Weight-finite modules over the quantum affine and double quantum affine algebras of type a 1
aa r X i v : . [ m a t h . QA ] J u l WEIGHT-FINITE MODULES OVER THE QUANTUM AFFINE AND DOUBLEQUANTUM AFFINE ALGEBRAS OF TYPE a E. MOUNZER AND R. ZEGERS
Abstract.
We define the categories of weight-finite modules over the type a quantum affine algebra ˙U q ( a )and over the type a double quantum affine algebra ¨U q ( a ) that we introduced in [MZ19]. In both cases, weclassify the simple objects in those categories. In the quantum affine case, we prove that they coincide withthe simple finite-dimensional ˙U q ( a )-modules which were classified by Chari and Pressley in terms of theirhighest (rational and ℓ -dominant) ℓ -weights or, equivalently, by their Drinfel’d polynomials. In the doublequantum affine case, we show that simple weight-finite modules are classified by their ( t -dominant) highest t -weight spaces, a family of simple modules over the subalgebra ¨U q ( a ) of ¨U q ( a ) which is conjecturallyisomorphic to a split extension of the elliptic Hall algebra. The proof of the classification, in the doublequantum affine case, relies on the construction of a double quantum affine analogue of the evaluation modulesthat appear in the quantum affine setting. Introduction
The representation theory of quantum affine algebras is a vast and extremely rich theory which is still thesubject of an intense research activity after more than three decades. The recent discovery of its relevance tothe monoidal categorification of cluster algebras provides one of the latest and most striking illustrations ofit – see [HL19] for a review on that subject. Probably standing as one of the most significant breakthroughsin the early days of this research area, the classification of the simple finite-dimensional modules over thequantum affine algebra of type a , U q ( ˙ a ), is due to Chari and Pressley [CP91]. It relies, on one hand, ona careful analysis of the ℓ -weight structure of those modules made possible by the existence of Drinfel’d’spresentation ˙U q ( a ) of U q ( ˙ a ) – see [Dam93] for the proof that ˙U q ( a ) ∼ = U q ( ˙ a ) – and, on the other hand, onthe existence of evaluation modules, proven earlier by Jimbo, [Jim86]. This seminal work paved the way for amore systematic study of the representation theory of quantum affine algebras of all Cartan types, leading tothe development of powerful tools such as q -characters, ( q, t )-characters and, consequently, to a much betterunderstanding of the categories FinMod of their finite-dimensional modules that recently culminated withthe realization that the Grothendieck rings of certain subcategories of the categories
FinMod actually havethe structure of a cluster algebra, [HL10].By contrast, it is fair to say that the representation theory of quantum toroidal algebras, which wereinitially introduced in type a n by Ginzburg, Kapranov and Vasserot [GKV95] and later generalized tohigher rank types, is significantly less well understood and remains, to this date, much more mysterious –although see [Her09] for a review and references therein. In our previous work, [MZ19], we constructed anew (topological) Hopf algebra ¨U q ( a ), called double quantum affinization of type a , and proved that itscompletion (in an appropriate topology) is bicontinuously isomorphic to (a corresponding completion) ofthe quantum toroidal algebra ˙U q ( ˙ a ). Whereas ˙U q ( ˙ a ) is naturally graded over Z × ˙ Q , where ˙ Q stands forthe root lattice of the untwisted affine root system ˙ a of type A (1)1 , ˙U q ( ˙ a ) is naturally graded over Z × Q ,where Q stands for the root lattice of the finite root system a of type A . Thus ¨U q ( a ) turns out to be to˙U q ( ˙ a ) what ˙U q ( a ) is to U q ( ˙ a ), i.e. its Drinfel’d presentation. The latter, in the quantum affine case, has a natural triangular decomposition which allows one to define an adapted class of highest weight modules,namely highest ℓ -weight modules, in which finite-dimensional modules are singled out by the particularform of their highest ℓ -weights. Therefore, it is only natural to ask the question of whether ¨U q ( a ) plays asimilar role for the representation theory of ˙U q ( ˙ a ), leading, in particular, to a new notion of highest weightmodules. We answer positively that question and introduce the corresponding notion of highest t -weightmodules. Schematically, whereas the transition from the classical Lie theoretic weights to ℓ -weights canbe regarded as trading numbers for (rational) functions, the transition from ℓ -weights to t -weights can beregarded as trading (rational) functions for entire modules over the non-commutative ¨U q ( a )-subalgebra of¨U q ( a ). That substitution can be interpreted from the perspective of a conjecture in [MZ19], stating that¨U q ( a ) is isomorphic to a split extension of the elliptic Hall algebra E q − ,q ,q which was initially defined byMiki, in [Mik07], as a ( q, γ )-analogue of the W ∞ algebra and reappeared later on in different guises; thequantum continuous gl ∞ algebra in [FFJ + gl in [FJMM12] and in subsequentworks by Feigin et al. Our conjecture is actually supported by the existence of an algebra homomorphismbetween E q − ,q ,q and ¨U q ( a ) which we promote, in the present paper, to a (continuous) homomorphismof (topological) Hopf algebras. Intuitively, the weights adapted to our new triangular decomposition cantherefore be regarded as representations of a quantized algebra of functions on a non-commutative 2-torus.On the other hand, unless the value of some scalar depending on the deformation parameter is takento be a root of unity, the question of the existence of finite-dimensional modules over quantum toroidalalgebras of type a n ≥ was already answered negatively by Varagnolo and Vasserot in [VV96]. However,it is possible to push further the analogy with the quantum affine situation by defining another type offiniteness condition, namely weight-finiteness. It turns out that, in type 1, i.e. when the central charges acttrivially, ¨U q ( a ) admits an infinite dimensional abelian subalgebra that, itself, admits as a subalgebra theCartan subalgebra U q ( a ) of the Drinfel’d-Jimbo quantum algebra U q ( a ) of type a . Hence, we can assignclassical Lie theoretic weights to the t -weight spaces of our modules and declare that a ¨U ′ q ( a )-module isweight-finite whenever it has only finitely many classical weights. The same notion is readily defined formodules over ˙U q ( a ) and we then focus on WFinMod˙ (resp.
WFinMod ), i.e. the full subcategory of thecategory
Mod˙ (resp.
Mod ) of ¨U q ( a )-modules (resp. ˙U q ( a )-modules) whose modules are weight-finite.Of course, the widely studied category FinMod of finite-dimensional ˙U q ( a )-modules is a full subcategoryof WFinMod . The main results of the present paper consist in showing that, on one hand, the simpleobjects in
WFinMod are all finite-dimensional and therefore coincide with the simple finite-dimensional˙U q ( a )-modules classified by Chari and Pressley, and, on the other hand, in classifying the simple objectsin WFinMod˙ in terms of their highest t -weight spaces. These results clearly establish WFinMod˙ as thenatural quantum toroidal analogue of
FinMod and suggest studying further its structure and, in particular,the structure of its Grothendieck ring. Another natural development at this point would be to generalize tothe quantum toroidal setting the interesting classes of ˙U q ( a )-modules outside of FinMod , for example byconstructing a quantum toroidal analogue of category O . We leave these questions for future work.The present paper is organized as follows. In section 2, we briefly review classic results about the quantumaffine algebra ˙U q ( a ) and its finite-dimensional modules. Then, we prove that simple objects in WFinMod are actually finite-dimensional. In section 3, we review the main relevant results of [MZ19] and establish afew new results, as relevant for the subsequent sections. We define highest t -weight modules in section 4 and,by thoroughly analyzing their structure, we establish one implication in our classification theorem, namelytheorem 4.21. The opposite implication is established in section 5 by explicitly constructing a quantum EIGHT-FINITE MODULES OVER ˙U q ( a ) AND ¨U q ( a ) 3 toroidal analogue of the quantum affine evaluation modules. That construction is obtained after proving theexistence of an evaluation homomorphism between ¨U q ( a ) and an evaluation algebra built as a double semi-direct product of ˙U q ( a ) with the completions of some Heisenberg algebras. The evaluation modules are thenobtained by pulling back induced modules over the evaluation algebra along the evaluation homomorphism. Notations and conventions.
We let N = { , , . . . } be the set of natural integers including 0. We denoteby N × the set N − { } . For every m ≤ n ∈ N , we denote by J m, n K = { m, m + 1 , . . . , n } . We also let J n K = J , n K for every n ∈ N . For every m, n ∈ N × , we let C m ( n ) := (cid:8) λ = ( λ , . . . , λ m ) ∈ (cid:0) N × (cid:1) m : λ + · · · + λ m = n (cid:9) , denote the set of m -compositions of n , i.e. of compositions of n having m summands.We let sign : Z → {− , , } be defined by setting, for any n ∈ Z ,sign( n ) = − n < n = 0;1 if n > K is an algebraically closed field of characteristic 0 and we let F := K ( q )denote the field of rational functions over K in the formal variable q . As usual, we let K × = K − { } and F × = F − { } . Whenever we wish to evaluate q to some element of K × , we shall always do so under therestriction that 1 / ∈ q Z × . For every m, n ∈ N , we define the following elements of F [ n ] q := q n − q − n q − q − , [ n ] ! q := [ n ] q [ n − q · · · [1] q if n ∈ N × ;1 if n = 0; (cid:18) nm (cid:19) q := [ n ] ! q [ m ] ! q [ n − m ] ! q . (1.1)We shall say that a polynomial P ( z ) ∈ F [ z ] is monic if P (0) = 1. For every rational function P ( z ) /Q ( z ),where P ( z ) and Q ( z ) are relatively prime polynomials, we denote by (cid:18) P ( z ) Q ( z ) (cid:19) | z |≪ (resp. (cid:18) P ( z ) Q ( z ) (cid:19) | z | − ≪ )the Laurent series of P ( z ) /Q ( z ) at 0 (resp. at ∞ ).We shall let a [ A, B ] b = aAB − bBA , for any symbols a , b , A and B provided the r.h.s of the above equations makes sense.The Dynkin diagrams and correponding Cartan matrices of the root systems a and ˙ a are reminded inthe following table. Type Dynkin diagram Simple roots Cartan matrix a { α } (2)˙ a { α , α } − − ! E. MOUNZER AND R. ZEGERS Weight-finite modules over the quantum affine algebra ˙U q ( a )2.1. The quantum affine algebra ˙U q ( a ) .Definition 2.1. The quantum affine algebra ˙U q ( a ) is the associative K ( q )-algebra generated by n D, D − , C / , C − / , k +1 ,n , k − , − n , x +1 ,m , x − ,m : m ∈ Z , n ∈ N o subject to the following relations C ± / is central C ± / C ∓ / = 1 D ± D ∓ = 1 (2.1) D k ± ( z ) D − = k ± ( zq − ) D x ± ( z ) D − = x ± ( zq − ) (2.2) k ± ( z ) k ± ( z ) = k ± ( z ) k ± ( z ) (2.3) k − ( z ) k +1 ( z ) = G − ( C − z /z ) G + ( Cz /z ) k +1 ( z ) k − ( z ) = 1 mod z /z (2.4) G ∓ ( C ∓ / z /z ) k +1 ( z ) x ± ( z ) = x ± ( z ) k +1 ( z ) (2.5) k − ( z ) x ± ( z ) = G ∓ ( C ∓ / z /z ) x ± ( z ) k − ( z ) (2.6)( z − q ± z ) x ± ( z ) x ± ( z ) = ( z q ± − z ) x ± ( z ) x ± ( z ) (2.7)[ x +1 ( z ) , x − ( z )] = 1 q − q − (cid:20) δ (cid:18) z Cz (cid:19) k +1 ( z C − / ) − δ (cid:18) z Cz (cid:19) k − ( z C − / ) (cid:21) (2.8)where we define the following ˙U q ( ˙ a )-valued formal distributions x ± ( z ) := X m ∈ Z x ± ,m z − m ∈ ˙U q ( ˙ a )[[ z, z − ]] ; (2.9) k ± ( z ) := X n ∈ N k ± , ± n z ∓ n ∈ ˙U q ( ˙ a )[[ z ∓ ]] , (2.10)for every i, j ∈ ˙ I , we define the following F -valued formal power series G ± ( z ) := q ± c ij + ( q − q − )[ ± c ij ] q X m ∈ N × q ± mc ij z m ∈ F [[ z ]] (2.11)and δ ( z ) := X m ∈ Z z m ∈ F [[ z, z − ]] (2.12)is an F -valued formal distribution. We denote by ˙U ′ q ( a ) the subalgebra of ˙U q ( a ) generated by n C / , C − / , k +1 ,n , k − , − n , x +1 ,m , x − ,m : m ∈ Z , n ∈ N o . We denote by ˙U q ( a ) the subalgebra of ˙U ′ q ( a ) generated by n C / , C − / , k +1 ,n , k − , − n : n ∈ N o . We let ˙U ≥ q ( a ) (resp. ˙U ≤ q ( a )) denote the subalgebra of ˙U ′ q ( a ) generated by n C / , C − / , k +1 ,n , k − , − n , x +1 ,m : m ∈ Z , n ∈ N o (resp. n C / , C − / , k +1 ,n , k − , − n , x − ,m : m ∈ Z , n ∈ N o ). We let ˙U q ( a )˘denote the F -algebra generated by the same generators as ˙U ′ q ( a ), subject to the relations(2.3 - 2.7) – i.e. we omit relation (2.8). We define the type a quantum loop algebra U q (L a ) as the quotient EIGHT-FINITE MODULES OVER ˙U q ( a ) AND ¨U q ( a ) 5 of ˙U ′ q ( a ) by its two-sided ideal ( C / −
1) generated by (cid:8) C / − , C − / − (cid:9) . Similarly, we let U ≥ q (L a ) =˙U ≥ q ( a ) / ( C / −
1) and U ≤ q (L a ) = ˙U ≤ q ( a ) / ( C / − q (L a ) = ˙U q ( a )˘ / ( C / − Proposition 2.2.
There exists a surjective F -algebra homomorphism ˘U q (L a ) → U q (L a ) . Finite dimensional ˙U ′ q ( a ) -modules. Let
Mod be the category of ˙U ′ q ( a )-modules. We denote by FinMod the full subcategory of
Mod whose objects are finite-dimensional. Following [CP91], we make thefollowing
Definition 2.3.
We shall say that a ˙U ′ q ( a )-module M is: • a weight module if k +1 , acts semisimply on M ; • of type C / acts on M as id; • highest ℓ -weight if it is of type 1 and there exists v ∈ M − { } such that x +1 ( z ) .v = 0 , k ± ( z ) .v = κ ± ( z ) v for some κ ± ( z ) ∈ F [[ z ∓ ]] and M = ˙U ′ q ( a ) .v . We shall refer to any such v as a highest ℓ -weightvector and to κ = ( κ + ( z ) , κ − ( z )) as the corresponding highest ℓ -weight.Clearly, type 1 ˙U ′ q ( a )-modules coincide with U q (L a )-modules. Definition 2.4.
For every κ ∈ F [[ z − ]] × F [[ z ]], we construct a one-dimensional U ≥ q (L a )-module F κ ∼ = F by setting x +1 ( z ) . , and k ± ( z ) . κ ± ( z ) . We then define the universal highest ℓ -weight ˙U ′ q ( a )-module with highest ℓ -weight κ by setting M ( κ ) := U q (L a ) ⊗ U ≥ q (L a ) F κ as U q (L a )-modules. Let N ( κ ) be the maximal U q (L a )-submodule of M ( κ ) such that N ( κ ) ∩ F κ = { } and set L ( κ ) := M ( κ ) /N ( κ ) . By construction, L ( κ ) is a simple highest ℓ -weight U q (L a )-module with highest ℓ -weight κ . It is unique upto isomorphisms.The simple objects in FinMod were classified by Chari and Pressley in [CP91]. The main result is thefollowing
Theorem 2.5 (Chari-Pressley) . The following hold:i. any simple finite-dimensional ˙U ′ q ( a ) -module M can be obtained by twisting a simple finite-dimensional ˙U ′ q ( a ) -module of type with an algebra automorphism of Aut( ˙U ′ q ( a )) ;ii. every simple finite dimensional ˙U ′ q ( a ) -module of type is highest ℓ -weight;iii. the simple highest ℓ -weight module L ( κ ) is finite-dimensional if and only if κ ± ( z ) = q deg( P ) (cid:18) P ( q − /z ) P (1 /z ) (cid:19) | z | ∓ ≪ , for some monic polynomial P (1 /z ) ∈ F [ z − ] called Drinfel’d polynomial of L ( κ ) .Proof. The proof can be found in [CP91]. (cid:3)
E. MOUNZER AND R. ZEGERS
Up to isomorphisms, the simple objects in
FinMod are uniquely parametrized by their Drinfel’d poly-nomials and we shall therefore denote by L ( P ) the (isomorphism class of the) simple ˙U ′ q ( a )-module withDrinfel’d polynomial P . Note that the roles of ˙U ≥ q ( a ) and ˙U ≤ q ( a ) in the above constructions are clearlysymmetrical and we could have equivalently considered lowest ℓ -weight modules. In particular, point iii. ofthe above theorem immediately translates into Proposition 2.6.
The simple lowest ℓ -weight module with lowest ℓ -weight κ = ( κ + ( z ) , κ − ( z )) ∈ F [[ z − ]] × F [[ z ]] is finite-dimensional if and only if κ ± ( z ) = q − deg( P ) (cid:18) P (1 /z ) P ( q − /z ) (cid:19) | z | ∓ ≪ , for some monic polynomial P (1 /z ) ∈ F [ z − ] . In the latter case, we denote it by ¯ L ( P ) . Weight-finite simple U q (L a ) -modules. We now wish to consider a slightly broader family of mod-ules over ˙U ′ q ( a ). In particular, we want to allow these modules to be infinite-dimensional, while retainingsome of the nice features of finite dimensional ˙U ′ q ( a )-modules such as the fact that they decompose into ℓ -weight spaces. This is achieved by introducing the following notion. Definition 2.7.
We shall say that a (not necessarily finite-dimensional) ˙U ′ q ( a )-module M is ℓ -weight if thereexists a countable set { M α : α ∈ A } of indecomposable locally finite-dimensional ˙U q ( a )-modules, called the ℓ -weight spaces of M , such that, as ˙U q ( a )-modules, M ∼ = M α ∈ A M α . We shall say that M is of type 1 if C / acts on M by id. Definition-Proposition 2.8.
Let M be an ℓ -weight ˙U ′ q ( a )-module. Then:i. C acts as id over M ;ii. for every ℓ -weight space M α , α ∈ A , of M , there exists κ α, ∈ F × and ( κ ± α, ± m ) m ∈ N × ∈ F N × such that M α ⊆ (cid:8) v ∈ M : ∃ n ∈ N × , ∀ m ∈ N (cid:0) k ± , ± m − κ ± α, ± m id (cid:1) n .v = 0 (cid:9) , where we have set κ ± α, = κ ± α, .We let Sp( M ) = { κ α, : α ∈ A } and refer to the formal power series κ ± α ( z ) = X m ∈ N κ ± α, ± m z ∓ m as the ℓ -weight of the ℓ -weight space M α . Proof.
Let M α be an ℓ -weight space of M and let v ∈ M α −{ } . By definition, there exists a finite dimensional˙U q ( a )-submodule ˜ M α of M α such that v ∈ ˜ M α . Over ˜ M α , C must admit an eigenvector and, since C iscentral, it follows that C acts over ˜ M α by a scalar mutliple of id. Assume for a contradiction that C − C − doesnot act by multiplication by zero. Then, it is possible to pull back ˜ M α into a finite-dimensional module overthe Weyl algebra A ( K ) = K h x, y i / ( xy − yx −
1) by the obvious algebra homomorphism A ( K ) ֒ → ˙U q ( a ).But the Weyl algebra is known to admit no finite-dimensional modules. A contradiction. It follows that C acts as id over ˜ M α . But this could be repeated for any non-zero vector in any ℓ -weight space of M .i. follows. As for ii., observe that, as a consequence of i. and of the defining relations (2.3) and (2.4), (cid:8) k +1 ,m , k − , − m : m ∈ N (cid:9) acts by a family of commuting linear operators over M . Thus ii. follows from thedecomposition of locally finite-dimensional vector spaces into the generalized eigenspaces of a commuting EIGHT-FINITE MODULES OVER ˙U q ( a ) AND ¨U q ( a ) 7 family of linear operators; the indecomposability of M α further imposing that it coincides with a single blockin a single generalized eigenspace. (cid:3) Remark . It is worth emphasizing that definition 2.7 and definition-proposition 2.8 straightforwardly gen-eralize to (topological) modules over any (topological) algebra A containing ˙U q ( a ) as a (closed) subalgebra. Definition 2.10.
We shall say that an ℓ -weight ˙U ′ q ( a )-module M is weight-finite if Sp ( M ) is a finite set.We let WFinMod denote the full subcategory of the category
Mod of ˙U ′ q ( a )-modules whose objects areweight-finite.Clearly, finite dimensional ˙U ′ q ( a )-modules are objects in WFinMod , but not every object in
WFinMod is in
FinMod . However we have
Theorem 2.11.
The following hold:i. every simple ℓ -weight ˙U ′ q ( a ) -module can be obtained by twisting a simple ℓ -weight ˙U ′ q ( a ) -module oftype with an algebra automorphism of Aut( ˙U q ( a )) ;ii. every weight-finite simple U q (L a ) -module is highest ℓ -weight;iii. every weight-finite simple U q (L a ) -module is finite dimensional.Proof. In view of definition-proposition 2.8, C acts as id over M . Since the latter is simple and since C / is central, it is clear that C acts over M either as id or as − id. In the former case, there is nothing to do;whereas in the latter, upon twisting as in the finite-dimensional case – see [CP91] –, we can ensure that C / acts as id. This proves i.. As for ii., the same proof as for part ii. of theorem 2.5 can be used. So,we eventually prove iii.. Let M be a weight-finite simple U q (L a )-module. By ii. it is highest ℓ -weight.Hence, there exists v ∈ M − { } such that M ∼ = U q (L a ) .v , x +1 ( z ) .v = 0 and k ± ( z ) .v = κ ± ( z ) v , for some κ ± ( z ) ∈ F [[ z ∓ ]] with res z ,z z − z − κ + ( z ) κ − ( z ) = 1 . The triangular decomposition of U q (L a ) impliesthat M = U − q (L a ) .v and, setting for every n ∈ N v ( z , . . . , z n ) = x − ( z ) · · · x − ( z n ) .v , (2.13)it is clear that (cid:26) v m ,...,m n = res z ,...,z n z − − m · · · z − − m n n v ( z , . . . , z n ) : n ∈ N , m , . . . , m n ∈ Z (cid:27) (2.14)is a spanning set of M . The defining relations (2.5) and (2.6) of U q (L a ) easily imply that, for every n ∈ N , k ± ( z ) .v ( z , . . . , z n ) = κ ± ( z ) n Y p =1 G ∓ (cid:16) ( z p /z ) ∓ (cid:17) v ( z , . . . , z n ) (2.15)and, in particular, k ± .v ( z , . . . , z n ) = ( κ +0 ) ± q − n v ( z , . . . , z n ) . Therefore, M being weight-finite, there must exist an N ∈ N such that x − ( z ) .v ( z , . . . , z N ) = 0 . (2.16)Making use of (2.8), one easily proves that, for every n ∈ J N K x +1 ( z ) .v ( z , . . . , z n ) = 1 q − q − n X p =0 δ (cid:16) z p z (cid:17) " κ + ( z ) n Y r = p +1 G − ( z r /z ) (2.17) − κ − ( z ) n Y r = p +1 G + ( z/z r ) v ( z , . . . , b z p , . . . , z n ) , E. MOUNZER AND R. ZEGERS where a hat over a variable indicates that that variable should be omitted. Combining (2.16) and (2.8), weget − x − ( z ) x +1 ( z ) .v ( z , . . . , z N ) = [ x +1 ( z ) , x − ( z )] .v ( z , . . . , z N )= 1 q − q − δ (cid:16) z z (cid:17) " κ + ( z ) N Y p =1 G − ( z p /z ) − κ − ( z ) N Y p =1 G + ( z/z p ) v ( z , . . . , z N ) . Making use of (2.17) and (2.13), the above equation eventually yields N X p =0 δ (cid:16) z p z (cid:17) " κ + ( z p ) N Y r = p +1 G − ( z r /z p ) − κ − ( z p ) N Y r = p +1 G + ( z p /z r ) v ( z , . . . , b z p , . . . , z N ) = 0 . Acting on the l.h.s of the above equation with x +1 ( ζ N ) · · · x +1 ( ζ ) and making repeated use of (2.17), oneeasily shows that X σ ∈ S N +1 N Y i =0 δ (cid:18) z i ζ σ ( i ) (cid:19) κ + ( z i ) N Y r = i +1 σ ( r ) >σ ( i ) G − ( z r /z i ) − κ − ( z i ) N Y r = i +1 σ ( r ) >σ ( i ) G + ( z i /z r ) v = 0 . (2.18)Since v = 0, its prefactor in the above equation must vanish. Now, it is clear that multiplication of the latterby Q N − j =0 ( z − ζ j ) annihilates all the summands with σ such that σ (0) = N . Similarly, multiplication by Q i =0 Q N − j =0 ( z i − ζ j ) annihilates all the summands with σ such that σ (0) = N and σ (1) = N −
1. Repeating theargument finitely many times, we arrive at the fact that multiplication by Q Ni =0 Q N − i − j =0 ( z i − ζ j ) annihilatesall the summands with σ = ( N, N − , . . . , N Y i =0 δ (cid:18) z i ζ N − i (cid:19) N − i − Y j =0 ( z i − ζ j ) (cid:2) κ + ( z i ) − κ − ( z i ) (cid:3) = N Y i =0 δ (cid:18) z i ζ N − i (cid:19) N − i − Y j =0 ( z i − z N − j ) (cid:2) κ + ( z i ) − κ − ( z i ) (cid:3) . Taking the zeroth order term in ζ j for j = 0 , . . . , N , we get0 = N Y i =0 N Y j = i +1 ( z i − z j ) (cid:2) κ + ( z i ) − κ − ( z i ) (cid:3) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ κ + ( z ) − κ − ( z )] [ κ + ( z ) − κ − ( z )] . . . [ κ + ( z N ) − κ − ( z N )] z [ κ + ( z ) − κ − ( z )] z [ κ + ( z ) − κ − ( z )] . . . z N [ κ + ( z N ) − κ − ( z N )]... ... . . . ... z N − [ κ + ( z ) − κ − ( z )] z N − [ κ + ( z ) − κ − ( z )] . . . z N − N [ κ + ( z N ) − κ − ( z N )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Hence, the rows of the matrix on the r.h.s. of the above equation are linearly dependent and it follows thatthere exists a P ( z ) ∈ F [ z ] − { } of degree at most N −
1, such that P ( z ) (cid:2) κ + ( z ) − κ − ( z ) (cid:3) = 0 . (2.19)As a consequence, there clearly exists Q ( z ) ∈ F [ z ] such that deg Q = deg P and κ ± ( z ) = (cid:18) Q ( z ) P ( z ) (cid:19) | z | ∓ ≪ . Now considering (2.17) with n = 0 and multiplying it by P ( z ) obviously yields x +1 ( z ) .P ( z ) v ( z ) = 0 . (2.20) EIGHT-FINITE MODULES OVER ˙U q ( a ) AND ¨U q ( a ) 9 Set for very m ∈ Z , w m = res z z − − m P ( z ) v ( z ). Then, (2.20), together with (2.15) for n = 1, implies that M m ∈ Z ˙U q ( a ) .w m is a strict submodule of the simple U q (L a )-module M and it follows that w m = 0 for every m ∈ Z . Allthe vectors in { v m : m ∈ Z } – see (2.14) – can therefore be expressed as linear combinations of the vectorsin, say, (cid:8) v , . . . , v deg( P ) (cid:9) and the linear span of { v m : m ∈ Z } turns out to be finite dimensional. Repeatingthat argument finitely many times for the linear spans of (cid:8) v m ,...,m p : m , . . . , m p ∈ Z (cid:9) with p = 1 , . . . , N eventually concludes the proof. (cid:3) Corollary 2.12.
Let M be a weight-finite simple highest (resp. lowest) ℓ -weight U q (L a ) -module. Then M ∼ = L ( P ) (resp. M ∼ = ¯ L ( P ) ), for some monic polynomial P .Proof. In the highest ℓ -weight case, this follows directly by the previous theorem and the classification ofthe simple finite dimensional U q (L a )-modules, theorem 2.5. In the lowest ℓ -weight case, see proposition2.6. (cid:3) Double quantum affinization of type a Definition of ¨U q ( a ) .Definition 3.1. The double quantum affinization ¨U q ( a ) of type a is defined as the F -algebra generated by { D , D − , D , D − , C / , C − / , c + m , c −− m , K +1 , ,m , K − , , − m , K +1 ,n,r , K − , − n,r , X +1 ,r,s , X − ,r,s : m ∈ N , n ∈ N × , r, s ∈ Z } subject to the relations C ± / and c ± ( z ) are central (3.1)res v,w vw c ± ( v ) c ∓ ( w ) = 1 , (3.2) D ± D ∓ = 1 D ± D ∓ = 1 D D = D D (3.3) D K ± , ± m ( z ) D − = q ± m K ± , ± m ( z ) D X ± ,r ( z ) D − = q r X ± ,r ( z ) , (3.4) D K ± , ± m ( z ) D − = K ± , ± m ( zq − ) D X ± ,r ( z ) D − = X ± ,r ( zq − ) , (3.5)res v,w vw K ± , ( v ) K ∓ , ( w ) = 1 , (3.6)( v − q ± z )( v − q m − n ∓ z ) K ± , ± m ( v ) K ± , ± n ( z ) = ( vq ± − z )( vq ∓ − q m − n ) z ) K ± , ± n ( z ) K ± , ± m ( v ) , (3.7)( C q − m ) v − w )( q n − v − C w ) K +1 ,m ( v ) K − , − n ( w ) = ( C q − m v − q w )( q n v − C q − w ) K − , − n ( w ) K +1 ,m ( v ) , (3.8)( v − q ± z ) K ± , ± m ( v ) X ± ,r ( z ) = ( q ± v − z ) X ± ,r ( z ) K ± , ± m ( v ) , (3.9)( C v − q m ∓ z ) K ± , ± m ( v ) X ∓ ,r ( z ) = ( C q ∓ v − q m z ) X ∓ ,r ( z ) K ± , ± m ( v ) , (3.10)( v − q ± w ) X ± ,r ( v ) X ± ,s ( w ) = ( vq ± − w ) X ± ,s ( w ) X ± ,r ( v ) , (3.11)[ X +1 ,r ( v ) , X − ,s ( z )] = 1 q − q − δ (cid:18) C vq r + s ) z (cid:19) | s | Y p =1 c − (cid:16) C − / q (2 p − s ) − z (cid:17) − sign( s ) K +1 ,r + s ( v ) − δ (cid:18) C − vq r + s ) z (cid:19) | r | Y p =1 c + (cid:16) C − / q (1 − p )sign( r ) − v (cid:17) sign( r ) K − ,r + s ( z ) , (3.12) where m, n ∈ N , r, s ∈ Z and we have set c ± ( z ) = X m ∈ N c ±± m z ∓ m , (3.13) K ± , ( z ) = X m ∈ N K ± , , ± m z ± m , (3.14)and, for every m ∈ N × and r ∈ Z , K ± , ± m ( z ) = X s ∈ Z K ± , ± m,s z − s , (3.15) X ± ,r ( z ) = X s ∈ Z X ± ,r,s z − s . (3.16)In (3.12), we further assume that K ± , ∓ m ( z ) = 0 for every m ∈ N × . Definition 3.2.
We denote by ¨U ′ q ( a ) the subalgebra of ¨U q ( a ) generated by { D , D − , C / , C − / , c + m , c −− m , K +1 , ,m , K − , , − m , K +1 ,n,r , K − , − n,r , X +1 ,r,s , X − ,r,s : m ∈ N , n ∈ N × , r, s ∈ Z } , i.e. the subalgebra generated by all the generators of ¨U q ( a ) except D and D − . We shall denote by : ¨U ′ q ( a ) ֒ → ¨U q ( a )the natural injective algebra homomorphism. Definition 3.3.
We denote by ¨U q ( a ) the subalgebra of ¨U q ( a ) generated by n C / , C − / , c + m , c −− m , K +1 , ,m , K − , , − m , K +1 ,n,r , K − , − n,r : m ∈ N , n ∈ N × , r ∈ Z o and by ¨U , q ( a ) the subalgebra of ¨U q ( a ) generated by n C / , C − / , c + m , c −− m , K +1 , ,m , K − , , − m : m ∈ N o . Similarly, we denote by ¨U ± q ( a ) the subalgebra of ¨U q ( a ) generated by (cid:8) X ± ,r,s : r, s ∈ Z (cid:9) . We eventuallydenote by ¨U ≥ q ( a ) (resp. ¨U ≤ q ( a )) the subalgebra of ¨U q ( a ) generated by n C / , C − / , c + m , c −− m , K +1 , ,m , K − , , − m , K +1 ,n,r , K − , − n,r , X +1 ,r,s : m ∈ N , n ∈ N × , r, s ∈ Z o (resp. n C / , C − / , c + m , c −− m , K +1 , ,m , K − , , − m , K +1 ,n,r , K − , − n,r , X − ,r,s : m ∈ N , n ∈ N × , r, s ∈ Z o ) Remark . Obviously, ¨U ± q ( a ) is graded over Q ± whereas ¨U q ( a ) is graded over the root lattice Q of a .¨U q ( a ) is also graded over Z = Z (1) × Z (2) ;¨U q ( a ) = M ( n ,n ) ∈ Z ¨U q ( a ) ( n ,n ) , where, for every ( n , n ) ∈ Z , we let¨U q ( a ) ( n ,n ) = n x ∈ ¨U q ( a ) : D x D − = q n x, D x D − = q n x o . Proposition 3.5.
The set n C / , C − / , K +1 , ,m , K − , , − m : m ∈ N o generates a subalgebra of ¨U , q ( a ) that isisomorphic to ˙U q ( a ) . EIGHT-FINITE MODULES OVER ˙U q ( a ) AND ¨U q ( a ) 11 Proof.
This can be directly checked from the defining relations. Otherwise, it suffices to observe that thealgebra isomorphism b Ψ : c ˙U q ( ˙ a ) → \ ¨U ′ q ( a ) – see theorem 3.22 – restricts on that set to b Ψ( C ± / ) = C ± / and b Ψ( k ± ( z )) = − K ∓ , ( C − / z ) . (cid:3) q ( a ) as a topological algebra. Because of relation (3.12), the definition of ¨U q ( a ) is not purelyalgebraic. Indeed, the r.h.s. of (3.12) involves two infinite series. One way to make sense of that relation isto equip ¨U q ( a ) – and, for later use, its tensor powers – with a topology, such that both series be convergentin the corresponding completion \ ¨U q ( a ) of ¨U q ( a ). Making use of the natural Z (2) -grading of the tensoralgebras ¨U q ( a ) ⊗ m , m ∈ N × , we let, for every n ∈ N ,˙Ω ( m ) n := M r ≥ ns ≥ n ¨U q ( a ) ⊗ m · (cid:16) ¨U q ( a ) ⊗ m (cid:17) − r · ¨U q ( a ) ⊗ m · (cid:16) ¨U q ( a ) ⊗ m (cid:17) s · ¨U q ( a ) ⊗ m . One easily checks that
Proposition 3.6.
The following hold true for every m ∈ N × :i. For every n ∈ N , ˙Ω ( m ) n is a two-sided ideal of ¨U q ( a ) ;ii. For every n ∈ N , ˙Ω ( m ) n ⊇ ˙Ω ( m ) n +1 ;iii. ˙Ω ( m )0 := S n ∈ N ˙Ω ( m ) n = ¨U q ( a ) ;iv. T n ∈ N ˙Ω ( m ) n = { } ;v. For every n, p ∈ N , ˙Ω ( m ) n + ˙Ω ( m ) p ⊆ ˙Ω min( n,p ) ;vi. For every n, p ∈ N , ˙Ω ( m ) n · ˙Ω ( m ) p ⊆ ˙Ω max( n,p ) .Proof. See [MZ19] for a proof in the ˙U q ( ˙ a ) case that can be transposed to the present situation. (cid:3) Definition-Proposition 3.7.
We endow ¨U q ( a ) with the topology τ whose open sets are either ∅ ornonempty subsets O ⊆ ¨U q ( a ) such that for every x ∈ O , x + ˙Ω n ⊆ O for some n ∈ N . Similarly, weendow each tensor power ¨U q ( a ) ⊗ m ≥ with the topology induced by { ˙Ω ( m ) n : n ∈ N } . These turn ¨U q ( a ) intoa (separated) topological algebra. We then let \ ¨U q ( a ) denote its completion and we extend by continuityto \ ¨U q ( a ) all the (anti)-automorphisms defined over ¨U q ( a ) and its subalgebras in the previous section Inparticular, we extend : ¨U ′ q ( a ) ֒ → ¨U q ( a ) into b : \ ¨U ′ q ( a ) ֒ → \ ¨U q ( a ) . Similarly, we denote with a hat the completion of any subalgebra of \ ¨U q ( a ), like e.g. \ ¨U − q ( a ), \ ¨U q ( a ) and \ ¨U + q ( a ). We eventually denote by ¨U q ( a ) b ⊗ m ≥ the corresponding completions of ¨U q ( a ) ⊗ m ≥ . Proof.
This was proven in [MZ19]. (cid:3)
Remark . As was noted in [MZ19], the above defined topology is actually ultrametrizable.3.3.
The double quantum loop algebra.
An alternative way to make sense of relations (3.12) consistsin observing that ¨U q ( a ) is proalgebraic . Indeed, for every N ∈ N , let ¨U q ( a ) ( N ) be the F -algebra generatedby { C / , C − / , c + n , c −− n , K +1 , ,m , K − , , − m , K +1 ,p,r , K − , − p,r , X +1 ,r,s , X − ,r,s : m ∈ N , n ∈ J , N K , p ∈ N × , r, s ∈ Z } subject to relations ((3.1) – (3.12)), where, this time, c ± ( z ) = N X m =0 c ±± m z ∓ m . (3.17)Now clearly, each ¨U q ( a ) ( N ) is algebraic since the sums on the r.h.s. of (3.12) are both finite – whenever c ± ( z ) − is involved, just multiply through by c ± ( z ) to get an equivalent algebraic relation. Moreover, letting I N be the two-sided ideal of ¨U q ( a ) ( N ) generated by { c + N , c −− N } (resp. { c +0 − , c − − } ) for every N > N = 0), we obviously have a surjective algebra homomorphism¨U q ( a ) ( N ) −→ ¨U q ( a ) ( N − ∼ = ¨U q ( a ) ( N ) I N (3.18)and we can define ¨U q ( a ) as the inverse limit¨U q ( a ) = lim ←− ¨U q ( a ) ( N ) of the system of algebras · · · −→ ¨U q ( a ) ( N ) −→ ¨U q ( a ) ( N − −→ · · · −→ ¨U q ( a ) (0) −→ ¨U q ( a ) ( − . Definition 3.9.
We shall refer to the quotient of ¨U q ( a ) ( − by the two-sided ideal generated by n C / − o as the double quantum loop algebra of type a and denote it by ¨L q ( a ). Correspondingly, we denote by¨L ± q ( a ) and ¨L q ( a ), the subalgebras of ¨L q ( a ) respectively generated by (cid:8) X ± ,r,s : r, s ∈ Z (cid:9) and (cid:8) K +1 , ,m , K − , , − m , K +1 ,n,r , K − , − n,r : m ∈ N , n ∈ N × , r ∈ Z (cid:9) . We denote by ¨L , q ( a ) the subalgebra of ¨L q ( a ) generated by (cid:8) K +1 , ,m , K − , , − m : m ∈ N (cid:9) . It is worth emphasizing that ¨L , q ( a ) is abelian.3.4. Triangular decomposition of \ ¨U ′ q ( a ) . In [MZ19], we proved that \ ¨U ′ q ( a ) has a triangular decompo-sition in the following sense. Definition 3.10.
Let A be a complete topological algebra with closed subalgebras A ± and A . We shall saythat ( A − , A , A + ) is a triangular decomposition of A if the multiplication induces a bicontinuous isomorphismof vector spaces A − b ⊗ A b ⊗ A + ∼ → A .Recalling the definitions of ¨U ± q ( a ) and ¨U q ( a ) from definition 3.1, we have Proposition 3.11. ( ¨U − q ( a ) , ¨U q ( a ) , ¨U + q ( a )) is a triangular decomposition of \ ¨U ′ q ( a ) and ¨U ± q ( a ) is bicon-tinuously isomorphic to the algebra generated by { X ± ,r,s : r, s ∈ Z } subject to relation (3.11).Proof. See [MZ19]. (cid:3)
The closed subalgebra \ ¨U q ( a ) as a topological Hopf algebra.Definition 3.12. In \ ¨U q ( a ), we define p ± ( z ) = X m ∈ N p ±± m z ∓ m = c ± ( z ) K ∓ , ( C − / z ) − K ∓ , ( C − / zq ) (3.19) EIGHT-FINITE MODULES OVER ˙U q ( a ) AND ¨U q ( a ) 13 and for every m ∈ N × , t +1 ,m ( z ) = X n ∈ N t +1 ,m,n z − n = − q − q − K +1 , ( zq − m ) − K +1 ,m ( z ) , (3.20) t − , − m ( z ) = X n ∈ N t − , − m,n z n = 1 q − q − K − , − m ( z ) K − , ( zq − m ) − . (3.21)Then, we let ¨U + q ( a ) be the subalgebra of \ ¨U q ( a ) generated by { C / , C − / , p + m , p −− m , t +1 ,p,n , t − , − p,n : m ∈ N , n ∈ Z , p ∈ N × } . and we let \ ¨U + q ( a ) be its completion in the inherited topology.Clearly, the closed subalgebra \ ¨U q ( a ) can be presented as in definition 3.3 or, equivalently, in terms of thegenerators in { C / , C − / , c + m , c −− m , p + m , p −− m , t +1 ,p,n , t − , − p,n : m ∈ N , n ∈ Z , p ∈ N × } . In section 3.10, we will endow \ ¨U ′ q ( a ) with a topological Hopf algebraic structure. It turns out that, for thatstructure, the closed subalgebra \ ¨U q ( a ) is not a closed Hopf subalgebra of \ ¨U q ( a ). However, it is possible toendow \ ¨U q ( a ) with its own topological Hopf algebraic structure as follows. Definition-Proposition 3.13.
We endow \ ¨U q ( a ) with:i. the comultiplication ∆ : \ ¨U q ( a ) → ¨U q ( a ) b ⊗ ¨U q ( a ) defined by∆ ( C ± / ) = C ± / ⊗ C ± / (3.22)∆ ( c ± ( z )) = c ± ( z C ± / ) ⊗ c ± ( z C ∓ / ) , (3.23)∆ ( p ± ( z )) = p ± ( z C ± / ) ⊗ p ± ( z C ∓ / ) , (3.24)∆ ( t +1 ,m ( z )) = t +1 ,m ( z ) ⊗ m Y k =1 p − ( zq − k C / ) b ⊗ t +1 ,m ( z C (1) ) − ( q − q − ) m − X k =1 m Y l = k +1 p − ( zq − l C / ) t +1 ,k ( z ) b ⊗ t +1 ,m − k ( zq − k C (1) ) , (3.25)∆ ( t − , − m ( z )) = t − , − m ( z C (2) ) b ⊗ m Y k =1 p + ( zq − k C / ) + 1 ⊗ t − , − m ( z )+( q − q − ) m − X k =1 t − , − ( m − k ) ( zq − k C (2) ) b ⊗ t − , − m ( z ) m Y l =+1 p + ( zq − l C / ) , (3.26)for every m ∈ N , where C ± / = C ± / ⊗ C ± / = 1 ⊗ C ± / ,ii. the counit ε ( C ) = ε ( c ± ( z )) = ε ( p ± ( z )) = 1, ε ( t ± , ± m ( z )) = 0, for every m ∈ N ,iii. and the antipode defined by S ( C ± / ) = C ∓ / , (3.27) S ( c ± ( z )) = c ± ( z ) − , (3.28) S ( p ± ( z )) = p ± ( z ) − , (3.29) S ( t +1 ,m ( z )) = − m Y k =1 p − ( zq − k C − / ) − m X n =1 X λ ∈ C n ( m ) ( − n − c m,λ t +1 ,λ ( z C − ) , (3.30) S ( t − , − m ( z )) = − m X n =1 X λ ∈ C n ( m ) c m,λ t − , − λ ( z C − ) m Y k =1 p + ( zq − k ) − , (3.31)where we have set, for every m ∈ N × and every λ ∈ C n ( m ), c m,λ = ( q − q − ) n − [ m + 1] q [ m − q n Y i =1 [ λ i − q [ λ i + 1] q and t +1 ,λ ( z C − ) = ←−− Y i ∈ J n K t +1 ,λ i ( zq − P nk = i +1 λ k C − ) , t − , − λ ( z C − ) = −−→ Y i ∈ J n K t − , − λ i ( zq − P nk = i +1 λ k C − ) . for every m ∈ N .With these operations, \ ¨U q ( a ) is a topological Hopf algebra. Proof.
One easily checks that ∆ as defined by (3.22 – 3.26) is compatible with the defining relations of \ ¨U q ( a ) and that S is compatible with both the multiplication and the comultiplication. (cid:3) In that presentation, one readily checks that
Proposition 3.14. \ ¨U + q ( a ) is a closed Hopf subalgebra of \ ¨U q ( a ) .Proof. \ ¨U + q ( a ) is a closed subalgebra of \ ¨U q ( a ) and it is clearly stable under ∆ and S . (cid:3) The closed subalgebra \ ¨U q ( a ) and the elliptic Hall algebra. As emphasized in [MZ19], anotherremarkable feature of \ ¨U ′ q ( a ) and, more particularly of its closed subalgebra \ ¨U q ( a ), is the existence of analgebra homomorphism onto it, from the elliptic Hall algebra that we now define. Definition 3.15.
Let q , q , q be three (dependent) formal variables such that q q q = 1. The elliptic Hallalgebra E q ,q ,q is the Q ( q , q , q )-algebra generated by (cid:8) C / , C − / , ψ + m , ψ −− m , e + n , e − n : m ∈ N , n ∈ Z (cid:9) , with ψ ± invertible, subject to the relations C ± / is central , (3.32) ψ ± ( z ) ψ ± ( w ) = ψ ± ( w ) ψ ± ( z ) , (3.33) g ( Cz, w ) g ( Cw, z ) ψ + ( z ) ψ − ( w ) = g ( z, Cw ) g ( w, Cz ) ψ − ( w ) ψ + ( z ) , (3.34) g ( C ± z, w ) ψ ± ( z ) e + ( w ) = − g ( w, C ± z ) e + ( w ) ψ ± ( z ) , (3.35) g ( w, C ∓ z ) ψ ± ( z ) e − ( w ) = − g ( C ∓ z, w ) e − ( w ) ψ ± ( z ) , (3.36)[ e + ( z ) , e − ( w )] = 1 g (1 , (cid:20) δ (cid:18) Cwz (cid:19) ψ + ( w ) − δ (cid:16) wCz (cid:17) ψ − ( z ) (cid:21) , (3.37) g ( z, w ) e + ( z ) e + ( w ) = − g ( w, z ) e + ( w ) e + ( z ) , (3.38) g ( w, z ) e − ( z ) e − ( w ) = − g ( z, w ) e − ( w ) e − ( z ) , (3.39)res v,w,z ( vwz ) m ( v + z )( w − vz ) e ± ( v ) e ± ( w ) e ± ( z ) = 0 , (3.40)where m ∈ Z and we have introduced g ( z, w ) = ( z − q w )( z − q w )( z − q w ) , (3.41) EIGHT-FINITE MODULES OVER ˙U q ( a ) AND ¨U q ( a ) 15 ψ ± ( z ) = X m ∈ N ψ ±± m z ∓ m , (3.42) e ± ( z ) = X m ∈ Z e ± m z − m . (3.43) Remark . The elliptic Hall algebra E q ,q ,q is Z -graded and can be equipped with a natural topologyalong the lines of what we did for ¨U q ( a ) in section 3.2. It then becomes a topological algebra and we denoteby \ E q ,q ,q its completion. Similar topologies can be constructed on its tensor powers. Definition-Proposition 3.17.
We endow \ E q ,q ,q with:i. the comultiplication ∆ E : \ E q ,q ,q → E q ,q ,q b ⊗E q ,q ,q defined by∆ E ( ψ ± ( z )) = ψ ± ( zC ± (2) ) ⊗ ψ ± ( zC ∓ (1) ) , (3.44)∆ E ( e + ( z )) = e + ( z ) ⊗ ψ − ( z ) b ⊗ e + ( zC (1) ) , (3.45)∆ E ( e − ( z )) = e − ( zC (2) ) b ⊗ ψ + ( z ) + 1 ⊗ e − ( z ) , (3.46)ii. the counit ε E : \ E q ,q ,q → F defined by ε E ( C ± / ) = ε E ( ψ ± ( z )) = 1, ε E ( e ± ( z )) = 0,iii. the antipode S E : \ E q ,q ,q → \ E q ,q ,q defined by S E ( ψ ± ( z )) = ψ ± ( zC − ) − , (3.47) S E ( e + ( z )) = − ψ − ( zC − ) − e + ( zC − ) , (3.48) S E ( e − ( z )) = − e − ( zC − ) ψ + ( zC − ) − . (3.49)As usual, we have set C ± / = C ± / ⊗ C ± / = 1 ⊗ C ± / . With the above defined operations, \ E q ,q ,q is a topological Hopf algebra. Proposition 3.18.
There exists a unique continuous Hopf algebra homomorphism f : \ E q − ,q ,q → \ ¨U + q ( a ) such that f ( C / ) = C / , (3.50) f ( ψ ± ( z )) = p ± ( C / zq − ) , (3.51) f ( e + ( z )) = t +1 , , ( z ) (3.52) f ( e − ( z )) = t − , − ( z )( q − q − ) . (3.53) Proof.
In [MZ19], we proved that the assignment C / C / ψ ± ( z ) ( q − q − ) p ± ( C / zq − ) , e ± ( z ) t ± , ± ( z )defined an F -algebra homomorphism. Hence, f , which is obtained from the above assignment by rescalingthe images of p ± ( z ) and e − ( z ), is obviously an F -algebra homomorphism. Moreover, it suffices to write(3.24), (3.25) and (3.26) with m = 1, to get∆ ( p ± ( z )) = p ± ( z C ± / ) ⊗ p ± ( z C ∓ / ) , ∆ ( t +1 , ( z )) = t +1 , ( z ) ⊗ p − ( zq − C / ) b ⊗ t +1 , ( z C (1) ) , ∆ ( t − , − ( z )) = t − , − ( z C (2) ) b ⊗ p + ( zq − C / ) + 1 ⊗ t − , − ( z ) , as well as (3.29), (3.30) and (3.31), with m = 1, to get S ( p ± ( z )) = p ± ( z ) − , S ( t +1 , ( z )) = − p − ( zq − C − / ) − t +1 , ( z C − ) ,S ( t − , − ( z )) = − t − , − ( z C − ) p + ( zq − ) − , and thus to prove that ( f b ⊗ f ) ◦ ∆ = ∆ E ◦ f and f ◦ S = S E ◦ f as claimed. (cid:3) Remark . Note that we have f ( ψ +0 ) f ( ψ − ) = f ( ψ − ) f ( ψ +0 ) = 1, meaning that f descends to the quotientof E q − ,q ,q by the two-sided ideal generated by { ψ +0 ψ − − , ψ − ψ +0 − } . That quotient is actually Miki’s( q, γ )-analogue of the W ∞ algebra [Mik07].3.7. The quantum toroidal algebra ˙U q ( ˙ a ) . Let ˙ I = { , } be a labeling of the nodes of the Dynkindiagram of type ˙ a and let ˙Φ = { α , α } be a choice of simple roots for the corresponding root system. Let˙ Q ± = Z ± α ⊕ Z ± α and let ˙ Q = Z α ⊕ Z α be the type ˙ a root lattice. Definition 3.20.
The quantum toroidal algebra ˙U q ( ˙ a ) is the associative F -algebra generated by the gener-ators n D, D − , C / , C − / , k + i,n , k − i, − n , x + i,m , x − i,m : i ∈ ˙ I, m ∈ Z, n ∈ N o subject to the following relations C ± / is central C ± / C ∓ / = 1 D ± D ∓ = 1 (3.54) D k ± i ( z ) D − = k ± i ( zq − ) D x ± i ( z ) D − = x ± i ( zq − ) (3.55) k ± i ( z ) k ± j ( z ) = k ± j ( z ) k ± i ( z ) (3.56) k − i ( z ) k + j ( z ) = G − ij ( C − z /z ) G + ij ( Cz /z ) k + j ( z ) k − i ( z ) = 1 mod z /z (3.57) G ∓ ij ( C ∓ / z /z ) k + i ( z ) x ± j ( z ) = x ± j ( z ) k + i ( z ) (3.58) k − i ( z ) x ± j ( z ) = G ∓ ij ( C ∓ / z /z ) x ± j ( z ) k − i ( z ) (3.59)( z − q ± c ij z ) x ± i ( z ) x ± j ( z ) = ( z q ± c ij − z ) x ± j ( z ) x ± i ( z ) (3.60)[ x + i ( z ) , x − j ( z )] = δ ij q − q − (cid:20) δ (cid:18) z Cz (cid:19) k + i ( z C − / ) − δ (cid:18) z Cz (cid:19) k − i ( z C − / ) (cid:21) (3.61) X σ ∈ S − cij − c ij X k =0 ( − k (cid:18) − c ij k (cid:19) q x ± i ( z σ (1) ) · · · x ± i ( z σ ( k ) ) x ± j ( z ) x ± i ( z σ ( k +1) ) · · · x ± i ( z σ (1 − c ij ) ) = 0 (3.62)where, for every i ∈ ˙ I , we define the following ˙U q ( ˙ a )-valued formal distributions x ± i ( z ) := X m ∈ Z x ± i,m z − m ∈ ˙U q ( ˙ a )[[ z, z − ]] ; (3.63) k ± i ( z ) := X n ∈ N k ± i, ± n z ∓ n ∈ ˙U q ( ˙ a )[[ z ∓ ]] , (3.64)for every i, j ∈ ˙ I , we define the following F -valued formal power series G ± ij ( z ) := q ± c ij + ( q − q − )[ ± c ij ] q X m ∈ N × q ± mc ij z m ∈ F [[ z ]] (3.65)is an F -valued formal distribution,Note that G ± ij ( z ) is invertible in F [[ z ]] with inverse G ∓ ij ( z ), i.e. G ± ij ( z ) G ∓ ij ( z ) = 1 , (3.66) EIGHT-FINITE MODULES OVER ˙U q ( a ) AND ¨U q ( a ) 17 and that it can be viewed as the power series expansion of a rational function of ( z , z ) ∈ C as | z | ≫ | z | ,which we shall denote as follows G ± ij ( z /z ) = (cid:18) z q ∓ c ij − z z − q ∓ c ij z (cid:19) | z |≫| z | . (3.67)Observe furthermore that we have the following useful identity in F [[ z, z − ]] G ± ij ( z /z ) − G ∓ ij ( z /z ) q − q − = [ ± c ij ] q δ (cid:18) z q ± c ij z (cid:19) . (3.68) Remark . In type a , ˙ I = { , } , c ij = 4 δ ij − G ± ( z ) = G ∓ ( z ).˙U q ( ˙ a ) is obviously a Z -graded algebra, i.e. we have˙U q ( ˙ a ) = M n ∈ Z ˙U q ( ˙ a ) n , where for all n ∈ Z ˙U q ( ˙ a ) n := { x ∈ ˙U q ( ˙ a ) : DxD − = q n x } . (3.69)It was proven in [Her05] to admit a triangular decomposition ( ˙U − q ( ˙ a ) , ˙U q ( ˙ a ) , ˙U + q ( ˙ a )), where ˙U ± q ( ˙ a ) and˙U q ( ˙ a ) are the subalgebras of ˙U q ( ˙ a ) respectively generated by n x ± i,m : i ∈ ˙ I, m ∈ Z o and n C / , C − / , D, D − , k + i,m , k − i,m : i ∈ ˙ I, m ∈ Z o . Observe that ˙U ± q ( ˙ a ) admits a natural gradation over ˙ Q ± that we shall denote by˙U ± q ( ˙ a ) = M α ∈ ˙ Q ± ˙U ± q ( ˙ a ) α . (3.70)Of course ˙U q ( ˙ a ) is graded over the root lattice ˙ Q . We finally remark that the two Dynkin diagram subal-gebras ˙U q ( a ) (0) and ˙U q ( a ) (1) of ˙U q ( ˙ a ) generated by n D, D − , C / , C − / , k + i,n , k − i, − n , x + i,m , x − i,m : m ∈ Z, n ∈ N o , with i = 0 and i = 1 respectively, are both isomorphic to ˙U q ( a ), thus yielding two injective algebrahomomorphisms ι ( i ) : ˙U q ( a ) ֒ → ˙U q ( ˙ a ). In [MZ19], making use of their natural Z -grading, ˙U q ( ˙ a ) and allits tensor powers were endowed with a topology along the lines of what we did in section 3.2 for ¨U q ( a ) andits tensor powers, and subsequently completed into \ ˙U q ( ˙ a ) and ˙U q ( ˙ a ) b ⊗ r . The main result in [MZ19] is thefollowing Theorem 3.22.
There exists a unique bicontinuous F -algebra isomorphism b Ψ : \ ˙U q ( ˙ a ) ∼ −→ \ ¨U ′ q ( a ) such that b Ψ( D ± ) = D ± b Ψ( C ± / ) = C ± / , b Ψ( k ± ( z )) = − c ± ( z ) K ∓ , ( C − / z ) − b Ψ( k ± ( z )) = − K ∓ , ( C − / z ) b Ψ( x +0 ( z )) = − c − ( C / z ) K +1 , ( z ) − X − , ( C z ) b Ψ( x − ( z )) = − X +1 , − ( C z ) c + ( C / z ) K − , ( z ) − b Ψ( x ± ( z )) = X ± , ( z ) . Proof.
See [MZ19] for a proof. (cid:3) q ( a ) subalgebras of ¨U q ( a ) . Interestingly, ¨U q ( a ) admits countably many embeddings of the quan-tum affine algebra ˙U q ( a ). This is the content of the following Proposition 3.23.
For every m ∈ Z , there exists a unique injective algebra homomorphism ι m : ˙U q ( a ) ֒ → \ ¨U ′ q ( a ) such that ι m ( C ± / ) = C ± / ι m ( D ± ) = D ± (3.71) ι m ( k ± ( z )) = − | m | Y p =1 c ± (cid:16) q (1 − p )sign( m ) − z (cid:17) sign( m ) K ∓ , ( C − / z ) , (3.72) ι m ( x ± ( z )) = X ± , ± m ( z ) . (3.73) Proof.
See [MZ19]. (cid:3)
We also have
Proposition 3.24.
For every i ∈ ˙ I = { , } , b Ψ ◦ ι ( i ) is an injective algebra homomorphism.Proof. This is obvious since b Ψ is an isomorphism and ι ( i ) is an injective algebra homomorphism. (cid:3) (Anti-)Automorphisms of \ ¨U ′ q ( a ) . \ ¨U ′ q ( a ) naturally inherits, through b Ψ, all the continuous (anti-)automorphisms defined over \ ˙U q ( ˙ a ). Proposition 3.25.
Conjugation by b Ψ clearly provides a group isomorphism Aut( \ ˙U q ( ˙ a )) ∼ = Aut( \ ¨U ′ q ( a )) .In particular, for every f ∈ Aut( \ ˙U q ( ˙ a )) , we let ˙ f = b Ψ ◦ f ◦ b Ψ − ∈ Aut( \ ¨U ′ q ( a )) . As an example, consider the Cartan anti-involution ϕ of ˙U q ( ˙ a ) defined in [MZ19]. It extends by continuityinto an anti-involution b ϕ over \ ˙U q ( ˙ a ) which eventually yields, upon conjugation by b Ψ, an anti-involution ˙ ϕ over \ ¨U ′ q ( a ). One can easily check – or take as a definition of ˙ ϕ the fact – that,˙ ϕ ( q ) = q − , ˙ ϕ ( D ± ) = D ∓ , ˙ ϕ ( C ± / ) = C ∓ / , ˙ ϕ ( c ± ( z )) = c ∓ (1 /z ) , ˙ ϕ ( K ± , ± m ( z )) = K ∓ , ∓ m (1 /z ) , ˙ ϕ ( X ± ,r ( z )) = X ∓ , − r (1 /z ) . for every m ∈ N and every r ∈ Z .In addition to the above, \ ¨U ′ q ( a ) also admits the following automorphisms that will prove useful in thestudy of its representation theory. Proposition 3.26. i. There exists a unique F -algebra automorphism τ of \ ¨U ′ q ( a ) such that, for every m ∈ N and every n ∈ Z , τ ( C ) = − C , τ ( c ± ( C − / z )) = c ± ( ∓ C − / z ) , τ ( K ± , ± m ( z )) = K ± , ± m ( ∓ z ) , τ ( X ± ,n ( z )) = X ± ,n ( ∓ z ) . ii. There exists a unique F -algebra automorphism σ of \ ¨U ′ q ( a ) such that σ ( C / ) = − C / , σ ( c ± ( z )) = c ± ( z ) , τ ( K ± , ± m ( z )) = K ± , ± m ( − z ) , τ ( X ± ,n ( z )) = X ± ,n ( − z ) . Proof.
It suffices to check the defining relations of \ ¨U q ( a ). (cid:3) Topological Hopf algebra structure on \ ¨U ′ q ( a ) .Definition 3.27. We endow the topological F -algebra \ ˙U q ( ˙ a ) with:i. the comultiplication ∆ : \ ˙U q ( ˙ a ) → ˙U q ( ˙ a ) b ⊗ ˙U q ( ˙ a ) defined by∆( C ± / ) = C ± / ⊗ C ± / , ∆( D ± ) = D ± ⊗ D ± , (3.74)∆( k ± i ( z )) = k ± i ( zC ± / ) ⊗ k ± i ( zC ∓ / ) , (3.75) EIGHT-FINITE MODULES OVER ˙U q ( a ) AND ¨U q ( a ) 19 ∆( x + i ( z )) = x + i ( z ) ⊗ k − i ( zC / ) b ⊗ x + i ( zC (1) ) , (3.76)∆( x − i ( z )) = x − i ( zC (2) ) b ⊗ k + i ( zC / ) + 1 ⊗ x − i ( z ) , (3.77)where C ± / = C ± / ⊗ C ± / = 1 ⊗ C ± / ;ii. the counit ε : \ ˙U q ( ˙ a ) → F , defined by ε ( D ± ) = ε ( C ± / ) = ε ( k ± i ( z )) = 1, ε ( x ± i ( z )) = 0 and;iii. the antipode S : \ ˙U q ( ˙ a ) → \ ˙U q ( ˙ a ), defined by S ( D ± ) = D ∓ , S ( C ± / ) = C ∓ / and S ( k ± i ( z )) = k ± i ( z ) − , S ( x + i ( z )) = − k − i ( zC − / ) − x + i ( zC − ) , S ( x − i ( z )) = − x − i ( zC − ) k + i ( zC − / ) − . With these operations so defined and the topologies defined in section 3.7, \ ˙U q ( ˙ a ) is a topological Hopfalgebra.In view of theorem 3.22, it is clear that \ ¨U q ( a ) inherits that topological Hopf algebraic structure. Definition-Proposition 3.28.
We define˙∆ = (cid:16) b Ψ b ⊗ b Ψ (cid:17) ◦ ∆ ◦ b Ψ − , (3.78)˙ S = b Ψ ◦ S ◦ b Ψ − , (3.79)˙ ε = ε ◦ b Ψ − . (3.80)Equipped with the above comultiplication, antipode and counit, \ ¨U ′ q ( a ) is a topological Hopf algebra.Before we move on to introducing t -weight ¨U ′ q ( a )-modules, we give the following Lemma 3.29.
For every m ∈ N × and every r ∈ Z , we havei. ˙∆( K ± , ± m ( z )) = ∆ ( K ± , ± m ( z )) mod ¨U q ( a )[[ z, z − ]] ;ii. ˙∆( X +1 ,r ( z )) ∈ (cid:16) ¨U >q ( a ) b ⊗ ¨U q ( a ) ⊕ ¨U q ( a ) b ⊗ ¨U >q ( a ) (cid:17) [[ z, z − ]] ;where we have set ¨U >q ( a ) = ¨U ≥ q ( a ) − ¨U ≥ q ( a ) ∩ ¨U q ( a ) and ¨U
q ( a )[[ z, z − ]] and it follows that i. holds for m = 1 and for upper choices of signs.Suppose it holds for upper choices of signs and for some m ∈ N × . Then, making use of (3.82), one easilychecks that i. holds for m + 1 and for upper choices of signs, which completes the proof of i. for upper choicesof signs. Now, i. for lower choices of signs follows after applying ˙ ϕ and observing that, indeed,˙∆ ◦ ˙ ϕ = (cid:0) ˙ ϕ b ⊗ ˙ ϕ (cid:1) ◦ ˙∆ cop , and ∆ ◦ ˙ ϕ | ¨U q ( a ) = (cid:0) ˙ ϕ b ⊗ ˙ ϕ (cid:1) | ¨U q ( a ) ◦ ∆ , cop . As for ii., we let, for every r ∈ Z , X +1 ,r ( z ) = b Ψ − ( X +1 ,r ( z )) . In [MZ19] – see definition 4.1 and proposition 4.8 –, we proved that X +1 ,r ( z ) could be defined recursively bysetting X +1 , ( z ) = x +1 ( z ) and letting, for every r ∈ N , (cid:2) ψ +1 , ( z ) , X +1 ,r ( v ) (cid:3) G − ( z/vq ) G − ( z/v ) = [2] q δ (cid:18) zvq (cid:19) X +1 ,r +1 ( z ) (3.83)and (cid:2) ψ − , − ( z ) , X +1 , − r ( v ) (cid:3) = [2] q δ (cid:18) Czv (cid:19) X +1 , − ( r +1) ( Cq − z ) ℘ + ( C / q − z ) , (3.84)where ψ − , − ( z ) = ϕ ( ψ +1 , (1 /z )) – see proposition 4.3 in [MZ19]. Observing that (cid:0) ϕ b ⊗ ϕ (cid:1) ◦ ∆ cop = ∆ ◦ ϕ , weclearly get (cid:16) b Ψ b ⊗ b Ψ (cid:17) ◦ ∆( ψ − , − ( z )) = ∆ ( t − , − ( z )) mod ¨U
q ( a )[[ z, z − ]] . Now, applying b Ψ b ⊗ b Ψ to (3.76) in definition 3.27 clearly proves ii. in the case r = 0. Assuming it holds for r ∈ N , it suffices to apply (cid:16) b Ψ b ⊗ b Ψ (cid:17) ◦ ∆ to (3.83) above to prove that it also holds for r + 1. Similarly, if ii.holds for some r ∈ − N , applying (cid:16) b Ψ b ⊗ b Ψ (cid:17) ◦ ∆ to (3.84) to prove that it also holds for r −
1. This concludesthe proof. (cid:3) t -weight ¨U q ( a ) -modules ℓ -weight modules over ¨U q ( a ) . Remember that ¨U , q ( a ) contains a subalgebra that is isomorphic to˙U q ( a ) – see proposition 3.5. Hence, in view of remark 2.9, we can repeat for modules over ¨U q ( a ) what wedid in section 2.3 for modules over ˙U q ( a ). We thus make the following Definition 4.1.
We shall say that a (topological) ¨U q ( a )-module M is ℓ -weight if there exists a countableset { M α : α ∈ A } of indecomposable locally finite-dimensional ¨U , q ( a )-modules called ℓ -weight spaces of M ,such that, as ¨U , q ( a )-modules, M ∼ = M α ∈ A M α . As in section 2.3, it follows that
Definition-Proposition 4.2.
Let M be an ℓ -weight ¨U q ( a )-module. Then:i. C acts on M by id; EIGHT-FINITE MODULES OVER ˙U q ( a ) AND ¨U q ( a ) 21 ii. for every ℓ -weight space M α , α ∈ A , of M , there exist κ α, ∈ F × and sequences ( κ ± α, ± m ) m ∈ N × ∈ F N × such that M α ⊆ (cid:8) v ∈ M : ∃ n ∈ N × , ∀ m ∈ N (cid:0) K ± , , ± m − κ ± α, ± m id (cid:1) n .v = 0 (cid:9) , (4.1)where we have set κ ± α, = κ ± α, .We let Sp( M ) = { κ α, : α ∈ A } and we shall refer to κ ± α ( z ) = X m ∈ N κ ± α, ± m z ± m as the ℓ -weight of the ℓ -weight space M α . We shall say that M is • of type C / acts by id over M ; • of type (1 , N ) for N ∈ N × if it is of type 1 and, for every m ≥ N , c ±± m acts by multiplication 0 over M ; • of type (1 ,
0) if it is of type (1 ,
1) and c ± acts by id over M . Proof.
The proof follows the same arguments as the proof of definition-proposition 2.8. (cid:3)
Proposition 4.3.
Let M be a type ℓ -weight ¨U q ( a ) -module and let M α and M β be two ℓ -weight spacesof M such that, for some m ∈ N × and some n ∈ Z , M α ∩ K ± , ± m,n .M β = { } . Then, there exists a unique a ∈ F × such that:i. the respective ℓ -weights κ εα ( z ) and κ εβ ( z ) of M α and M β be related by κ εα ( z ) = κ εβ ( z ) H εm,a ( z ) ± , where ε ∈ {− , + } and H ± m,a ( z ) = (cid:18) (1 − q − a/z )(1 − q − m − a/z )(1 − q a/z )(1 − q − m +1) a/z ) (cid:19) | z | ± ≪ ; (4.2) ii. ( z − a ) N M α ∩ K ± , ± m ( z ) .M β = { } for some N ∈ N × .Proof. There clearly exist two bases { v i : i = 1 , . . . , dim M α } and { w i : i = 1 , . . . , dim M β } of M α and M β respectively, in which ∀ i ∈ J dim M α K , K ± , ( z ) .v i = κ ± α ( z ) dim M α X k = i η ± α,i,k ( z ) v k , ∀ j ∈ J dim M β K , K ± , ( z ) .w j = κ ± β ( z ) dim M β X l = j η ± β,j,l ( z ) w l , for some η ± α,i,k ( z ) , η ± β,j,l ( z ) ∈ F [[ z ± ]], with i, k ∈ J dim M α K and j, l ∈ J dim M β K , such that η ± α,i,i ( z ) = 1 forevery i ∈ J dim M α K and η ± β,j,j ( z ) = 1 for every j ∈ J dim M β K .Now, if M α ∩ K ± , ± m,n .M β = { } , there must exist a largest nonempty subset J ⊆ J dim M β K such that, forevery j ∈ J , M α ∩ K ± , ± m ( z ) .w j = { } . Let j ∗ = max J . Obviously, for every j ∈ J , there must exist a largestnonempty subset I ( j ) ⊆ J dim M α K such that, for every j ∈ J and every i ∈ I ( j ), F v i ∩ K ± , ± m ( z ) .w j = { } .Let i ∗ ( j ) = min I ( j ) and let for simplicity i ∗ = i ∗ ( j ∗ ). Then, for every j ∈ J , M α ∩ K ± , ± m ( z ) .w j = X i ∈ I ( j ) ξ ± m,j,i ( z ) v i , for some ξ ± m,j,i ( z ) ∈ F [[ z, z − ]] − { } . When needed, we shall extend by zero the definition of ξ ± m,j,i ( z ) outsideof the set of pairs { ( j, i ) : j ∈ J, i ∈ I ( j ) } . Making use of the relations in ¨U q ( a ) – namely (3.7) and (3.8) –,we get, for ε ∈ {− , + } ,( z − q ± z )( z − q m ∓ z ) K ± , ± m ( z ) K ε , ( z ) .w j = ( z q ± − z )( z q ∓ − q m z ) K ε , ( z ) K ± , ± m ( z ) .w j . The latter easily implies that, for every j ∈ J and every i ∈ I ( j ),( z − q ± z )( z − q m ∓ z ) κ εβ ( z ) X l ∈ Jl ≥ j η εβ,j,l ( z ) ξ ± m,l,i ( z )= ( z q ± − z )( z q ∓ − q m z ) κ ± α ( z ) X k ∈ I ( j ) k ≤ i η εα,k,i ( z ) ξ ± m,j,k ( z ) . (4.3)Taking i = i ∗ and j = j ∗ in the above equation immediately yields h ( z − q ± z )( z − q m ∓ z ) κ εβ ( z ) − ( z q ± − z )( z q ∓ − q m z ) κ εα ( z ) i ξ ± m,j ∗ ,i ∗ ( z ) = 0 . The latter is equivalent to the fact that, for every p ∈ Z , (cid:0) ξ ± m,j ∗ ,i ∗ ,p q m z + ξ ± m,j ∗ ,i ∗ ,p +2 (cid:1) (cid:2) κ εβ ( z ) − κ εα ( z ) (cid:3) = ξ ± m,j ∗ ,i ∗ ,p +1 z h ( q m ∓ + q ± ) κ εβ ( z ) − ( q m ± + q ∓ ) κ εα ( z ) i . (4.4)where, as usual, we have set ξ ± m,j ∗ ,i ∗ ,p = res z z p − ξ ± m,j ∗ ,i ∗ ( z ) . Since ξ ± m,j ∗ ,i ∗ ( z ) = 0, there must exist a p ∈ Z such that ξ ± m,j ∗ ,i ∗ ,p = 0. Assuming that ξ ± m,j ∗ ,i ∗ ,p +1 = 0, oneeasily obtains that, on one hand κ εβ ( z ) = κ εα ( z ) and that, on the other hand, h ( q m ∓ + q ± ) κ εβ ( z ) − ( q m ± + q ∓ ) κ εα ( z ) i = 0 . A contradiction. By similar arguments, one eventually proves that ξ ± m,j ∗ ,i ∗ ,p = 0 for every p ∈ Z . But thendividing (4.4) by ξ ± m,j ∗ ,i ∗ ,p we get (cid:0) q m z + a (cid:1) (cid:2) κ εβ ( z ) − κ εα ( z ) (cid:3) − az h ( q m ∓ + q ± ) κ εβ ( z ) − ( q m ± + q ∓ ) κ εα ( z ) i = 0 . where we have set, for every p ∈ Z , ξ ± m,j ∗ ,i ∗ ,p +1 /ξ ± m,j ∗ ,i ∗ ,p = a ∈ F × and, consequently, ξ ± m,j ∗ ,i ∗ ,p +2 /ξ ± m,j ∗ ,i ∗ ,p = a . i. follows. Moreover, we clearly have ξ ± m,j ∗ ,i ∗ ( z ) = A ± m,j ∗ ,i ∗ δ ( z/a ) , for some A ± m,j ∗ ,i ∗ ∈ F × . More generally, we claim that, ∀ j ∈ J , ∀ i ∈ I ( j ) , ξ ± m,j,i ( z ) = N ( i,j ) X p =0 A ± m,j,i,p δ ( p ) ( z/a ) , (4.5) EIGHT-FINITE MODULES OVER ˙U q ( a ) AND ¨U q ( a ) 23 for some A ± m,j,i,p ∈ F and some N ( i, j ) ∈ N . This is proven by a finite induction on j and i . Indeed, makinguse of (4.2), we can rewrite (4.3) as( z − q ± z )( z − q m ∓ z )( z − q ± a )( z − q − m ± a ) X l ∈ Jl ≥ j η εβ,j,l ( z ) ξ ± m,l,i ( z )= ( z q ± − z )( z q ∓ − q m z )( z − q ∓ a )( z − q − m ∓ a ) X k ∈ I ( j ) k ≤ i η εα,k,i ( z ) ξ ± m,j,k ( z ) , (4.6)for every j ∈ J and every i ∈ I ( j ). Now, assume that (4.5) holds for every pair in { ( j, i ) : j ∈ J, i ∈ I ( j ) , j > j } ∪ { ( j , i ) : i ∈ I ( j ) , i ≤ i } , for some j ∈ J and some i ∈ I ( j ) such that i < max I ( j ). Let i ′ be the smallest element of I ( j ) suchthat i < i ′ . It suffices to write (4.6) for j = j and i = i ′ , to get( z − a ) z ( z a − q m z )( q ∓ + q − m ∓ − q ± − q − m ± ) ξ ± m,j ,i ′ ( z )= − ( z − q ± z )( z − q m ∓ z )( z − q ± a )( z − q − m ± a ) X l ∈ Jl>j η εβ,j ,l ( z ) ξ ± m,l,i ′ ( z )+( z q ± − z )( z q ∓ − q m z )( z − q ∓ a )( z − q − m ∓ a ) X k ∈ I ( j ) k ≤ i η εα,k,i ′ ( z ) ξ ± m,j ,k ( z ) . (4.7)Combining the recursion hypothesis and lemma A.1 from the appendix, one easily concludes that (4.5) holdsfor the pair ( j , i ′ ). Repeating the argument finitely many times, we get that it actually holds for all thepairs in { ( j, i ) : j ∈ J, i ∈ I ( j ) , j ≥ j } . Now, either j = min J and we are done; or j > min J and thereexists a largest j ′ ∈ J such that j > j ′ . Writing (4.6) for j = j ′ and i = i ∗ ( j ′ ), we get( z − a ) z ( z a − q m z )( q ∓ + q − m ∓ − q ± − q − m ± ) ξ ± m,j ′ ,i ∗ ( j ′ ) ( z )= − ( z − q ± z )( z − q m ∓ z )( z − q ± a )( z − q − m ± a ) X l ∈ Jl ≥ j η εβ,j ′ ,l ( z ) ξ ± m,l,i ∗ ( j ′ ) ( z ) . Combining again the recursion hypothesis and lemma A.1, we easily get that (4.19) holds for ( j ′ , i ∗ ( j ′ )).It is now clear that the claim holds for every j ∈ J and every i ∈ I ( j ). Letting N = max { N ( i, j ) : j ∈ J, i ∈ I ( j ) } , ii. follows. Furthermore, for every b ∈ F − { a } and every n ∈ N , we obviously have( z − b ) n M α ∩ K ± , ± m ( z ) .M β = { } , thus making a the only element of F satisfying ii.. This concludes theproof. (cid:3) We let ω denote the fundamental weight of a and we let P = Z ω be the corresponding weight lattice. Inview of proposition 4.3, it is natural to make the following Definition 4.4.
Let M be a type (1 , ℓ -weight ¨U q ( a )-module and let { M α : α ∈ A } be the countable setof its ℓ -weight spaces. We shall say that M is rational if, for every α ∈ A , there exist relatively prime monicpolynomials P α (1 /z ) , Q α (1 /z ) ∈ F [ z − ], called Drinfel’d polynomials of M , such that the ℓ -weight κ ± α ( z ) of M α be given by κ ± α ( z ) = − q deg( P α ) − deg( Q α ) (cid:18) P α ( q − /z ) Q α (1 /z ) P α (1 /z ) Q α ( q − /z ) (cid:19) | z | ± ≪ . With each rational ℓ -weight κ ± α ( z ) of a rational ¨U q ( a )-module M , we associate an integral weight λ α ∈ P ,by setting λ α = [deg( P α ) − deg( Q α )] ω . We shall say that M is ℓ -dominant (resp. ℓ -anti-dominant ) if it is rational and there exists N ∈ N × suchthat, for every α ∈ A , deg( P α ) = N and deg( Q α ) = 0 (resp. deg( P α ) = 0 and deg( Q α ) = N ). Remark . The classical weight
N ω (resp. − N ω ) associated with any ℓ -dominant (resp. ℓ -anti-dominant)type 1 ℓ -weight rational ¨U q ( a )-module M is a dominant (resp. anti-dominant) integral weight. Note thatthe converse need not be true. Remark . The data of the ℓ -weights of a rational ¨U q ( a )-module is equivalent to the data of its Drinfel’dpolynomials { ( P α , Q α ) : α ∈ A } which, in turn, is equivalent to the data of their finite multisets of roots { ( ν + α , ν − α ) : α ∈ A } . The latter are finitely supported maps ν ± α : F × → N such that, for every α ∈ A , P α (1 /z ) = Y x ∈ F × (1 − x/z ) ν + α ( x ) and Q α (1 /z ) = Y x ∈ F × (1 − x/z ) ν − α ( x ) . Note that, in the above formulae, since ν ± α is finitely supported, the products only run through the finitelymany numbers in the support supp( ν ± α ) of ν ± α . Moreover, since P α and Q α are relatively prime for every α ∈ A , we have supp( ν + α ) ∩ supp( ν − α ) = ∅ . We denote by N F × f , the set of finitely supported N -valued maps over F × . As is customary in the theory of q -characters, we associate with every ℓ -weight given by a pair ( P α , Q α )of Drinfel’d polynomials or, equivalently, by a pair ( ν + α , ν − α ) with ν + α , ν − α ∈ N F × f and supp( ν + α ) ∩ supp( ν − α ) = ∅ ,a monomial m α = Y ν + − ν − = Y x ∈ F × Y ν + α ( x ) − ν − α ( x ) x ∈ Z [ Y a , Y − a ] a ∈ F × . Definition 4.7.
Let M be an ℓ -dominant ¨U q ( a )-module and let M α and M β be any two ℓ -weight spacesof M with respective ℓ -weights κ ± α ( z ) = − q deg( P α ) (cid:18) P α ( q − /z ) P α (1 /z ) (cid:19) | z | ± ≪ and κ ± β ( z ) = − q deg( P β ) (cid:18) P β ( q − /z ) P β (1 /z ) (cid:19) | z | ± ≪ , where P α (1 /z ) , P β (1 /z ) ∈ F [ z − ] are two monic polynomials. By proposition 4.3. i. , if M α ∩ K ± , ± m ( z ) .M β = { } for some m ∈ N × , then there exists a unique a ∈ F × such that κ εα ( z ) = κ εβ ( z ) H εm,a ( z ) ± , where ε ∈ {− , + } . We shall say that M is t -dominant if, under the same assumptions, we have, in addition,that P β (1 /aq − ( m ± m ) ) = P β (1 /aq − ( m ± m ) ) = 0 . For every a ∈ F × , we let δ a ∈ N F × f be defined by δ a ( x ) = x = a ;0 otherwise. EIGHT-FINITE MODULES OVER ˙U q ( a ) AND ¨U q ( a ) 25 For every a ∈ F × , we let N F × a = n ν ∈ N F × f : { a, aq } ⊆ supp( ν ) o and we define, for every m ∈ Z , an operatorΓ m,a : N F × aq − m → N F × a by letting , for every ν ∈ N F × aq − m ,Γ m,a ( ν ) = ν − δ aq − m − δ aq − m + δ a + δ aq . Γ m,a is obviously invertible, with inverse Γ − m,a : N F × a → N F × aq − m given by Γ − m,a = Γ − m,aq − m . Note that, forevery a ∈ F × , Γ ,a = id over N F × a . Given two finite multisets ν, ν ′ ∈ N F × f , we we shall say that they are equivalent and write ν ∼ ν ′ iff ν = Γ m ,a ◦ · · · ◦ Γ m n ,a n ( ν ′ ) , (4.8)for some n ∈ N , m , . . . , m n ∈ Z n and some a , . . . , a n ∈ F × . In writing (4.8), it is assumed that, for every p = 2 , . . . , n , Γ m p ,a p ◦ · · · ◦ Γ m n ,a n ( ν ′ ) ∈ N F × a p − q − mp − . It is clear that ∼ is an equivalence relation and wedenote by [ ν ] ∈ N F × f / ∼ the equivalence class of ν in N F × f . Following remark 4.6, we naturally extend theaction of Γ m,a to Z [ Y b , Y − b ] b ∈ F × , by settingΓ m,a ( Y ν ) = Y Γ m,a ( ν ) . The equivalence relation ∼ similarly extends from N F × f to Z [ Y b , Y − b ] b ∈ F × . Note that, setting H m,a = Y − aq − m Y − aq − m Y a Y aq ∈ Z [ Y b , Y − b ] b ∈ F × , for every a ∈ F × and every m ∈ Z , we have, for every ν ∈ N F × a Γ m,a ( Y ν ) = H m,a Y ν . Corollary 4.8.
Let M be a simple t -dominant ¨U q ( a ) -module. Then there exists a multiset ν ∈ N F × f suchthat all the monomials associated with the ℓ -weights of M be in the equivalence class of Y ν .Proof. By proposition 4.3, for any two ℓ -weight spaces, M α and M β , of an ℓ -dominant ¨U q ( a )-module M ,with respective ℓ -weights κ ± α ( z ) = − q deg( P α ) (cid:18) P α ( q − /z ) P α (1 /z ) (cid:19) | z | ± ≪ and κ ± β ( z ) = − q deg( P β ) (cid:18) P β ( q − /z ) P β (1 /z ) (cid:19) | z | ± ≪ , if M α ∩ K ± , ± m,n .M β = { } for some m ∈ N × and some n ∈ Z , then we must have P α ( q − /z ) P α (1 /z ) = P β ( q − /z ) P β (1 /z ) (cid:18) (1 − q − a/z )(1 − a/z )(1 − a/z )(1 − q a/z ) (1 − q − m a/z )(1 − q − m − a/z )(1 − q − m +1) a/z )(1 − q − m a/z ) (cid:19) ± , (4.9)for some a ∈ F × . Now, assuming m >
1, it is clear that:- for the upper choice of sign on the right hand side of the above equation, the last fraction linemust completely cancel against factors in the first one, whereas the second one survives, eventuallyreplacing the cancelled factors;- for the lower choice of sign, the second fraction line must cancel against factors in the first one,whereas the last one survives, eventually replacing the cancelled factors.If on the other hand m = 1, since M is t -dominant, we have, by definition, that aq ∓ is a root of P β (1 /z ).In any case, denoting by ν α (resp. ν β ) the multiset of roots of P α (1 /z ) (resp. P β (1 /z )), it is clear that ν α ∼ ν β and hence Y ν α ∼ Y ν β . Since M is simple, there can be no non-zero ℓ -weight space M β of M suchthat M α ∩ K ± , ± m,n .M β = { } for every ℓ -weight space M α of M , every m ∈ N × and every n ∈ Z . (cid:3) Although the definition of Γ ± ± ,a easily extends to n ν ∈ N F × f : aq − (1 ± ∈ supp( ν ) o , we will not make use of that extensionand exclusively regard Γ ± ± ,a as a map N F × aq − ± → N F × aq ∓ . In view of definition-proposition 4.2, we can make the following
Definition 4.9.
For every monic polynomial P (1 /z ) ∈ F [ z − ], denote by F P the one-dimensional ¨U , q ( a )-module such that K ± , ( z ) .v = − q deg( P ) (cid:18) P ( q − /z ) P (1 /z ) (cid:19) | z | ± ≪ v , for every v ∈ F P . There exists a universal ¨U q ( a )-module M ( P ) ∼ = ¨U q ( a ) b ⊗ ¨U , q ( a ) F P that admits the ℓ -weightassociated with P . Denoting by N ( P ) the maximal ¨U q ( a )-submodule of M ( P ) such that N ( P ) ∩ F P = { } , we define the unique – up to isomorphisms – simple ¨U q ( a )-module L ( P ) = M ( P ) /N ( P ). Proposition 4.10.
For every simple ℓ -dominant ¨U q ( a ) -module M , there exists a monic polynomial P (1 /z ) ∈ F [ z − ] such that M ∼ = L ( P ) .Proof. Obviously, for every v ∈ M − { } , we have M ∼ = ¨U q ( a ) .v . Now since M is ℓ -dominant, v can bechosen as an ℓ -weight vector, i.e. K ± , ( z ) .v = − q deg( P ) (cid:18) P ( q − /z ) P (1 /z ) (cid:19) | z | ± ≪ v for some monic polynomial P (1 /z ) ∈ F [ z − ]. (cid:3) Remark . The above proof makes it clear that if { P α : α ∈ A } is the set of Drinfel’d polynomials of asimple ℓ -dominant ¨U q ( a )-module M , then, for every α ∈ A , M ∼ = L ( P α ). Theorem 4.12.
For every monic polynomial P (1 /z ) ∈ F [ z − ] , L ( P ) is t -dominant.Proof. We postpone the proof of this theorem until section 5, where we construct L ( P ) for every P anddirectly check that it is indeed t -dominant. (cid:3) Proposition 4.13.
Any topological \ ¨U q ( a ) -module pulls back to a module over the elliptic Hall algebra E q − ,q ,q .Proof. It suffices to make use of the Hopf algebra homomorphism E q − ,q ,q f −→ ¨U + q ( a ) ֒ → \ ¨U q ( a ) , where f is defined in proposition 3.18 and the second arrow is the canonical injection into \ ¨U q ( a ) of its Hopfsubalgebra ¨U + q ( a ) – see proposition 3.12. (cid:3) Remark . It is worth mentioning that, as an example of the above proposition, ℓ -anti-dominant ¨U q ( a )-modules pullback to a family of E q − ,q ,q -modules that were recently introduced in [DK19]. It might beinteresting to investigate further the class of E q − ,q ,q -modules obtained by pulling back other (rational)¨U q ( a )-modules.We conclude the present subsection by proving the following Lemma 4.15.
Let M be an ℓ -dominant ¨U q ( a ) -module. Suppose that, for any two ℓ -weight spaces M α and M β of M , with respective ℓ -weights κ ± α ( z ) and κ ± β ( z ) , such that M α ∩ K ± , ± ( z ) .M β = { } , the unique a ∈ F × such that κ εα ( z ) = κ εβ ( z ) H ε ,a ( z ) ± , for every ε ∈ {− , + } , and ( z − a ) N M α ∩ K ± , ± ( z ) .M β = { } for some N ∈ N × – see proposition 4.3 – also satisfies P β (1 /a ) = 0 . Then M is t -dominant. EIGHT-FINITE MODULES OVER ˙U q ( a ) AND ¨U q ( a ) 27 Proof.
Let M be as above and let M α and M β be two ℓ -weight spaces of M with respective ℓ -weights κ ± α ( z ) = − q deg( P α ) (cid:18) P α ( q − /z ) P α (1 /z ) (cid:19) | z | ± ≪ and κ ± β ( z ) = − q deg( P β ) (cid:18) P β ( q − /z ) P β (1 /z ) (cid:19) | z | ± ≪ . Suppose that M α ∩ K ± , ± m ( z ) .M β = { } for some m ∈ N × . If m >
1, writing down κ εα ( z ) = κ εβ ( z ) H εm,a ( z ) ± ,we obtain equation (4.9) as in the proof of corollary 4.8. By the same discussion as the one following equation(4.9), we conclude that P β (1 /aq − ( m ± m ) ) = P β (1 /aq − ( m ± m ) ) = 0, as needed – see definition 4.7. Finally, if m = 1, writing down κ εα ( z ) = κ εβ ( z ) H ε ,a ( z ) ± , we obtain P α ( q − /z ) P α (1 /z ) = P β ( q − /z ) P β (1 /z ) (cid:18) (1 − a/z )(1 − q a/z ) (1 − q − a/z )(1 − q − a/z ) (cid:19) ± . Then, it is clear that: • for the upper choice of sign on the right hand side of the above equation, the last fraction linemust completely cancel against factors in the first one, whereas the second one survives, eventuallyreplacing the cancelled factors; • for the lower choice of sign on the right hand side of the above equation, the second fraction linemust completely cancel against factors in the first one, whereas the last one survives, eventuallyreplacing the cancelled factors.In any case, it follows that P β (1 /aq ∓ ) = 0. But by our assumptions on M , we also have that P β (1 /a ) = 0and the t -dominance of M follows – see definition 4.7. (cid:3) t -weight ¨U q ( a ) -modules.Definition 4.16. For every N ∈ N × , we shall say that a (topological) module M over ¨U ′ q ( a ) is of type (1 , N )if: i. C ± / acts as id on M ;ii. c ±± m acts by multiplication by 0 on M , for every m ≥ N .We shall say that M is of type (1 ,
0) if points i. and ii. above hold for every m > c ± actsas id on M . Remark . Let N ∈ N . Then the ¨U ′ q ( a )-modules of type (1 , N ) are in one-to-one correspondence withthe ¨U q ( a ) ( N ) / ( C / − q ( a ) ( N ) . Obviously ¨U q ( a )-modulesof type (1 ,
0) descend to modules over the double quantum loop algebra of type a , ¨L q ( a ). Definition 4.18.
We shall say that a (topological) ¨U q ( a )-module M is a t -weight module if there exists acountable set { M α : α ∈ A } of indecomposable ℓ -weight ¨U q ( a )-modules, called t -weight spaces of M , suchthat, as (topological) ¨U q ( a )-modules, M ∼ = M α ∈ A M α . (4.10)We shall say that M is weight-finite if, regarding it as a completely decomposable ¨U q ( a )-module, its Sp( M )is finite – see definition-proposition 4.2 for the definition of Sp. A vector v ∈ M − { } is a highest t -weightvector of M if v ∈ M α for some α ∈ A and, for every r, s ∈ Z , X +1 ,r,s .v = 0 . (4.11)We shall say that M is highest t -weight if M ∼ = ¨U q ( a ) .v for some highest t -weight vector v ∈ M − { } . Definition-Proposition 4.19.
Let M be a t -weight ¨U q ( a )-module that admits a highest t -weight vector v ∈ M − { } . Denote by M the t -weight space of M containing v . Then M = ¨U q ( a ) .v and, for every r, s ∈ Z , X +1 ,r,s .M = { } . (4.12)We shall say that M is a highest t -weight space of M . If in addition M is simple, then it admits a unique –up to isomorphisms of ¨U q ( a )-modules – highest t -weight space M . Proof.
It is an easy consequence of the triangular decomposition of ¨U q ( a ) – see proposition 3.11 – andof the root grading of ¨U q ( a ) that, indeed, X +1 ,r,s . (cid:16) ¨U q ( a ) .v (cid:17) = { } , for every r, s ∈ Z . Now since M ishighest t -weight, we have M ∼ = ¨U q ( a ) .v . By proposition 3.11, M ⊂ M ∼ = ¨U − q ( a ) ¨U q ( a ) .v and it followsthat M ∼ = ¨U q ( a ) .v . Now, assuming that M is simple and that it admits highest t -weight spaces M and M ′ , we have that ¨U − q ( a ) .M ∼ = M ∼ = ¨U − q ( a ) .M ′ as ¨U q ( a )-modules. In particular, M ∼ = M ′ as¨U q ( a )-modules. (cid:3) In view of the triangular decomposition of ¨U q ( a ) – see proposition 3.11 –, the above proposition impliesthat any highest t -weight ¨U q ( a )-modules M is entirely determined as M ∼ = ¨U − q ( a ) .M , by the data of itshighest t -weight space M , a cyclic ℓ -weight ¨U q ( a )-module. Now for any v ∈ M − { } such that M ∼ =¨U q ( a ) .v , let N be the maximal ¨U q ( a )-submodule of M not containing v and set L = M /N . Then, byconstruction, L is a simple ¨U q ( a )-module such that, as ¨U q ( a )-modules, M ∼ = ¨U − q ( a ) .L mod ¨U − q ( a ) .N .We therefore make the following Definition 4.20.
We extend every simple (topological) ℓ -weight ¨U q ( a )-module M into a ¨U ≥ q ( a )-moduleby setting X +1 ,r,s .M = { } for every r, s ∈ Z . This being understood, we define the universal highest t -weight¨U ′ q ( a )-module with highest t -weight space M by setting M ( M ) = \ ¨U ′ q ( a ) b ⊗ ¨U ≥ q ( a ) M as ¨U ′ q ( a )-modules. Denoting by N ( M ) the maximal (closed) ¨U q ( a )-submodule of M ( M ) such that M ∩ N ( M ) = { } , we define the simple highest t -weight ¨U q ( a )-module L ( M ) with highest t -weight space M by setting L ( M ) ∼ = M ( M ) / N ( M ). It is unique up to isomorphisms.Classifying simple highest t -weight ¨U q ( a )-modules therefore amounts to classifying those simple ℓ -weight¨U q ( a )-modules M that appear as their highest t -weight spaces. In the case of weight-finite ¨U q ( a )-modules,this is achieved by the following Theorem 4.21.
The following hold:i. Every weight-finite simple ¨U ′ q ( a ) -module M is highest t -weight and can be obtained by twisting atype (1,0) weight-finite simple ¨U q ( a ) -module with an algebra automorphism from the subgroup of Aut( ¨U ′ q ( a )) generated by the algebra automorphisms τ and σ of proposition 3.26.ii. The type (1,0) simple highest t -weight ¨U ′ q ( a ) -module L ( M ) is weight-finite if and only if its highest t -weight space M is a simple t -dominant ¨U q ( a ) -module – see proposition-definition 4.4.Proof. Let M be a weight-finite simple t -weight ¨U q ( a )-module and assume for a contradiction that, forevery w ∈ M − { } , there exist r, s ∈ Z such that X +1 ,r,s .w = 0. Then, there must exist two sequences N clearly does not depend on the chosen generator v . Indeed, if N contained a generator v ′ of M , it would contain all theothers, including v . It follows that N and hence L are both independent of v . EIGHT-FINITE MODULES OVER ˙U q ( a ) AND ¨U q ( a ) 29 ( r n ) n ∈ N , ( s n ) n ∈ N ∈ Z N , such that 0 / ∈ (cid:8) w n = X + r ,s . . . X + r n ,s n .w : n ∈ N (cid:9) . Choosing w ∈ M −{ } to be an eigenvector of K +1 , , with eigenvalue λ ∈ F × – see definition-proposition 4.2 forthe existence of such a vector –, one easily sees from the relations that, for every n ∈ N , K +1 , , .w n = λq n w n .It follows – see definition-proposition 4.2 – that { λq n : n ∈ N } ⊆ Sp( M ). A contradiction with the weight-finiteness of M . Thus, we conclude that there exists a highest t -weight vector v ∈ M − { } such that K ± , , .v = κ ± v for some κ ∈ F × . Obviously, M ∼ = ¨U q ( a ) .v , for ¨U q ( a ) .v = { } is a submodule of thesimple ¨U q ( a )-module M . Thus M is highest t -weight. Denote by M = ¨U q ( a ) .v its highest t -weight space.The latter is an ℓ -weight ¨U q ( a )-module. As such, it completely decomposes into countably many locallyfinite-dimensional indecomposable ¨U , q ( a )-modules that constitute its ℓ -weight spaces. Over any of these, C / must admit an eigenvector. But since M is simple and C / is central, the latter acts over M by a scalarmultiple of id. It follows from definition-proposition 4.2 that C acts over M by id or − id. In the former case,there is nothing to do; whereas in the latter, it is quite clear from proposition 3.26 that, twisting the ¨U q ( a )action on M by τ , we can ensure that C acts by id. It follows that C / acts by id or − id. Again, in theformer case, there is nothing to do; whereas in the latter, twisting by σ , we can ensure that C / acts by id.Similarly, for every m ∈ N , c ±± m must admit an eigenvector over any locally finite-dimensional ℓ -weight spaceof M . But again, since M is simple and c ±± m is central, the latter must act over M by a scalar multiple ofid.In any case, in view of (3.7) and (3.8), K ± , , commutes with all the other generators of ¨U q ( a ) and, since M = ¨U q ( a ) .v , we have K ± , , .w = κ ± w for every w ∈ M . Moreover, M turns out to be a type 1 ℓ -weight¨U q ( a )-module and, by definition-proposition 4.2, M ⊆ M α ∈ A (cid:8) v ∈ M : K ± , , .v = κ ± v and ∃ n ∈ N × , ∀ m ∈ N × (cid:0) K ± , , ± m − κ ± α, ± m id (cid:1) n .v = 0 (cid:9) for some countable set of sequences n ( κ ± α, ± m ) m ∈ N × ∈ F N × : α ∈ A o . By proposition 4.19, X +1 ,r,s .M = { } , (4.13)for every r, s ∈ Z . Pulling back with ι (0) and ι (1) respectively, we can simultaneously regard M as aU q (L a )-module for both of its Dynkin diagram subalgebras U q (L a ) (0) and U q (L a ) (1) – see section 3. Let v ∈ M −{ } be a simultaneous eigenvector of the pairwise commuting linear operators in (cid:8) K ± , , ± m : m ∈ N (cid:9) .Equation (4.13) implies that x +1 ( z ) .v = x − ( z ) .v = 0. Thus v is a highest (resp. lowest) ℓ -weight vector of˙U q ( a ) (1) .v (resp. ˙U q ( a ) (0) .v ). The weight finiteness of M now allows us to apply corollary 2.12 to prove thatthe respective simple quotients of U q (L a ) (0) .v and U q (L a ) (1) .v containing v are both finite-dimensional andisomorphic to a unique simple highest (resp. lowest) ℓ -weight module L ( P ) (resp. ¯ L ( P )). As a consequenceof theorem 2.5 and of proposition 2.6, we conclude that k ± ( z ) .v = q − deg( P ) (cid:18) P (1 /z ) P ( q − /z ) (cid:19) | z | ∓ ≪ v and k ± ( z ) .v = q deg( P ) (cid:18) P ( q − /z ) P (1 /z ) (cid:19) | z | ∓ ≪ v , for some monic polynomials P and P . On the other hand, pulling back with ι m for every m ∈ Z , we canregard M as a U q (L a )-module in infinitely many independent ways. Again, for every m ∈ Z , v turns outto be a highest ℓ -weight vector for a unique simple weight finite, hence finite dimensional U q (L a )-module.As such, it satisfies ι m ( k +1 ( z )) .v = q deg( Q m ) (cid:18) Q m ( q − /z ) Q m (1 /z ) (cid:19) | z | ∓ ≪ v , for some monic polynomial Q m . Now since ι m ( k ± ( z )) = − | m | Y p =1 c ± (cid:16) q (1 − p )sign( m ) − z (cid:17) sign( m ) K ∓ , ( C − / z )and Ψ( k ± ( z ) k ± ( z )) = c ± ( z ), we must have q deg( Q m ) (cid:18) Q m ( q − /z ) Q m (1 /z ) (cid:19) | z | ∓ ≪ = q deg( P )+ m (deg( P ) − deg( P )) (cid:18) P ( q − /z ) P (1 /z ) (cid:19) | z | ∓ ≪ × | m | Y p =1 (cid:18) P ( q (2 p − m ) − /z ) P ( q (2 p − m )+1 /z ) P ( q (2 p − m )+1 /z ) P ( q (2 p − m ) − /z ) (cid:19) | z | ∓ ≪ (4.14)for every m ∈ Z × . In the limit as z − →
0, this implies q deg( Q m ) = q deg( P )+ m (deg( P ) − deg( P )) for every m ∈ Z and, consequently, deg( P ) = deg( P ) = deg( Q m ). After obvious simplifications, (4.14) becomes (cid:18) Q m ( q − /z ) Q m (1 /z ) (cid:19) | z | ∓ ≪ = (cid:18) P ( q − /z ) P ( q − (1 ± /z ) P (1 /z ) P ( q − (1 ± /z ) P ( q m − (1 ± /z ) P ( q m − (1 ± /z ) (cid:19) | z | ∓ ≪ (4.15)for every m ∈ Z × . Now, z − = 0 is not a root of P (1 /z ) for any monic polynomial P . Moreover, q being aformal parameter – in case q is regarded as a complex number, we shall assume that 1 / ∈ q Z × –, it follows thatthe map z − q m z − has no fixed points over the set of roots of a monic polynomial. Thus, for | m | largeenough, the respective sets of roots of P ( q − /z ) P ( q − (1 ± /z ) and P ( q m − (1 ± /z ) are disjoint. Similarly,for | m | large enough, the respective sets of roots of P (1 /z ) P ( q − (1 ± /z ) and P ( q m − (1 ± /z ) are disjoint.It follows that, for | m | large enough, on the r.h.s. of (4.15), cancellations can only occur between factors onopposite sides of the same fraction line. Now, either P = P or P = P , in which case P (1 /z ) P (1 /z ) = n Y p =1 − α p /z − β p /z , for some n ∈ N × such that n ≤ deg( P ) = deg( P ) and some n -tuples ( α p ) p ∈ J n K , ( β p ) p ∈ J n K ∈ F n such that { α p : p ∈ J n K } ∩ { β p : p ∈ J n K } = ∅ . But then, we should have, for | m | large enough, (cid:18) Q m ( q − /z ) Q m (1 /z ) (cid:19) | z | ∓ ≪ = P ( q − /z ) P (1 /z ) n Y p =1 − α p q − (1 ± /z − β p q − (1 ± /z n Y p =1 − β p q m − (1 ± /z − α p q m − (1 ± /z ! | z | ∓ ≪ , where, on the r.h.s., cancellations can only occur across the leftmost fraction line. A contradiction. i. follows.As for part of ii., we shall prove it in section 5. (cid:3) Although we must postpone the proof of part ii. of theorem 4.21, the proof above still makes it clear that
Proposition 4.22.
If a type (1,0) simple highest t -weight ¨U q ( a ) -module L ( M ) is weight-finite, then itshighest t -weight space M is a simple ℓ -dominant ¨U q ( a ) -module. Proposition 4.23.
Let M be a t -weight ¨U q ( a ) -module and let M α and M β be two ℓ -weight spaces of M such that, for some m, n ∈ Z , M α ∩ X ± ,m,n .M β = { } . Then, there exists a unique a ∈ F × such that:i. the respective ℓ -weights κ εα ( z ) and κ εβ ( z ) of M α and M β be related by κ εα ( z ) = κ εβ ( z ) A εa ( z ) ± , (4.16) EIGHT-FINITE MODULES OVER ˙U q ( a ) AND ¨U q ( a ) 31 where ε ∈ {− , + } and A ± a ( z ) = q (cid:18) − q − a/z − q a/z (cid:19) | z | ± ≪ ; ii. ( z − a ) N M α ∩ X ± ,m ( z ) .M β = { } for some N ∈ N × .Proof. We keep the same notations as in the proof of proposition 4.3. More specifically, we have two bases { v i : i = 1 , . . . , dim( M α ) } and { w j : j = 1 , . . . , dim( M β ) } of M α and M β respectively, in which ∀ i ∈ J dim M α K , K ± , ( z ) .v i = κ ± α ( z ) dim M α X k = i η ± α,i,k ( z ) v k , ∀ j ∈ J dim M β K , K ± , ( z ) .w j = κ ± β ( z ) dim M β X l = j η ± β,j,l ( z ) w l , for some η ± α,i,k ( z ) , η ± β,j,l ( z ) ∈ F [[ z ± ]], with i, k ∈ J dim M α K and j, l ∈ J dim M β K , such that η ± α,i,i ( z ) = 1 forevery i ∈ J dim M α K and η ± β,j,j ( z ) = 1 for every j ∈ J dim M β K .Now, if M α ∩ X ± ,m,n .M β = { } , there must exist a largest nonempty subset J ⊆ J dim M β K such that, forevery j ∈ J , M α ∩ X ± ,m ( z ) .w j = { } . Let j ∗ = max J . Obviously, for every j ∈ J , there must exist a largestnonempty subset I ( j ) ⊆ J dim M α K such that, for every j ∈ J and every i ∈ I ( j ), F v i ∩ X ± ,m ( z ) .w j = { } .Let i ∗ ( j ) = min I ( j ) and let for simplicity i ∗ = i ∗ ( j ∗ ). Then, for every j ∈ J , M α ∩ X ± ,m ( z ) .w j = X i ∈ I ( j ) ξ ± m,j,i ( z ) v i , for some ξ ± m,j,i ( z ) ∈ F [[ z, z − ]] − { } . When needed, we shall extend by zero the definition of ξ ± m,j,i ( z ) outsideof the set of pairs { ( j, i ) : j ∈ J, i ∈ I ( j ) } . Making use of the relations in ¨U q ( a ) – namely (3.9) and (3.10)–, we get, for every j ∈ J and every ε ∈ {− , + } ,( z − q ± z ) X ± ,m ( z ) K ε , ( z ) .w j = ( z q ± − z ) K ε , ( z ) X ± ,m ( z ) .w j . The latter easily implies that, for every j ∈ J and every i ∈ I ( j ),( z − q ± z ) κ εβ ( z ) X l ∈ Jl ≥ j η εβ,j,l ( z ) ξ ± m,l,i ( z ) = ( z q ± − z ) κ ± α ( z ) X k ∈ I ( j ) k ≤ i η εα,k,i ( z ) ξ ± m,j,k ( z ) . (4.17)Taking i = i ∗ and j = j ∗ in the above equation immediately yields (cid:2) ( z − q ± z ) κ εβ ( z ) − ( z q ± − z ) κ εα ( z ) (cid:3) ξ ± m,j ∗ ,i ∗ ( z ) = 0 . The latter is equivalent to the fact that, for every p ∈ Z , ξ ± m,j ∗ ,i ∗ ,p z (cid:0) q ± κ εβ ( z ) − κ εα ( z ) (cid:1) = ξ ± m,j ∗ ,i ∗ ,p +1 (cid:0) κ εβ ( z ) − q ± κ εα ( z ) (cid:1) , (4.18)where, as usual, we have set ξ ± m,j ∗ ,i ∗ ,p = res z z p − ξ ± m,j ∗ ,i ∗ ( z ) . Since ξ ± m,j ∗ ,i ∗ ( z ) = 0, there exists at least one - p ∈ Z such that ξ ± m,j ∗ ,i ∗ ,p = 0. Assuming that ξ ± m,j ∗ ,i ∗ ,p +1 = 0,one easily derives a contradiction from (4.18) and, repeating the argument, one proves that ξ ± m,j ∗ ,i ∗ ,p = 0 forevery p ∈ Z . Dividing (4.18) by ξ ± m,j ∗ ,i ∗ ,p , one gets z (cid:0) q ± κ εβ ( z ) − κ εα ( z ) (cid:1) = a (cid:0) κ εβ ( z ) − q ± κ εα ( z ) (cid:1) , where we have set, for every p ∈ Z , ξ ± m,j ∗ ,i ∗ ,p +1 /ξ ± m,j ∗ ,i ∗ ,p = a ∈ F × . i. now follows. Moreover, we clearlyhave ξ ± m,j ∗ ,i ∗ ( z ) = A ± m,j ∗ ,i ∗ δ ( z/a ) , for some A ± m,j ∗ ,i ∗ ∈ F × . More generally, we claim that, ∀ j ∈ J , ∀ i ∈ I ( j ) , ξ ± m,j,i ( z ) = N ( i,j ) X p =0 A ± m,j,i,p δ ( p ) ( z/a ) , (4.19)for some A ± m,j,i,p ∈ F and some N ( i, j ) ∈ N . This is proven by a finite induction on j and i . Indeed, makinguse of (4.16), we can rewrite (4.17) as( z − q ± z )( z − q ± a ) X l ∈ Jl ≥ j η εβ,j,l ( z ) ξ ± m,l,i ( z ) = ( z q ± − z )( q ± z − a ) X k ∈ I ( j ) k ≤ i η εα,k,i ( z ) ξ ± m,j,k ( z ) , (4.20)for every j ∈ J and every i ∈ I ( j ). Now, assume that (4.19) holds for every pair in { ( j, i ) : j ∈ J, i ∈ I ( j ) , j > j } ∪ { ( j , i ) : i ∈ I ( j ) , i ≤ i } , for some j ∈ J and some i ∈ I ( j ) such that i < max I ( j ). Let i ′ be the smallest element of I ( j ) suchthat i < i ′ . It suffices to write (4.20) for j = j and i = i ′ , to get( z − a ) z (1 − q ± ) ξ ± m,j ,i ′ ( z ) = − ( z − q ± z )( z − q ± a ) X l ∈ Jl>j η εβ,j ,l ( z ) ξ ± m,l,i ′ ( z )+( z q ± − z )( q ± z − a ) X k ∈ I ( j ) k ≤ i η εα,k,i ′ ( z ) ξ ± m,j ,k ( z ) . (4.21)Combining the recursion hypothesis and lemma A.1 from the appendix, one easily concludes that (4.19)holds for the pair ( j , i ′ ). Repeating the argument finitely many times, we get that it actually holds for allthe pairs in { ( j, i ) : j ∈ J, i ∈ I ( j ) , j ≥ j } . Now, either j = min J and we are done; or j > min J andthere exists a largest j ′ ∈ J such that j > j ′ . Writing (4.20) for j = j ′ and i = i ∗ ( j ′ ), we get( z − a ) z (1 − q ± ) ξ ± m,j ′ ,i ∗ ( j ′ ) ( z ) = − ( z − q ± z )( z − q ± a ) X l ∈ Jl ≥ j η εβ,j ′ ,l ( z ) ξ ± m,l,i ∗ ( j ′ ) ( z ) . Combining again the recursion hypothesis and lemma A.1, we easily get that (4.19) holds for ( j ′ , i ∗ ( j ′ )).It is now clear that the claim holds for every j ∈ J and every i ∈ I ( j ). Letting N = max { N ( i, j ) : j ∈ J, i ∈ I ( j ) } , ii. follows. Furthermore, for every b ∈ F − { a } and every n ∈ N , we obviously have( z − b ) n M α ∩ X ± ,m ( z ) .M β = { } , thus making a the unique element of F satisfying ii. . (cid:3) Corollary 4.24.
The ℓ -weights of any type (1 , weight-finite simple ¨U q ( a ) -module are all rational – seedefinition 4.4.Proof. Let M be a type (1 ,
0) weight-finite simple ¨U q ( a )-module. By proposition 4.22, its highest t -weightspace M is an ℓ -dominant simple ¨U q ( a )-module. Hence, M ∼ = L ( M ) ∼ = ¨U − q ( a ) .M and it easily followsby proposition 4.23 that all the ℓ -weights of L ( M ) are of the form κ ± α ( z ) N Y p =1 A ± a p ( z ) − , EIGHT-FINITE MODULES OVER ˙U q ( a ) AND ¨U q ( a ) 33 for some N ∈ N , some a , . . . , a N ∈ F × and κ ± α ( z ) = − q deg P α (cid:18) P α ( q − /z ) P α (1 /z ) (cid:19) | z | ± ≪ , for some monic polynomial P α (1 /z ) ∈ F [ z − ]. Now, observe that A ± a ( z ) − = q − (cid:18) − q a/z − q − a/z (cid:19) | z | ± ≪ = q − (cid:18) − q a/z − a/z (cid:19) | z | ± ≪ q − (cid:18) − a/z − q − a/z (cid:19) | z | ± ≪ . Hence, all the ℓ -weights of L ( M ) are of the form κ ± β ( z ) = − q deg( P β ) − deg( Q β ) (cid:18) P β ( q − /z ) Q β (1 /z ) P β (1 /z ) Q β ( q − /z ) (cid:19) | z | ± ≪ , (4.22)for some relatively prime monic polynomials P β (1 /z ) , Q β (1 /z ) ∈ F [ z − ], which concludes the proof. (cid:3) In view of remark 4.6, we can therefore associate with any weigh-finite simple ¨U q ( a )-module a q -characterdefined as the (formal) sum of the monomials corresponding to all its rational ℓ -weights. Proposition 4.25.
Let M and N be two t -dominant simple ¨U q ( a ) -modules such that M b ⊗ N be simple.Then:i. M b ⊗ N is a simple t -dominant ¨U q ( a ) -module of type (1 , ;ii. there exists a short exact sequence of ¨U q ( a ) -modules { } → N → L ( M ) b ⊗L ( N ) → L ( M b ⊗ N ) → { } ; iii. if, in addition, L ( M ) b ⊗L ( N ) is simple, then L ( M ) b ⊗L ( N ) ∼ = L ( M b ⊗ N ) . Proof. M and N are both of type (1 ,
0) and (3.22) and (3.23) respectively imply that so is M b ⊗ N .Similarly, they are both ℓ -weight and ℓ -dominant. Combining eqs. (3.19), (3.20), (3.21), (3.24), (3.25) and(3.26), we easily prove that∆ ( K +1 ,m ( z )) = − m X k =0 m Y l = k +1 c − ( zq − l C / ) K +1 ,k ( z ) b ⊗ K +1 ,m − k ( zq − k C (1) ) , (4.23)∆ ( K − , − m ( z )) = − m X k =0 K − , − ( m − k ) ( zq − k C (2) ) b ⊗ K − , − k ( z ) m Y l = k +1 c + ( zq − l C / ) , (4.24)for every m ∈ N . In particular, taking m = 0, we have ∆ ( K ± , ( z )) = − K ± , ( z C ∓ (2) ) ⊗ K ± , ( z C ± (1) ). It followsthat, if { M ,α : α ∈ A } and { N ,β : β ∈ B } are the countable sets of ℓ -weights of M and N respectively,with respective Drinfel’d polynomials { P α : α ∈ A } and { P β : β ∈ B } , then { M ,α ⊗ N ,β : α ∈ A , β ∈ B } is the countable set of ℓ -weights of M b ⊗ N . Moreover, the latter is obviously ℓ -dominant since its Drinfel’dpolynomials are in { P α P β : α ∈ A β ∈ B } . Now let α, α ′ ∈ A , β, β ′ ∈ B and let P α , P α ′ , P β and P β ′ bethe Drinfel’d polynomials of M ,α , M ,α ′ , N ,β and N ,β ′ respectively and assume that( M ,α ⊗ N ,β ) ∩ ∆ ( K ± , ± ( z )) . ( M ,α ′ ⊗ N ,β ′ ) = { } . (4.25)Then, writing (4.23) and (4.24) above with m = 1, we get∆ ( K +1 , ( z )) = − c − ( zq − C / ) K +1 , ( z ) b ⊗ K +1 , ( z C (1) ) − K +1 , ( z ) b ⊗ K +1 , ( zq − C (1) ) , ∆ ( K − , − ( z )) = − K − , − ( z C (2) ) b ⊗ K − , ( z ) c + ( zq − C / ) − K − , ( zq − C (2) ) b ⊗ K − , − ( z ) . Since both M ,α ′ and N ,β ′ are ℓ -weight spaces, it follows that∆ ( K ± , ± ( z )) . ( M ,α ′ ⊗ N ,β ′ ) ⊆ (cid:0) K ± , ± ( z ) .M ,α ′ ⊗ N ,β ′ (cid:1) ⊕ (cid:0) M ,α ′ ⊗ K ± , ± ( z ) .N ,β ′ (cid:1) , Therefore, condition (4.25) holds only if the direct sum on the r.h.s. above has a non-vanishing intersectionwith M ,α ⊗ N ,β . But since the latter is an ℓ -weight space, this happens only if either M ,α ∩ K ± , ± ( z ) .M ,α ′ = { } or N ,β ∩ K ± , ± ( z ) .N ,β ′ = { } . The t -dominance of M and N implies that for the only a ∈ F × suchthat , either P α ′ (1 /a ) = 0 or P β ′ (1 /a ) = 0. In any case, P α ′ (1 /a ) P β ′ (1 /a ) = 0 and M b ⊗ N is t -dominant.i. follows. By lemma 3.29, it is clear that ˙∆( X +1 ,r ( z )) . (cid:0) M b ⊗ N (cid:1) = { } . Hence M b ⊗ N is a highest t -weightspace in L ( M ) b ⊗L ( N ). Let N denote the largest closed \ ¨U ′ q ( a )-submodule of L ( M ) b ⊗L ( N ) such that N ∩ (cid:0) M b ⊗ N (cid:1) = { } . ii. obviously follows. iii. is clear. (cid:3) An evaluation homomorphism and evaluation modules
In this section, we construct an evaluation algebra b A t and an F -algebra homomorphism ev : ¨U q ( a ) → b A t ,that we shall refer to as the evaluation homomorphism.5.1. The quantum Heisenberg algebras H + t and H − t .Definition 5.1. The quantum Heisenberg algebra H ± t is the Hopf algebra generated over K ( t ) by n γ / , γ − / , α ± , α − ± , α ± ,m : m ∈ Z × o , subject to the relations, γ / , γ − / , α ± , α − ± are central,[ α ± , − m , α ± ,n ] = − δ m,n m [2 m ] t γ m − γ − m t − t − , for every m, n ∈ Z × , with comultiplication ∆ defined by setting∆( γ / ) = γ / ⊗ γ / , ∆( γ − / ) = γ − / ⊗ γ − / , ∆( α ± ) = α ± ⊗ α ± , ∆( α − ± ) = α − ± ⊗ α − ± , ∆( α ± ,m ) = α ± ,m ⊗ γ | m | / + γ −| m | / ⊗ α ± ,m , for every m, n ∈ Z × , antipode S defined by setting S ( γ / ) = γ − / , S ( γ − / ) = γ / ,S ( α ± ) = α − ± , S ( α − ± ) = α ± S ( α ± ,m ) = − α ± ,m , and counit ε defined by setting ε ( γ / ) = ε ( γ − / ) = ε ( α ± ) = ε ( α − ± ) = ε (1) = 1 ,ε ( α ± ,m ) = 0 . Definition 5.2. In H + t , we let L + ( z ) = 1 + X m ∈ N × L + − m z m = exp " − ( t − t − ) X m ∈ N × α + , − m ( t z ) m , R + ( z ) = α + X m ∈ N × R + m z − m ! = α + exp " ( t − t − ) X m ∈ N × α + ,m ( t − z ) − m . EIGHT-FINITE MODULES OVER ˙U q ( a ) AND ¨U q ( a ) 35 Similarly, in H − t , we let L − ( z ) = α − X m ∈ N × L −− m z m ! = α − exp " − ( t − t − ) X m ∈ N × α − , − m ( t − z ) m , R − ( z ) = 1 + X m ∈ N × R − m z − m = exp " ( t − t − ) X m ∈ N × α − ,m ( t z ) − m . Then, we have the following equivalent presentation of H ± t . Proposition 5.3. H ± t is the Hopf algebra generated over K ( t ) by { γ / , γ − / , L ±− m , R ± m : m ∈ N } subject to the relations [ L ± ( v ) , L ± ( z )] = [ R ± ( v ) , R ± ( z )] = 0 , R ± ( v ) L ± ( z ) = θ ± ( z/v ) L ± ( z ) R ± ( v ) , where we have defined θ ± ( z ) ∈ Z ( H t )[[ z ]] , by setting θ ± ( z ) = (cid:18) (1 − t ± γz )(1 − t ± − γ − z )(1 − t ± − γz )(1 − t ± γ − z ) (cid:19) | z |≪ . Furthermore, we have ∆( L ± ( z )) = L ± ( zγ / ) ⊗ L ± ( zγ − / ) , ∆( R ± ( z )) = R ± ( zγ − / ) ⊗ R ± ( zγ / ) , where, by definition, γ / = γ / ⊗ , γ − / = γ − / ⊗ , γ / = 1 ⊗ γ / , γ − / = 1 ⊗ γ − / and S ( L ± ( z )) = L ± ( z ) − , S ( R ± ( z )) = R ± ( z ) − . Finally, ε ( L ± ( z )) = ε ( R ± ( z )) = 1 .Proof. This is an easy consequence of the definition of H ± t . (cid:3) Remark . Observe that θ + ( z ) and θ − ( z ) are not independent and that we actually have θ − ( z ) = θ + ( t − z ).5.2. A PBW basis for H ± t . For every n ∈ N × , we let Λ n := { λ = ( λ , . . . , λ n ) ∈ ( N × ) n : λ ≥ · · · ≥ λ n } denote the set of n -partitions. We adopt the convention that Λ = {∅} reduces to the empty partition andwe let Λ = S n ∈ N Λ n be the set of all partitions. Proposition 5.5.
Define, for every λ ∈ Λ , L ± λ = L ±− λ · · · L ±− λ n , (5.1) R ± λ = R ± λ · · · R ± λ n , (5.2) with the convention that L ±∅ = R ±∅ = 1 . Then, n Φ ± λ,µ = L ± λ R ± µ : λ, µ ∈ Λ o (5.3) is a K ( t )[ γ / , γ − / ] -basis for H ± t . Proof.
The relations in H ± t read, for every m, n ∈ N ,[ L ±− m , L ±− n ] = [ R ± m , R ± n ] = 0 ,R ± m L ±− n = L ±− n R ± m + min( m,n ) X p =1 θ ± p L ± p − n R ± m − p , where, for every p ∈ N , θ ± p ∈ K ( t )[ γ / , γ − / ] can be obtained from θ ± ( z ) = 1 + X p ∈ N × θ ± p z p . It is clear that any monomial in { L ±− m , R ± m : m ∈ N } can therefore be rewritten as a linear combination withcoefficients in K ( t )[ γ / , γ − / ] of elements in { φ ± λ,µ : λ, µ ∈ Λ } . The independence of the latter is clear. (cid:3) A convenient way to encode the above basis elements is through H ± t -valued symmetric formal distributions.Let indeed, for every n + , n − , m + , m − ∈ N , every n ± -tuple z ± = ( z ± , . . . , z ± n ± ) and every m ± -tuple ζ ± =( ζ ± , . . . , ζ ± m ± ) of formal variables, Φ ± ( z ± , ζ ± ) = L ± ( z ± ) R ± ( ζ ± ) , where we have set L ± ( z ± ) = n ± Y p =1 L ± ( z ± p ) , R ± ( ζ ± ) = m ± Y p =1 R ± ( ζ ± p ) , with the convention that if n ± (resp. m ± = 0), then L ± ( ∅ ) = 1 (resp. R ± ( ∅ ) = 1). It turns out thatΦ ± ( z ± , ζ ± ) ∈ H ± t [[ z ± , ( ζ ± ) − ]] S n ± × S m ± . Indeed, owing to the commutation relations in H ± t , the formal distribution Φ ± ( z ± , ζ ± ) is symmetric in eachof its argument tuples, z ± and ζ ± respectively; i.e. it is invariant under the natural action of S n ± × S m ± onits arguments. It is also clear that, for every λ ± ∈ Λ n ± and µ ± ∈ Λ m ± ,Φ ± λ ± ,µ ± = res z ± , ζ ± ( z ± ) − − λ ± ( ζ ± ) − µ ± Φ ± ( z ± , ζ ± ) , where we have set ( z ± ) − − λ ± = n ± Y p =1 ( z ± p ) − − λ ± p and ( ζ ± ) − µ ± = m ± Y p =1 ( ζ ± p ) − µ ± p . The dressing factors L ± m ( z ) and R ± m ( z ) .Definition 5.6. For every m ∈ Z × , we let L ± m ( z ) = | m | Y p =1 L ± ( zt ± − p )sign( m )+2 ) ± sign( m ) (5.4) R ± m ( z ) = | m | Y p =1 R ± ( zt ± − p )sign( m )+2 ) ± sign( m ) (5.5)It easily follows that EIGHT-FINITE MODULES OVER ˙U q ( a ) AND ¨U q ( a ) 37 Proposition 5.7. In H ± t , for every m, n ∈ Z × , we have [ L ± m ( v ) , L ± n ( z )] = [ R ± m ( v ) , R ± n ( z )] = 0 , R ± m ( v ) L ± n ( z ) = θ ± m,n ( z/v ) L ± n ( z ) R ± m ( v ) , where we have set θ ± m,n ( z ) = | m | Y r =1 | n | Y s =1 θ ± ( zt ± − s )sign( n ) ∓ − r )sign( m ) ) sign( mn ) . Furthermore, we have, for every m ∈ Z × , ∆( L ± m ( z )) = L ± m ( zγ / ) ⊗ L ± m ( zγ − / ) , ∆( R ± m ( z )) = R ± m ( zγ − / ) ⊗ R ± m ( zγ / ) . It is worth emphasizing that the L ± m ( z ) are not indepedent for all values of m ∈ Z × and that neither arethe R ± m ( z ). Indeed, we have Lemma 5.8.
For every m, n ∈ Z × , L ±− m ( z ) − = L ± m ( zt ± m ) (5.6) R ±− m ( z ) − = R ± m ( zt ± m ) (5.7) L ± m ( zt ± m ) L ± n ( z ) = L ± m + n ( zt ± m ) (5.8) R ± m ( zt ± m ) R ± n ( z ) = R ± m + n ( zt ± m ) (5.9)5.4. The algebra B t . Remember the Hopf algebra ˘U q (L a ) from definition 3.20. It has an invertibleantipode and we denote by ˘U q (L a ) cop its coopposite Hopf algebra. Proposition 5.9.
The quantum Heisenberg algebra H + t (resp. H − t ) is a left ˘U t (L a ) -module algebra (resp.a left ˘U t (L a ) cop -module algebra) with k ε ( v ) ⊲ γ / = k ε ( v ) ⊲ γ − / = 0 , k ε ( v ) ⊲ L ± ( z ) = λ ε, ± ( v, z ) L ± ( z ) , k ε ( v ) ⊲ R ± ( z ) = ρ ε, ± ( v, z ) R ± ( z ) , x ε ( v ) ⊲ γ / = x ε ( v ) ⊲ γ − / = x ε ( v ) ⊲ L ± ( z ) = x ε ( v ) ⊲ R ± ( z ) = 0 , for ε ∈ {− , + } and where we have set λ ε, ± ( v, z ) = (cid:18) t ∓ v − t − ± zv − t ± z (cid:19) | z/v | ε ≪ and ρ ε, ± ( v, z ) = (cid:18) t ± v − zt ± v − t − (2 ± z (cid:19) | z/v | ε ≪ . Proof.
One readily checks the compatibility with the defining relations of H ± t and ˘U t (L a ). (cid:3) Proposition 5.10.
For every m ∈ Z × and every ε ∈ {− , + } , we have k ε ( v ) ⊲ L ± m ( z ) = λ ε, ± m ( v, z ) L ± m ( z ) , k ε ( v ) ⊲ R ± m ( z ) = ρ ε, ± m ( v, z ) R ± m ( z ) , x ε ( v ) ⊲ L ± m ( z ) = x ε ( v ) ⊲ R ± m ( z ) = 0 , where we have set λ ε, ± m ( v, z ) = (cid:18) t − ∓ m v − t ± − ± m zv − t ± z (cid:19) | z/v | ε ≪ and ρ ε, ± m ( v, z ) = (cid:18) t ± v − zt ± − ∓ m v − t − ± m z (cid:19) | z/v | ε ≪ . Proof.
This is readily checked making use of definition 5.6, of the Hopf algebraic structures of ˘U t (L a )and ˘U t (L a ) cop , of the ˘U t (L a )-module algebra structures of H + t and of the ˘U t (L a ) cop -module algebrastructure of H − t . (cid:3) Definition-Proposition 5.11.
We denote by H + t ⋊ ˘U t (L a ) cop ⋉ H − t the associative F -algebra obtainedby endowing H + t ⊗ ˘U t (L a ) ⊗ H − t with the multiplication given by setting, for every h + , h ′ + ∈ H + t , every h − , h ′− ∈ H − t and every x, x ′ ∈ ˘U t (L a ),( h + ⊗ x ⊗ h − ) . (cid:0) h ′ + ⊗ x ′ ⊗ h ′− (cid:1) = X h + (cid:0) x (1) ⊲ h ′ + (cid:1) ⊗ x (2) x ′ ⊗ h − (cid:0) x (3) ⊲ h ′− (cid:1) , – see proposition 5.10 for the definition of the ˘U t (L a )-module algebra structure of H + t and of the ˘U t (L a ) cop -module algebra structure of H − t . In that algebra, { γ / − t, γ − / − t − } generates a left ideal. The lat-ter is actually a two-sided ideal since γ ± / is central and, denoting it by ( γ / − t ), we can set ˘ B t = H + t ⋊ ˘U t (L a ) cop ⋉ H − t / ( γ / − t ). Proof.
Making use of the coassociativity of the comultiplication ∆, it is very easy to prove that, with theabove defined multiplication, H + t ⋊ ˘U t (L a ) cop ⋉ H − t is actually an associative F -algebra. (cid:3) Proposition 5.12.
Setting x ⊗ x ⊗ , for every x ∈ ˘U t (L a ) , defines a unique injective K ( t ) -algebrahomomorphism ˘U t (L a ) ֒ → ˘ B t . Similarly, h h ⊗ ⊗ and h ⊗ ⊗ h define unique injective K ( t ) -algebra homomorphisms H + t ֒ → ˘ B t and H − t ֒ → ˘ B t respectively.Remark . We shall subsequently identify ˘U t (L a ), H + t and H − t with their respective images in ˘ B t underthe injective algebra homomorphisms of the above proposition. Proposition 5.14. In ˘ B t , for every m ∈ Z × and every ε ∈ {− , + } , we have the following relations ( v − t ± z )( v − t ± m − n ) z ) R ± m ( v ) L ± n ( z ) = ( v − t ± − n ) z )( v − t ± m ) z ) L ± n ( z ) R ± m ( v ) , (5.10)( zt ± − v ) k ε ( v ) L ± m ( z ) = ( zt ± − ± m − vt − ∓ m ) L ± m ( z ) k ε ( v ) , (5.11)( zt ± − v ) x ± ( v ) L ± m ( z ) = ( zt ± − ± m − vt − ∓ m ) L ± m ( z ) x ± ( v ) , (5.12) x ± ( v ) L ∓ m ( z ) = L ∓ m ( z ) x ± ( v ) , (5.13)( zt − ± m − vt ± − ∓ m ) k ε ( v ) R ± m ( z ) = ( z − vt ± ) R ± m ( z ) k ε ( v ) , (5.14)( zt − ± m − vt ± − ∓ m ) x ± ( v ) R ± m ( z ) = ( z − vt ± ) R ± m ( z ) x ± ( v ) , (5.15) x ± ( v ) R ∓ m ( z ) = R ∓ m ( z ) x ± ( v ) , (5.16) Proof.
In order to prove (5.10), it suffices to check that θ ± ( z ) = (cid:18) (1 − z )(1 − t ± z )(1 − t ± z ) (cid:19) | z |≪ mod I and that subsequently, for every m, n ∈ Z × , θ ± m,n ( z ) = (cid:18) (1 − t ± − n ) z )(1 − t ± m ) z )(1 − t ± z )(1 − t ± m − n ) z ) (cid:19) | z |≪ mod I . EIGHT-FINITE MODULES OVER ˙U q ( a ) AND ¨U q ( a ) 39 As for the equations (5.11 – 5.16), they immediately follow from the definitions of H + t ⋊ ˘U t (L a ) cop ⋉ H − t and of the actions ⊲ of ˘U t (L a ) on H + t and H − t – see proposiiton 5.10. E.g., we have, by definition, x +1 ( v ) L + m ( z ) = (cid:0) ⊗ x +1 ( v ) ⊗ (cid:1) (cid:0) L + m ( z ) ⊗ ⊗ (cid:1) = X (cid:0) x +1 ( v ) (1) ⊲ L + m ( z ) (cid:1) ⊗ x +1 ( v ) (2) ⊗ (cid:0) x +1 ( v ) (3) ⊲ (cid:1) = (cid:0) x +1 ( v ) ⊲ L + m ( z ) (cid:1) ⊗ ⊗ (cid:0) k − ( v ) ⊲ L + m ( z ) (cid:1) ⊗ x +1 ( v ) ⊗ (cid:0) k − ( v ) ⊲ L + m ( z ) (cid:1) ⊗ k − ( v ) ⊗ ε ( x +1 ( v ))1= λ + m ( v, z ) L + m ( z ) x +1 ( v ) , and x +1 ( v ) L − m ( z ) = (cid:0) ⊗ x +1 ( v ) ⊗ (cid:1) (cid:0) ⊗ ⊗ L − m ( z ) (cid:1) = X (cid:0) x +1 ( v ) (1) ⊲ (cid:1) ⊗ x +1 ( v ) (2) ⊗ (cid:0) x +1 ( v ) (3) ⊲ L − m ( z ) (cid:1) = ε ( x +1 ( v ))1 ⊗ ⊗ (cid:0) ⊲ L − m ( z ) (cid:1) + 1 ⊗ x +1 ( v ) ⊗ (cid:0) ⊲ L − m ( z ) (cid:1) + 1 ⊗ (cid:0) x +1 ( v ) ⊲ L − m ( z ) (cid:1) ⊗ k − ( v )= L − m ( z ) x +1 ( v ) , as claimed. (cid:3) Remark . In addition to the above, we obviously have in ˘ B t , all the relations of its subalgebra ˘U t (L a )and all the relations of its subalgebras H + t and H − t modulo ( γ / − t ). Definition-Proposition 5.16.
Let I be the left ideal of ˘ B t generated by (cid:26) res z ,z z − m z − n (cid:18) [ x +1 ( z ) , x − ( z )] − q − q − δ (cid:18) z z (cid:19) (cid:2) k +1 ( z ) − k − ( z ) (cid:3)(cid:19) : m, n ∈ Z (cid:27) . Then I . ˘ B t ⊆ I and I is a two-sided ideal of ˘ B t . Set B t = ˘ B t / I . Proof.
In order to prove that I . ˘ B t ⊆ I , it suffices to prove that, for any x ∈ ˘ B t , (cid:18) [ x +1 ( z ) , x − ( z )] − q − q − δ (cid:18) z z (cid:19) (cid:2) k +1 ( z ) − k − ( z ) (cid:3)(cid:19) x ∈ I . The latter easily follows by inspection, making use of the relevant relations in ˘ B t and ˘U t (L a ), namely (5.11- 5.16) and (2.3 - 2.7). (cid:3) Remark . Thus, in addition to the relations in ˘ B t , we have, in B t ,[ x +1 ( z ) , x − ( z )] = 1 q − q − δ (cid:18) z z (cid:19) (cid:2) k +1 ( z ) − k − ( z ) (cid:3) . The completion b B t of B t . Making use of its natural Z -grading, we endow B t with a topology, inthe same way as we endowed ¨U q ( a ) with its topology in section 3. We denote by b B t the correspondingcompletion. Consequently, its subalgebra H ± t inherits a topology and we denote by b H ± t its correspondingcompletion in that topology.5.6. The shift factors.Definition 5.18. In b H ± t , we define, H ± ( z ) = L ± ( z ) R ± ( z ) . Similarly, for every m ∈ Z × , we let H ± m ( z ) = | m | Y p =1 H ± ( zt ± − p )sign( m )+2 ) ± sign( m ) . Lemma 5.19.
For every m, n ∈ Z × , H ±− m ( z ) − = H ± m ( zt ± m ) (5.17) H ± m ( zt ± m ) H ± n ( z ) = H ± m + n ( zt ± m ) (5.18) Proof.
Follows directly from the definition in the same way as lemma 5.8. (cid:3)
Proposition 5.20. In b H ± t , we have, for every m, n ∈ Z × , H ± m ( z ) H ± n ( v ) = Θ ± m,n ( z, v ) H ± n ( v ) H ± m ( z ) , where Θ ± m,n ( z, v ) = ( v − t ± z )( v − t ± n − m ) z )( t ± − n ) v − z )( t ± m ) v − z )( z − t ± v )( z − t ± m − n ) v )( t ± − m ) z − v )( t ± n ) z − v ) . Proof.
In view of definition 5.18 and of the relations in proposition 5.7, it is clear that commuting H ± m ( z )and H ± n ( v ) amounts to commuting, on one hand L ± m ( z ) and R ± n ( v ) and, on the other hand, R ± m ( z ) and L ± n ( v ). The result follows. (cid:3) Proposition 5.21.
For every m ∈ Z × and every ε ∈ {− , + } , we have k ε ( v ) ⊲ H ±± m ( z ) = H εm,z ( v ) ± H ±± m ( z ) , (5.19) x ε ( v ) ⊲ H ± m ( z ) = 0 . (5.20) Proof.
The left ˘U t (L a )-module algebra (resp. a left ˘U t (L a ) cop -module algebra) structure of H + t (resp. H − t ) – see proposition 5.9 – is extended by continuity to b H + t (resp. b H − t ) Then, it suffices to check that, forevery m ∈ Z × and every ε ∈ {− , + } , k ε ( v ) ⊲ H ±± m ( z ) = λ ε, ±± m ( v, z ) ρ ε, ±± m ( v, z ) H ±± m ( z ) , and that H εm,z ( v ) ± = λ ε, ±± m ( v, z ) ρ ε, ±± m ( v, z ) . (cid:3) Corollary 5.22.
For every m ∈ Z , every p ∈ N and every ε ∈ {− , + } , we have p +1 Y k =1 (cid:2) k ε ( v k ) − H εm,z ( v k ) ± (cid:3) ⊲ ∂ p H ±± m ( z ) = 0 , Proof.
It suffices to differentiate (5.19) p times with respect to z to obtain (cid:2) k ε ( v ) − H εm,z ( v ) ± id (cid:3) ⊲ ∂ p H ±± m ( z ) = p − X k =0 (cid:18) pk + 1 (cid:19) ∂ k +1 ∂z k +1 (cid:2) H εm,z ( v ) ± (cid:3) ∂ p − k − H ±± m ( z ) . The claim immediately follows. (cid:3)
Proposition 5.23. In b B t , we have, for every m, n ∈ Z × , H + m ( z ) H − n ( v ) = H − n ( v ) H + m ( z ) , ( zt ± − v )( zt − ± m − vt ± − ∓ m ) k ε ( v ) H ± m ( z ) = ( z − vt ± )( zt ± − ± m − vt − ∓ m ) H ± m ( z ) k ε ( v ) , ( zt ± − v )( zt − ± m − vt ± − ∓ m ) x ± ( v ) H ± m ( z ) = ( z − vt ± )( zt ± − ± m − vt − ∓ m ) H ± m ( z ) x ± ( v ) , x ± ( v ) H ∓ m ( z ) = H ∓ m ( z ) x ± ( v ) . Proof.
This follows immediately from [ L ± ( z ) , L ∓ ( v )] = [ L ± ( z ) , R ∓ ( v )] = [ R ± ( z ) , R ∓ ( v )] = 0. (cid:3) EIGHT-FINITE MODULES OVER ˙U q ( a ) AND ¨U q ( a ) 41 The evaluation algebra b A t .Definition-Proposition 5.24. Let J denote the closed left ideal of b B t generated by n res z z m (cid:2) H − ( z ) (cid:0) k +1 ( zt − ) − k − ( zt − ) (cid:1) − H + ( z ) − (cid:0) k +1 ( z ) − k − ( z ) (cid:1)(cid:3) : m ∈ Z o . (5.21)Then, J . b B t ⊆ J , making J a closed two-sided ideal of b B t , and we let b A t = b B t / J . Proof.
In order to prove that J . b B t ⊆ J , it suffices to check that, for every x ∈ b B t , (cid:2) H − ( z ) (cid:0) k +1 ( zt − ) − k − ( zt − ) (cid:1) − H + ( z ) − (cid:0) k +1 ( z ) − k − ( z ) (cid:1)(cid:3) x ∈ J . The latter easily follows by inspection, making use of the relevant relations in b B t , namely (5.10–5.16) inproposition 5.14. (cid:3) Proposition 5.25.
For every m ∈ Z , the following relation holds in b A t , H −− m ( z ) (cid:2) k +1 ( zt − m ) − k − ( zt − m ) (cid:3) = H + m ( z ) − (cid:2) k +1 ( z ) − k − ( z ) (cid:3) . (5.22) Proof.
We prove (5.22) for m ∈ N × by induction. The case m = 1 corresponds to the vanishing of thegenerators of the ideal J , see (5.21). Assuming the result holds for some m ∈ N × , we have H −− ( m +1) ( z ) h k +1 ( zt − m +1) ) − k − ( zt − m +1) ) i = H − ( z ) H −− m ( zt − ) h k +1 ( zt − m +1) ) − k − ( zt − m +1) ) i = H − ( z ) H + m ( zt − ) − (cid:2) k +1 ( zt − ) − k − ( zt − ) (cid:3) = H + m ( zt − ) − H − ( z ) (cid:2) k +1 ( zt − ) − k − ( zt − ) (cid:3) = H + m ( zt − ) − H + ( z ) − (cid:2) k +1 ( z ) − k − ( z ) (cid:3) = H + m +1 ( z ) − (cid:2) k +1 ( z ) − k − ( z ) (cid:3) The cases with m ∈ − N × follow by rewriting the above equation for m ∈ N × as H + m ( z ) (cid:2) k +1 ( zt − m ) − k − ( zt − m ) (cid:3) = H −− m ( z ) − (cid:2) k +1 ( z ) − k − ( z ) (cid:3) and making use of lemma 5.8. (cid:3) Remark . In addition to the above relation, b A t obviously inherits the relations in b B t modulo J . Inparticular, all the relations in proposition 5.14 hold in b A t .5.8. The evaluation homomorphism.Proposition 5.27.
There exists a unique continuous K -algebra homomorphism ev : ¨U q ( a ) ( − → b A t suchthat, for every m ∈ N × and every n ∈ Z , ev( q ) = t , (5.23)ev( K ± , ( z )) = − k ∓ ( z ) , (5.24)ev( K ± , ± m ( z )) = H ±± m ( z ) (cid:2) k ± ( zt − m ) − k ∓ ( zt − m ) (cid:3) , (5.25)ev( X ± ,n ( z )) = H ± n ( z ) x ± ( zt ∓ n ) . (5.26) We shall refer to ev as the evaluation homomorphism . It is such that ev ◦ ι = id over U t (L a ) .Proof. It suffices to check all the defining relations of ¨U q ( a ). E.g. we have, for every m, n ∈ Z , (cid:2) ev( X + m ( v )) , ev( X − n ( z )) (cid:3) = 1 t − t − δ (cid:16) vzt m + n ) (cid:17) H + m ( v ) H − n ( z ) (cid:2) k +1 ( vt − m ) − k − ( zt n ) (cid:3) . (5.27) If m + n = 0, making use of (5.22), we are done. Assuming that m + n >
0, lemma 5.8 allows us to write H + m ( zt m + n ) ) H − n ( z ) (cid:2) k +1 ( zt n ) − k − ( zt n ) (cid:3) = H + m ( zt m + n ) ) H + − n ( z ) − (cid:2) k +1 ( z ) − k − ( z ) (cid:3) = H + m ( zt m + n ) ) H + n ( zt n ) (cid:2) k +1 ( z ) − k − ( z ) (cid:3) = H + m + n ( zt m + n ) ) (cid:2) k +1 ( z ) − k − ( z ) (cid:3) so that, eventually, (cid:2) ev( X + m ( v )) , ev( X − n ( z )) (cid:3) = 1 t − t − δ (cid:16) vzt m + n ) (cid:17) ev( K + m + n ( zt m + n ) )) . A similar argument proves the case m + n < (cid:3) We have the following obvious
Corollary 5.28.
For every N ∈ N there exists an algebra homomorphism ev ( N ) : ¨U q ( a ) ( N ) → b A t makingthe following diagram commutative. · · · ¨U q ( a ) ( N ) ¨U q ( a ) ( N − · · · ¨U q ( a ) ( − b A t ev ( N ) ev ( N − ev We can furthermore define the algebra homomorphism ev ( ∞ ) : ¨U q ( a ) → b A t by ev ( ∞ ) = lim ←− ev ( N ) . Evaluation modules.
Remember the surjective algebra homomorphism ˘U q (L a ) → U q (L a ) fromproposition 2.2. It allows us to pull back any simple U q (L a )-module M into a simple ˘U q (L a )-module.With that construction in mind, we have Proposition 5.29.
Let M be a simple finite dimensional U q (L a ) -module. Then,i. b H + t ⊗ M ⊗ b H − t is a H + t ⋊ ˘U t (L a ) cop ⋉ H − t -module with the action defined by setting, for every h + , h ′ + ∈ H + t , every h − , h ′− ∈ H − t , every x ∈ ˘U t (L a ) and every v ∈ M , ( h + ⊗ x ⊗ h − ) . ( h ′ + ⊗ v ⊗ h ′− ) = X h + (cid:0) x (1) ⊲ h ′ + (cid:1) ⊗ x (2) .v ⊗ h − (cid:0) x (3) ⊲ h ′− (cid:1) and extending by continuity.ii. b H + t ⊗ M ⊗ b H − t descends to a B t -module.iii. (cid:16) b H + t ⊗ M ⊗ b H − t (cid:17) / J . (cid:16) b H + t ⊗ M ⊗ b H − t (cid:17) is an b A t -module. It pulls back along ev to a ¨U ′ q ( a ) -modulethat we denote by ev ∗ ( M ) .iv. As a ¨U ′ q ( a ) -module, ev ∗ ( M ) is weight-finite.v. For any highest ℓ -weight vector v ∈ M − { } , the ¨U q ( a ) -module ˜ M ∼ = (cid:16) b H + t ⊗ F v ⊗ b H − t (cid:17) / J . (cid:16) b H + t ⊗ F v ⊗ b H − t (cid:17) , is a highest t -weight space of ev( M ) . We denote by M the simple quotient of ˜ M containing v andwe let ev ∗ ( M ) = ¨U ′ q ( a ) .M .vi. M is t -dominant.Proof. i. is readily checked. As for ii., it suffices to check that I . (cid:16) b H + t ⊗ M ⊗ b H − t (cid:17) = { } . But the latteris clear when M is obtained by pulling back a U q (L a )-module over which the relation generating I is EIGHT-FINITE MODULES OVER ˙U q ( a ) AND ¨U q ( a ) 43 automatically satisfied. iii. is obvious. It easily follows from proposition 5.23 that, for every m ∈ Z × ,[ k ε , , H ± m ( z )] = 0. Hence, Sp(ev ∗ ( M )) = Sp( M ) and the weight finiteness of ev ∗ ( M ) follows from that of M ,which proves iv.. It is clear that, for every r ∈ Z , we haveev( X +1 ,r ( z )) . (cid:0) H + t ⊗ v ⊗ H − t (cid:1) = H + r ( z ) x +1 ( zt − r ) . (cid:16) b H + t ⊗ v ⊗ b H − t (cid:17) = X H + r ( z ) (cid:16) x +1 ( zt − r ) (1) ⊲ b H + t (cid:17) ⊗ x +1 ( zt − r ) (2) .v ⊗ (cid:16) x +1 ( zt − r ) (3) ⊲ b H − t (cid:17) = 0 . v. follows. Denote by P (1 /z ) ∈ F [ z − ] the Drinfel’d polynomial associated with v and let ν ∈ N F × f denotethe multiset of its roots. Then, k ± ( z ) .v = − κ ∓ ( z ) v , where κ ∓ ( z ) = − t P ) (cid:18) P ( t − /z ) P (1 /z ) (cid:19) | z | ∓ ≪ . (5.28)Moreover, the partial fraction decomposition P ( t − /z ) P (1 /z ) = Y a ∈ F × − a/z ) ν ( a ) − ν ( at ) = C + X a ∈ F × ν ( a ) − ν ( at ) X p =1 C p ( a )(1 − a/z ) p , in which C , C p ( a ) ∈ F and the product and sum over a ∈ F × are always finite since P only has finitelymany roots, allows us to write (cid:2) k +1 ( z ) − k − ( z ) (cid:3) .v = t P ) X a ∈ F × ν ( a ) − ν ( at ) − X p =0 ( − p +1 C p +1 ( a ) p ! a p +1 δ ( p ) (cid:16) za (cid:17) v . Letting ˜ C p ( a ) = ( − p +1 t P ) C p +1 ( a ) a − p − /p ! for every a ∈ F × and every p ∈ J , ν ( a ) − ν ( at ) − K , itfollows that, for every m ∈ N × ,ev( K +1 ,m ( z )) . (1 ⊗ v ⊗
1) = t P ) X a ∈ F × ν ( at − m ) − ν ( at − m ) ) − X p =0 ˜ C p ( at − m ) δ ( p ) (cid:16) za (cid:17) (cid:0) H + m ( z ) ⊗ v ⊗ (cid:1) , (5.29)ev( K − , − m ( z )) . (1 ⊗ v ⊗
1) = − t P ) X a ∈ F × ν ( at − m ) − ν ( at − m ) ) − X p =0 ˜ C p ( at − m ) δ ( p ) (cid:16) za (cid:17) (cid:0) ⊗ v ⊗ H −− m ( z ) (cid:1) . (5.30)Now, making use of (5.24), (5.28) and of corollary 5.22, one easily shows that, for every p ∈ N and every a ∈ F × , p +1 Y k =1 (cid:2) ev( K ± , ( z k )) − H ± m,a ( z k ) κ ± ( z k ) id (cid:3) . (cid:0) ∂ p H + m ( a ) ⊗ v ⊗ (cid:1) = 0 , p +1 Y k =1 (cid:2) ev( K ± , ( z k )) − H ± m,a ( z k ) − κ ± ( z k ) id (cid:3) . (cid:0) ⊗ v ⊗ ∂ p H −− m ( a ) (cid:1) = 0 , thus proving that ∂ p H + m ( a ) ⊗ v ⊗ ⊗ v ⊗ ∂ p H −− m ( a )) is an ℓ -weight vector in the ℓ -weightspace ev ∗ ( M ) κ (+ ,m,a ) (resp. ev ∗ ( M ) κ ( − ,m,a ) ) of ev ∗ ( M ) with ℓ -weight κ ± (+ ,m,a ) ( z ) = κ ± ( z ) H ± m,a ( z ) (resp. κ ± ( − ,m,a ) ( z ) = κ ± ( z ) H ± m,a ( z ) − ), as expected from proposition 4.3. On the other hand, (cid:8) H − ( z ) (cid:2) k +1 ( zt − ) − k − ( zt − ) (cid:3) − H + ( z ) − (cid:2) k +1 ( z ) − k − ( z ) (cid:3)(cid:9) . (1 ⊗ v ⊗ X a ∈ F × ν ( at − ) − ν ( a ) − X p =0 ˜ C p ( at − ) δ ( p ) (cid:16) za (cid:17) (cid:0) ⊗ v ⊗ H − ( z ) (cid:1) − ν ( a ) − ν ( at ) − X p =0 ˜ C p ( a ) δ ( p ) (cid:16) za (cid:17) (cid:0) H + ( z ) − ⊗ v ⊗ (cid:1) . Thus, modulo J , we have, for every a ∈ F × , ν ( at − ) − ν ( a ) − X p =0 ˜ C p ( at − ) δ ( p ) (cid:16) za (cid:17) (cid:0) ⊗ v ⊗ H − ( z ) (cid:1) = ν ( a ) − ν ( at ) − X p =0 ˜ C p ( a ) δ ( p ) (cid:16) za (cid:17) (cid:0) H + ( z ) − ⊗ v ⊗ (cid:1) . The above equation makes it clear that every a ∈ F × such that ν ( at − ) > ν ( a ) is a zero of order at least ν ( at − ) − ν ( a ) + ν ( at ) of 1 ⊗ v ⊗ H − ( z ), unless ν ( at − ) − ν ( a ) ≤ ν ( a ) − ν ( at ). Hence, in view of (5.30), wehave ev ∗ ( M ) κ ( − , ,a ) ∩ ev( K − , − ( z )) . (1 ⊗ v ⊗
1) = 0 unless a ∈ D ( ν ) = { x ∈ F × : ν ( xt − ) > ν ( x ) > ν ( xt ) } .But the latter implies that P (1 /a ) = 0. A similar reasoning applies to any ℓ -weight vector in ˜ M and˜ M is t -dominant by lemma 4.15. Taking the quotient of ˜ M to M clearly preserves t -dominance and vi.follows. (cid:3) By the universality of M ( M ) – see definition 4.20 – and the above proposition, there must exist asurjective ¨U ′ q ( a )-module homomorphism π : M ( M ) ։ ev ∗ ( M ). Restricting the latter to the (closed)¨U ′ q ( a )-submodule N ( M ) of M ( M ), we get the surjective ¨U ′ q ( a )-module homomorphism π |N ( M ) , whoseimage naturally injects as a ¨U ′ q ( a )-submodule in ev ∗ ( M ). The canonical short exact sequence involving N ( M ), M ( M ) and the simple quotient L ( M ) – see definition 4.20 – allows us to define a surjective¨U ′ q ( a )-module homomorphism ˜ π to get the following commutative diagram, { } N ( M ) M ( M ) L ( M ) { }{ } π ( N ( M )) ev ∗ ( M ) ev ∗ ( M ) /π ( N ( M )) { }{ } { } { } π |N ( M π ˜ π where columns and rows are exact. It is obvious that ˜ π is not identically zero and, by the simplicity of L ( M ), we must have ker(˜ π ) = { } . Hence, ˜ π is a ¨U ′ q ( a )-module isomorphism and we have constructed thesimple weight-finite ¨U ′ q ( a )-modules L ( M ) as a quotient of the evaluation module ev ∗ ( M ). To see that allthe simple weight-finite ¨U ′ q ( a )-modules L ( M ) can be obtained in this way, it suffices to observe that, byproposition 4.10, all the simple ℓ -dominant ¨U q ( a )-modules are of the form L ( P ) for some monic polynomial P and that, in the construction above, one can choose any P , simply by choosing the corresponding simplefinite-dimensional U q (L a )-module M . Therefore, as a consequence of the above proposition, the highest t -weight space of any simple weight-finite ¨U ′ q ( a )-modules L ( M ) is t -dominant. This concludes the proof ofpart ii. of theorem 4.21 as well as that of theorem 4.12. EIGHT-FINITE MODULES OVER ˙U q ( a ) AND ¨U q ( a ) 45 Appendix A. A lemma about formal distributions
In this short appendix, we prove the following
Lemma A.1.
Let m ∈ { , } and n ∈ N , let A ( v ) ∈ F [[ v ]] − { } be a non-zero formal power series and let F ( z ) ∈ F [[ z, z − ]] be a formal distribution such that ( z − a )( z − v ) m A ( v ) F ( z ) + n X p =0 B p ( v ) δ ( p ) ( z/a ) = 0 , (A.1) for some non-zero scalar a ∈ F × and some formal power series B ( v ) , . . . , B n ( v ) ∈ F [[ v ]] . Then, F ( z ) = n +1 X p =0 f p δ ( p ) ( z/a ) , for some scalars f , . . . , f n +1 ∈ F .Proof. Consider first the case where m = 0. Then, multiplying (A.1) by ( z − a ) n +1 , we get( z − a ) n +2 A ( v ) F ( z ) = 0 . Since A ( v ) = 0, we can specialize at a non-zero v -mode and it follows that F ( z ) = n +1 X p =0 f p δ ( p ) ( z/a )for some scalars f , . . . , f n +1 ∈ F . Now consider the case where m = 1. It follows from (A.1) that( z − a ) A ( v ) F ( z ) + (cid:18) z − v (cid:19) | v/z |≪ n X p =0 B p ( v ) δ ( p ) ( z/a ) = C ( z ) δ ( z/v ) , for some formal distribution C ( z ) ∈ F [[ z, z − ]]. But specializing the l.h.s. of the above equation to any v -mode of the form v − p with p ∈ N × , we immediately get that C ( z ) = 0. We are thus back to the previouscase. (cid:3) References [CP91] V. Chari and A. Pressley,
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