ] v − C + [2][ B j,l +1 , Θ i,k − k ] v − . On the RHS of (4.11) above, there is exactly one term involving B i,l +1 and one term involving B i,l − , with opposite coefficients, just as those appearing in Y k − k +1 (4.10). This allows usto rewrite the RHS of (4.11) in terms of X k − k − | l , Y k − k +1 and the terms involving B j,l only. The terms involving B j,l can then be simplified by some direct computation (withoutusing any relations) to the formula stated in the lemma. (cid:3) Now we can show that (4.2) ≤ k (together with (3.11) and (3.36)–(3.37)) implies (4.1) ≤ k . Proposition 4.15. Let k ≥ . If (4.2) ≤ k holds, then (4.1) ≤ k holds. Proof. We prove by induction on k . The identity (4.1) k with k = 0 is trivial. The case(4.1) k +1 (i.e., Y k +1 = 0) follows by comparing the identities in Lemma 4.13 and Lemma 4.14(where k = k − k ≥ ≤ k +1 .(See the proof of Proposition 4.12 for more details of the same type of arguments.) (cid:3) Proposition 4.16. The relations (3.34) and (3.38) for c ij = − hold in e U ı .Proof. These relations follow by the equivalent identities (4.1)–(4.2), which have been estab-lished inductively and simultaneously in Proposition 4.12 and Proposition 4.15. (cid:3) Variants of Drinfeld type presentations In this section, we formulate several variants of the Drinfeld type presentation for e U ı , onein generating function formalism, one in a more symmetrized form, and another via differentimaginary root vectors. We also deduce a Drinfeld type presentation for the ı quantum group U ı ς with parameters ς = ( ς i ) i ∈ I .5.1. Presentation via generating functions. Recall the generating functions B i ( z ) , Θ i ( z )and ∆ ( z ) from (3.41). Theorem 5.1. Dr e U ı is generated by K ± i , C ± , Θ i,k and B i,l ( i ∈ I , k ≥ , l ∈ Z ) , subjectto the following relations, for i, j ∈ I : K i are central, Θ i ( z ) Θ j ( w ) = Θ j ( w ) Θ i ( z ) , (5.1) Θ i ( z ) B j ( w ) = (1 − v − c ij zw − )(1 − v c ij zwC )(1 − v c ij zw − )(1 − v − c ij zwC ) B j ( w ) Θ i ( z ) , (5.2) B i ( w ) B j ( z ) = B j ( z ) B i ( w ) , if c ij = 0 , (5.3) ( v c ij z − w ) B i ( z ) B j ( w ) + ( v c ij w − z ) B j ( w ) B i ( z ) = 0 , if i = j, (5.4) ( v z − w ) B i ( z ) B i ( w ) + ( v w − z ) B i ( w ) B i ( z )(5.5) = v − v − v − K i ∆ ( zw ) (cid:0) ( v z − w ) Θ i ( w ) + ( v w − z ) Θ i ( z ) (cid:1) , B i ( w ) B i ( w ) B j ( z ) − [2] B i ( w ) B j ( z ) B i ( w ) + B j ( z ) B i ( w ) B i ( w ) + { w ↔ w } (5.6) = − K i ∆ ( w w ) v − v − (cid:16) [ Θ i ( w ) , B j ( z )] v − [2] zw − − v w w − + [ B j ( z ) , Θ i ( w )] v − w w − − v w w − (cid:17) + { w ↔ w } , if c ij = − . Proof. We simply rewrite the relations (3.33)–(3.38) in Theorem 3.10 by using the generatingfunctions (3.41). The relation (5.1) is clear. As formulated in Proposition 3.12, the relation(5.2) is equivalent to (3.34). The relation (5.5) is obtained from (3.37) by multiplying bothside of the relation (3.37) by v z r +1 w s +1 and summing over r, s ∈ Z . Similarly, the relations(5.3) and (5.4) are equivalent to the relations (3.35) and (3.36), respectively. Finally, therelation (5.6) is obtained by multiplying both side of the relation (3.38) by w k w k z r andsumming over k , k , r ∈ Z . (cid:3) DRINFELD TYPE PRESENTATION OF AFFINE ı QUANTUM GROUPS, I 35 A symmetrized presentation. We add two central generators C ± such that C C − =1 = C − C and ( C ) = C . Definition 5.2. Let DR e U ı be the Q ( v ) -algebra generated by K ± i , C ± , b H i,m and B i,l , where i ∈ I , m ≥ , l ∈ Z , subject to the following relations, for m, n ≥ and k, l ∈ Z : K i , C ± are central, [ b H i,m , b H j,n ] = 0 , (5.7) [ b H i,m , B jl ] = [ mc ij ] m B j,l + m C − m − [ mc ij ] m B j,l − m C m , (5.8) [ B i,k , B j,l +1 ] v − cij − v − c ij [ B i,k +1 , B j,l ] v cij = 0 , if i = j, (5.9) [ B i,k , B j,l ] = 0 , if c ij = 0 , (5.10) [ B i,k , B i,l +1 ] v − − v − [ B i,k +1 , B i,l ] v (5.11) = K i C k + l +12 (cid:16) v − b Θ i,l − k +1 − v − b Θ i,l − k − + v − b Θ i,k − l +1 − v − b Θ i,k − l − (cid:17) ,B i,k B i,k B j,l − [2] B i,k B j,l B i,k + B j,l B i,k B i,k + { k ↔ k } (5.12) = K i C k k (cid:16) − X p ≥ v p [2][ b Θ i,k − k − p − , B j,l − ] v − C − X p ≥ v p − [2][ B j,l , b Θ i,k − k − p ] v − + v [ B j,l , b Θ i,k − k ] v − (cid:17) + { k ↔ k } , if c ij = − . Here b Θ i,m , for i ∈ I and m ≥ , are defined by the following equation: X m ≥ ( v − v − ) b Θ i,m u m = exp (cid:0) ( v − v − ) ∞ X m =1 b H i,m u m (cid:1) ;(5.13) we also set b Θ i, = v − v − , b Θ i,m = 0 for m < . We enlarge Dr e U ı to a Q ( v )-algebra Dr e U ı [ C ± ] := Dr e U ı ⊗ Q ( v )[ C ± ] Q ( v )[ C ± ]; recall itcontains the generators H m . For i ∈ I and m ≥ 1, we identify b H i,m = H i,m C − m , b Θ i,m = Θ i,m C − m . (5.14)Then (5.13) holds. Now such an identification (5.14) leads to an isomorphism DR e U ı ∼ = Dr e U ı [ C ± ] in the proposition below, which also identifies elements in the same notation. Weskip the detail as the verification is straightforward. Proposition 5.3. We have a natural Q ( v ) -algebra isomorphism DR e U ı ∼ = Dr e U ı [ C ± ] . Presentation via different root vectors. Recall starting from the rank 1 casetreated in Section 2, we have preferred the imaginary root vectors Θ m over ´Θ m , becauseof a consideration from Hall algebra [LRW20]. Choosing ´Θ m will lead to the following pre-sentation for e U ı . Theorem 5.4. The algebra e U ı admits a presentation with the same set of generators of Dr e U ı as in Definition 3.10, subject to the relations (3.33) – (3.36) for Dr e U ı and the following (5.15) – (5.16) (in place of (3.37) – (3.38) for Dr e U ı ): [ B i,r , B i,s +1 ] v − − v − [ B i,r , B i,s ] v = v − Θ i,s − r +1 C r K i − v − Θ i,s − r − C r +1 K i (5.15) + v − Θ i,r − s +1 C s K i − v − Θ i,r − s − C s +1 K i ,B i,k B i,k B j,l − [2] B i,k B j,l B i,k + B j,l B i,k B i,k + { k ↔ k } (5.16) = − C k K i (cid:16) P p ≥ ( v − p − + v p +1 )[Θ i,k − k − p − , B j,l − ] v − C p +1 + P p ≥ ( v − p + v p )[ B j,l , Θ i,k − k − p ] v − C p + [ B j,l , Θ i,k − k ] v − (cid:17) + { k ↔ k } , if c ij = − . Proof. In this proof, we shall denote the generators Θ i,k in the proposition by ´Θ i,k , in orderto distinguish the Θ i,k used for Dr e U ı in Definition 3.10.By Theorem 3.13, it suffices to show the algebra with presentation given by the propositionis isomorphic to Dr e U ı . The isomorphism is given by match generators in the same notationand in addition imposing the relation between Θ i,k and ´Θ i,k as follows (cf. (2.31)): Θ i ( z ) = 1 − Cz − v − Cz ´ Θ i ( z ) . Then the equivalence between the relations (3.37) and (5.15) follows from the equivalencebetween (2.18) and (2.35). Finally, the equivalence between the relations (3.38) and (5.16)follows by a direct computation, which we omit here. (cid:3) Presentation of affine ı quantum groups with parameters. Fix ς = ( ς i ) i ∈ I , where ς i ∈ Q ( v ) × , for each i . A quantum symmetric pair ( U , U ı ς ) was first formulated by G. Letzterin finite type and then generalized by Kolb [Ko14]. The ı quantum group U ı ς is the Q ( v )-algebra with generators B i ( i ∈ I ), and it can be obtained from e U ı by a central reduction[LW19a] (recall ˜ k i was used loc. cit. and it is related to K i in this paper by K i = − v ˜ k i ): U ı ς = e U ı / ( K i + v ς i | i ∈ I ) . (5.17)For δ = P i ∈ I a i α i , we define δ ς = Y i ∈ I ( − v ς i ) a i ∈ Q ( v ) . From (5.17) we obtain a natural surjective homomorphism of Q ( v )-algebra π : e U ı −→ U ı ς , B i B i ( i ∈ I ) . By abuse of notations, we shall keep using the same notations for the images under π of various elements such as B i,k , Θ i,m , H i,m , for i ∈ I , k ∈ Z , m ≥ 1. The algebra U ı ς has a Serre-type presentation with generators B i , for i ∈ I , and defining relations (3.10)–(3.12), where K i is replaced by − v ς i , cf. [BB10, Ko14]. Below we present a Drinfeld typepresentation for U ı ς . Theorem 5.5. Let ς = ( ς i ) i ∈ I ∈ Q ( v ) × , I . Then the Q ( v ) -algebra U ı ς has a presentation withgenerators H i,m and B i,l , where i ∈ I , m ≥ , l ∈ Z and the following relations, for r, s ∈ Z , DRINFELD TYPE PRESENTATION OF AFFINE ı QUANTUM GROUPS, I 37 m, n ≥ , i, j ∈ I : [ H i,m , H j,n ] = 0 , (5.18) [ H i,m , B j,l ] = [ mc ij ] m B j,l + m − [ mc ij ] m δ m ς B j,l − m , (5.19) [ B i,k , B j,l ] = 0 , if c ij = 0 , (5.20) [ B i,k , B j,l +1 ] v − cij − v − c ij [ B i,k , B j,l ] v cij = 0 , if i = j, (5.21) [ B i,k , B i,l +1 ] v − − v − [ B i,k , B i,l ] v = − δ k ς ς i Θ i,l − k +1 + v − δ k +1 ς ς i Θ i,l − k − (5.22) − δ l ς ς i Θ i,k − l +1 + v − δ l +1 ς ς i Θ i,k − l − ,B i,k B i,k B j,l − [2] B i,k B j,l B i,k + B j,l B i,k B i,k + { k ↔ k } (5.23)= v ς i δ k ς X p ≥ v p [2] δ p +1 ς [Θ i,k − k − p − , B j,l − ] v − + X p ≥ v p − [2] δ p ς [ B j,l , Θ i,k − k − p ] v − + [ B j,l , Θ i,k − k ] v − ! + { k ↔ k } , if c ij = − . 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Watanabe, Classical weight modules over ı quantum groups , arXiv:1912.11157 Department of Mathematics, Sichuan University, Chengdu 610064, P.R.China E-mail address : [email protected] Department of Mathematics, University of Virginia, Charlottesville, VA 22904, USA E-mail address ::