aa r X i v : . [ m a t h . QA ] A ug THE FORMAL SHIFT OPERATOR ON THE YANGIAN DOUBLE
CURTIS WENDLANDT
Abstract.
Let g be a symmetrizable Kac–Moody algebra with associatedYangian Y ~ g and Yangian double D Y ~ g . An elementary result of fundamentalimportance to the theory of Yangians is that, for each c ∈ C , there is anautomorphism τ c of Y ~ g corresponding to the translation t t + c of thecomplex plane. Replacing c by a formal parameter z yields the so-called formalshift homomorphism τ z from Y ~ g to the polynomial algebra Y ~ g [ z ].We prove that τ z uniquely extends to an algebra homomorphism Φ z fromthe Yangian double D Y ~ g into the ~ -adic closure of the algebra of Laurentseries in z − with coefficients in the Yangian Y ~ g . This induces, via evaluationat any point c ∈ C × , a homomorphism from D Y ~ g into the completion of theYangian with respect to its grading. We show that each such homomorphismgives rise to an isomorphism between completions of D Y ~ g and Y ~ g and, as acorollary, we find that the Yangian Y ~ g can be realized as a degeneration ofthe Yangian double D Y ~ g . Using these results, we obtain a Poincar´e–Birkhoff–Witt theorem for D Y ~ g applicable when g is of finite type or of simply-lacedaffine type. Contents
1. Introduction 12. Yangians and Yangian doubles 63. Derived subalgebras and classical limits 104. Extending the shift automorphism 145. Isomorphism with completed Yangian 216. D Y ~ g as a flat deformation 29Appendix A. Grading completions 36References 381. Introduction Y ~ g associated to a symmetriz-able Kac–Moody algebra g , after Khoroshkin and Tolstoy [24], by taking the ap-proach that it should be characterized in terms of the underlying Yangian Y ~ g . Ourmain results realize such a characterization by showing that D Y ~ g can be viewedas both a dense subalgebra of the completion of the Yangian Y ~ g with respect toits N -grading, and as the closure of a Z -graded subalgebra of the space of formalLaurent series in z − with coefficients in Y ~ g . As a particular consequence of thisdescription, we obtain a uniform Poincar´e–Birkhoff–Witt theorem for the Yangian Mathematics Subject Classification.
Primary 17B37; Secondary 17B67, 81R10. double D Y ~ g of an arbitrary finite-dimensional or simply laced affine Kac–Moodyalgebra. These results are based on the construction of an extension Φ z of theformal shift homomorphism τ z on the Yangian to the Yangian double D Y ~ g . Herewe recall that τ z is a graded algebra embedding τ z : Y ~ g → Y ~ g [ z ]which gives rise to an action of the group of translations of the complex plane on Y ~ g ; see (2.15) and (2.16). This action dates back to the foundational work ofDrinfeld [2] and has become ubiquitous in the theory of Yangians.1.2. For the purpose of motivating our construction, let us first consider its clas-sical counterpart with g taken to be a complex semisimple Lie algebra. Under thisassumption, the Yangian Y ~ g and Yangian double D Y ~ g are graded deformationsof the enveloping algebras for the current algebra g [ t ] and loop algebra g [ t, t − ],respectively, and τ z provides a quantization of the embedding γ z : g [ t ] → g [ t, z ]sending any polynomial f ( t ) to its translate f ( t + z ). Note that this is a gradedhomomorphism, provided t and z are both given degree 1. As t + z is an invertibleelement in the ring C [ t ][ z ; z − ]] of Laurent series in z − with coefficients in C [ t ], γ z uniquely extends to a graded Lie algebra homomorphismΥ z : g [ t, t − ] → M n ∈ Z z n g [[ t/z ]] ⊂ g [ t ][ z ; z − ]] , where g [[ t/z ]] is the image of the embedding g [[ w ]] → g [ t ][ z ; z − ]] sending f ( w ) to f ( t/z ). This homomorphism is injective and possesses a number of propertieswhich elucidate the intimate connection shared by g [ t ] and g [ t, t − ]. For instance,the formal parameter z may be evaluated to any c ∈ C × to yield a family of Liealgebra embeddings Υ c : g [ t, t − ] → g [[ t ]] . Each member Υ c of this family restricts to an automorphism of g [ t ] and uniquelyextends to an isomorphism b Υ c : \ g [ t, t − ] ∼ −→ g [[ t ]]when g [ t, t − ] is completed with respect to the descending filtration given by thelower central series for the evaluation ideal J c = ( t − c ) g [ t, t − ]. In addition, Υ c induces an isomorphism of N -graded Lie algebrasgr(Υ c ) : gr( g [ t, t − ]) ∼ −→ M n ≥ t n g [[ t ]] /t n +1 g [[ t ]] ∼ = g [ t ]which realizes g [ t ] as a degeneration of g [ t, t − ].The results of the present paper provide a quantization Φ z of Υ z admitting coun-terparts to each of the above properties and satisfying the commutative diagram(1.1) D Y ~ g L d Y ~ g z Y ~ g Φ z ı τ z HE FORMAL SHIFT OPERATOR ON THE YANGIAN DOUBLE 3 where ı is a quantization of the natural inclusion g [ t ] ⊂ g [ t, t − ] and L d Y ~ g z is a sub-algebra of the ~ -adic completion of Y ~ g [ z ; z − ]], described explicitly in Proposition4.2, which plays the role of the graded Lie algebra L n ∈ Z z n g [[ t/z ]].1.3. When g is an infinite-dimensional symmetrizable Kac–Moody algebra, theYangian Y ~ g and Yangian double D Y ~ g no longer deform the enveloping algebrasof the respective Lie algebras g [ t ] and g [ t, t − ]. They do, however, deform theenveloping algebras of semidirect products s ⋊ ¨ h and t ⋊ ¨ h , where ¨ h is a finite-dimensional abelian subalgebra of g , and s and t are perfectLie algebras which project onto the current algebra ˙ g [ t ] and loop algebra ˙ g [ t, t − ],respectively, of the derived subalgebra ˙ g = [ g , g ]. In general, the kernel of theseprojections is large and one cannot a priori put too much stock in the classicalstory outlined above. Despite this fact, our construction of Φ z remains completelyvalid, and we exploit it as an effective and simple algebraic tool for studying theYangian double D Y ~ g in full generality. In fact, one of the main goals of our workis to further develop the algebraic theory of D Y ~ g when g is an untwisted affineLie algebra. In this case, s and t are non-trivial central extensions of ˙ g [ t ] and˙ g [ t, t − ], respectively, which admit entirely concrete descriptions, and D Y ~ g may beviewed as a rational, level zero, analogue of the so-called quantum toroidal algebraassociated to g .The associated affine Yangians were first studied in detail in the work of Guay[10–12] in type A , which in particular illuminated their connection to both rationaland trigonometric Cherednik algebras, as well as deformed double current alge-bras. They have since been afforded a more general treatment in what is now arapidly growing body of literature. This includes, but is not limited to, the list ofcontributions [1, 14, 16, 25–27, 32, 33, 35–37].1.4. Let us now outline our main results in detail. Let g be a symmetrizable Kac–Moody algebra, and let d Y ~ g denote the formal completion of the Yangian Y ~ g withrespect to its N -grading. In this article, we prove the following theorem. Theorem 1.1.
There is a unique algebra homomorphism Φ z : D Y ~ g → L d Y ~ g z satisfying the commutative diagram (1.1) . Moreover: (1) Φ z is a Z -graded algebra homomorphism. (2) Φ z evaluates at any z = c ∈ C × to an algebra homomorphism Φ c : D Y ~ g → d Y ~ g which uniquely extends the shift automorphism τ c of Y ~ g . (3) Each specialization Φ c of Φ z determines an isomorphism b Φ c : \ D Y ~ g c ∼ −→ d Y ~ g , where D Y ~ g is completed with respect to its evaluation ideal at t = c . (4) Φ z induces an isomorphism of N -graded algebras gr(D Y ~ g ) ∼ −→ Y ~ g , where D Y ~ g is filtered by powers of its evaluation ideal at t = 1 . C. WENDLANDT
This theorem is the amalgamation of two of the three main results establishedin this paper. Our first main result, Theorem 4.3, outputs our main tool: a uniqueextension Φ z of τ z satisfying (1) and (2). Our second main result, Theorem 5.5,then proves that Φ = Φ induces an isomorphism b Φ : \ D Y ~ g ∼ −→ d Y ~ g , where \ D Y ~ g is the completion of D Y ~ g with respect to its evaluation ideal J at t = 1. This is precisely the assertion of (3) in the special case where c = 1, andis generalized to an arbitrary evaluation point c ∈ C × in Corollary 5.7, using thatΦ may be transformed into Φ c by conjugating by a gradation automorphism, asproven in Proposition 4.7.Part (4) of the above theorem provides the Yangian double analogue of Drinfeld’sresult [3], proven by Guay and Ma in [13], that the Yangian Y ~ g may be realized asa degeneration of the quantum loop algebra U ~ ( L g ). It is established in Corollary5.10 as an application of Theorem 5.5. As another byproduct of Theorem 5.5, wefind in Corollary 5.9 that ı extends to an isomorphism b ı : d Y ~ g × ∼ −→ \ D Y ~ g , where d Y ~ g × is the completion of Y ~ g with respect to its own evaluation ideal at t = 1. As explained in Section 5.3, this affords the completed Yangian double arather precise description.Our third main result is provided by Theorem 6.2, which outputs the followingPoincar´e–Birkhoff–Witt theorem for D Y ~ g . Theorem 1.2.
Let g be a symmetrizable Kac–Moody algebra of finite type or ofsimply-laced affine type. Then: (1) Φ z and Φ c are injective for each c ∈ C × . (2) D Y ~ g is a flat deformation of U ( t ⋊ ¨ h ) over C [[ ~ ]] . In particular, there is anisomorphism of C [[ ~ ]] -modules D Y ~ g ∼ = U ( t ⋊ ¨ h )[[ ~ ]] . When g is finite-dimensional, the abelian Lie algebra ¨ h vanishes and t coincideswith the loop algebra g [ t, t − ]. In this case, Part (2) of this theorem improves upon[4, Thm. 1.5], which established that the positive part D Y + ~ g of the Yangian doubleD Y ~ g is topologically free. It is also a close relative of [5, Prop. 5.4] which, inthe particular setting outlined in [5, Rem. 8], shows that a quantum algebra closelyrelated to the so-called centrally extended Yangian double [23] has a similar flatnessproperty. When g is taken to be a classical Lie algebra of type B , C or D , Part (2)of Theorem 1.2 is in fact a consequence of Theorems 3.4 and 6.2 from the recentarticle [19]. In the type A setting, this should instead follow from the Poincar´e–Birkhoff–Witt result established in Theorem 2.2 of [18] (see also [31, Thm. 15.3])for the Yangian double of gl N in its R -matrix presentation, and the identificationobtained in [17, Cor. 3.5]. The proof given in the present paper does not rely onthese results, and applies uniformly in all Dynkin types.In the affine setting, there does not appear to be any counterpart to either partof Theorem 1.2 which exists in the literature. Our proof applies the recent resultsof [16] and [37], and ultimately reduces to a detailed computation of the classicallimit of Φ, which we prove is injective under the more general hypothesis that g is HE FORMAL SHIFT OPERATOR ON THE YANGIAN DOUBLE 5 of untwisted affine type with underlying simple Lie algebra ¯ g ≇ sl . Our argumentsexploit the fact that, for any such g , the Lie algebras s and t admit perfectly tangibledescriptions. Namely, due to a result of Moody, Rao and Yokonuma [30], one has s ∼ = uce ( ˙ g [ t ]) ∼ = uce (¯ g [ v ± , t ]) and t κ ∼ = uce ( ˙ g [ t, t − ]) ∼ = uce (¯ g [ v ± , t ± ]) , where t κ is a one-dimensional central extension of t , and uce ( a ) denotes the universalcentral extension of a given perfect Lie algebra a . Universal central extensions ofthis type were realized concretely in the work of Kassel [21], and this description isrecalled in the course of our proof of Theorem 6.2: see Sections 6.3–6.5.1.5. The results obtained in this paper, coupled with the findings of [9], lay thefoundation for a uniform proof of a conjecture from the pioneering work [24] ofKhoroshkin and Tolstoy. This is the assertion that, when g is finite-dimensional,D Y ~ g coincides with the restricted quantum double of the Yangian Y ~ g .Our interest in this conjecture stems, in part, from a desire to understand theuniversal R -matrix of the Yangian from a more familiar Hopf-theoretic point ofview. This is a remarkable formal series R ( z ) ∈ ( Y ~ g ⊗ Y ~ g )[[ z − ]], introduced byDrinfeld in [2], which has played a central role in many of the developments at theheart of the representation theory of Yangians. It is not, however, understood tobe a universal R -matrix in the traditional sense and, in particular, has not beenshown to arise as the canonical tensor associated to a Hopf pairing. On the otherhand, the universal R -matrix R associated to the restricted quantum double of theYangian Y ~ g has these properties by construction.In the sequel [34] to this paper, we will show that there is a unique Hopf algebrastructure on D Y ~ g preserved by Φ z and that, when equipped with this structure,D Y ~ g is isomorphic to the restricted quantum double of Y ~ g , as conjectured in [24].Using this identification, we will establish that R and R ( z ) are in fact one and thesame. More precisely, one has the equality(Φ v ⊗ Φ z ) R = R ( v − z ) ∈ ( Y ~ g ⊗ Y ~ g )[ v ][[ z − ]] . GTL : U ~ ( L g ) → d Y ~ g which has several remarkable properties. In particular, when g is finite-dimensional,it induces isomorphisms b Φ GTL : \ U ~ ( L g ) ∼ −→ d Y ~ g and gr(Φ GTL ) : gr( U ~ ( L g )) ∼ −→ Y ~ g , where U ~ ( L g ) is both completed and filtered with respect to its evaluation ideal at t = 1: see Theorem 6.2 and Proposition 6.5 of [7]. Combining Theorem 1.1 withthe results of [7], we obtain an algebra homomorphismΨ = b Φ − ◦ Φ GTL : U ~ ( L g ) → \ D Y ~ g which extends to an isomorphism between the evaluation completions of U ~ ( L g ) andD Y ~ g . It may be viewed as a filtered map with associated graded map providing anisomorphism between gr( U ~ ( L g )) and gr(D Y ~ g ), both of which may be identifiedwith the Yangian Y ~ g . As U ~ ( L g ) and D Y ~ g both deform the enveloping algebra of C. WENDLANDT the loop algebra g [ t, t − ], it is perhaps natural to speculate on whether or not thiscomposition can be viewed as an isomorphism between U ~ ( L g ) and D Y ~ g , withoutany completions at play. Though we do not consider Ψ in any detail in the presentarticle, we note in passing that this is easily seen not to be the case, even afterreducing modulo ~ .1.7. Outline.
In Section 2, we review the definitions and basic properties of theYangian Y ~ g and Yangian double D Y ~ g associated to a symmetrizable Kac–Moodyalgebra g . Our preliminary overview continues in Section 3, where we introducethe Yangian Y ~ ˙ g and Yangian double D Y ~ ˙ g of ˙ g = [ g , g ], in addition to the Liealgebras s ⋊ ¨ h and t ⋊ ¨ h . In Section 4, we construct the unique extension Φ z of τ z and its specialization Φ c at any invertible complex number c . We then show inSection 5 that each homomorphism Φ c induces an isomorphism between the eval-uation completion of D Y ~ g at the point c and the completion of Y ~ g with respectto its natural N -grading. In Section 6, we prove our final main result, which si-multaneously establishes the injectivity of Φ z and Φ c , for any c ∈ C × , and thePoincar´e–Birkhoff–Witt theorem for D Y ~ g , when g is of finite type or simply-lacedaffine type. Finally, Appendix A contains the proof of a technical result on gradingcompletions used in the proof of Lemma 4.1 of Section 4.1.1.8. Acknowledgments.
The author gratefully acknowledges the support of theNatural Sciences and Engineering Research Council of Canada (NSERC) providedvia the postdoctoral fellowship (PDF) program. He would also like to thank SachinGautam for several helpful comments and insightful discussions.2.
Yangians and Yangian doubles
Let g be a symmetrizable Kac–Moody algebra with indecomposable Cartan ma-trix A = ( a ij ) i ∈ I . We fix a realization ( h , { α i } i ∈ I , { α ∨ i } i ∈ I ) of A as in [20, § h is a Cartan subalgebra of g , { α i } i ∈ I ⊂ h ∗ is the set of simple roots,and { α ∨ i } i ∈ I ⊂ h the set of simple coroots, so that α j ( α ∨ i ) = a ij for all i, j ∈ I .Let Q = L i ∈ I Z α i ⊂ h ∗ be the associated root lattice, and let ( , ) be a standardinvariant form on g , as in [20, § h ∗ . Set d ij = ( α i , α j )2 and d i = d ii ∀ i, j ∈ I . By [20, § , ) is normalized so that { d i } i ∈ I are positive,relatively prime, integers.Let ˙ g denote the derived subalgebra [ g , g ]. The notation N and N + will be usedto denote the sets of non-negative and strictly positive integers, respectively. Allof this data shall remain fixed throughout the course of this paper, unless specifiedotherwise.2.1. The Yangian Y ~ g . We begin by recalling the definition of the Yangian asso-ciated to g . Let S m denote the symmetric group on { , . . . , m } . HE FORMAL SHIFT OPERATOR ON THE YANGIAN DOUBLE 7
Definition 2.1.
The Yangian Y ~ g is the unital associative C [ ~ ]-algebra generatedby h ∈ h and { x ± ir , h ir } i ∈ I ,r ∈ N , subject to the following relations for i, j ∈ I , r, s ∈ N and h, h ′ ∈ h : h i = d i α ∨ i , (2.1) [ h ir , h js ] = 0 , [ h ir , h ] = 0 , [ h, h ′ ] = 0 , (2.2) [ h, x ± js ] = ± α j ( h ) x ± js , (2.3) [ x + ir , x − js ] = δ ij h i,r + s , (2.4) [ h i,r +1 , x ± js ] − [ h ir , x ± j,s +1 ] = ± ~ d ij ( h ir x ± js + x ± js h ir ) , (2.5) [ x ± i,r +1 , x ± js ] − [ x ± ir , x ± j,s +1 ] = ± ~ d ij ( x ± ir x ± js + x ± js x ± ir ) , (2.6) X π ∈ S m h x ± i,r π (1) , h x ± i,r π (2) , · · · , h x ± i,r π ( m ) , x ± js i · · · ii = 0 , (2.7)where in the last relation i = j , m = 1 − a ij and r , . . . , r m ∈ N .The Yangian Y ~ g is an N -graded algebra with deg ~ = 1, deg h = 0, anddeg x ± ir = deg h ir = r ∀ i ∈ I , r ∈ N . The k -th graded component of Y ~ g will be denoted Y ~ g k , so that Y ~ g = M k ∈ N Y ~ g k . As a C [ ~ ]-algebra, Y ~ g is generated by its degree zero and one subspaces. Moreprecisely, we have the following standard result. Lemma 2.2. Y ~ g is generated by h ∪ { x ± i , h i } i ∈ I . Explicitly, for s > , x ± is and h i,s +1 are determined by x ± is = ± d i (cid:2) t i , x ± i,s − (cid:3) , where t i = h i − ~ h i ,h i,s +1 = [ x + is , x − i ] . Let { e i , f i } i ∈ I denote the Chevalley generators of g , as in [20, § h i = d i α ∨ i , x + i = p d i e i , x − i = p d i f i ∀ i ∈ I . These normalized generators satisfy ( x + i , x − i ) = 1 and h i = [ x + i , x − i ] for all i ∈ I ,and the relations (2.1)–(2.7) imply that the assignment x ± i x ± i , h i h i , h h ∀ i ∈ I and h ∈ h , determines a C -algebra homomorphism U ( g ) → Y ~ g .2.2. Generating series and shift automorphisms.
We now spell out a moreefficient presentation of Y ~ g , which can be deduced from [8, Prop. 2.3]. Proposition 2.3.
For each i ∈ I , define x ± i ( u ) , h i ( u ) ∈ Y ~ g [[ u − ]] by x ± i ( u ) = X r ≥ x ± ir u − r − and h i ( u ) = X r ≥ h ir u − r − . C. WENDLANDT
Then the defining relations (2.1) – (2.7) of Y ~ g are equivalent to the following rela-tions for i, j ∈ I and h, h ′ ∈ h : h i = d i α ∨ i , (2.8) [ h i ( u ) , h j ( v )] = 0 , [ h, h j ( u )] = 0 , [ h, h ′ ] = 0 , (2.9) [ h, x ± j ( u )] = ± α j ( h ) x ± j ( u ) , (2.10) ( u − v ∓ ~ d ij ) h i ( u ) x ± j ( v )= ( u − v ± ~ d ij ) x ± j ( v ) h i ( u ) ± d ij x ± j ( v ) − [ h i ( u ) , x ± j ] , (2.11) ( u − v ∓ ~ d ij ) x ± i ( u ) x ± j ( v )= ( u − v ± ~ d ij ) x ± j ( v ) x ± i ( u ) + [ x ± i , x ± j ( v )] − [ x ± i ( u ) , x ± j ] , (2.12) ( u − v )[ x + i ( u ) , x − j ( v )] = δ ij ( h i ( v ) − h i ( u )) , (2.13) X π ∈ S m (cid:2) x ± i ( u π (1) ) , (cid:2) x ± i ( u π (2) ) , · · · , (cid:2) x ± i ( u π ( m ) ) , x ± j ( v ) (cid:3) · · · (cid:3)(cid:3) = 0 , (2.14) where in the last relation i = j and m = 1 − a ij . Remark 2.4.
Since x ± i ( u ) , h i ( u ) ∈ u − Y ~ g [[ u − ]], the relations (2.8)–(2.13) can(and will) be viewed as identities in the algebra Y ~ g [[ u − , v − ]]. Similarly, the Serrerelations (2.14) should be understood as identities in Y ~ g [[ u − , . . . , u − m , v − ]].The Yangian Y ~ g admits a family of automorphisms { τ c } c ∈ C defined by(2.15) τ c ( h ) = h ∀ h ∈ h ,τ c ( x ± i ( u )) = x ± i ( u − c ) , τ c ( h i ( u )) = h i ( u − c ) ∀ i ∈ I . This is readily verified using the relations of Proposition 2.3. In terms of thegenerators x ± ir and h ir , the above formulas read as τ c ( x ± ir ) = r X k =0 (cid:18) rk (cid:19) x ± ik c r − k , τ c ( h ir ) = r X k =0 (cid:18) rk (cid:19) h ik c r − k ∀ i ∈ I and r ∈ N . Each τ c is called a shift automorphism. Replacing c by a formal variable z , weobtain the formal shift homomorphism(2.16) τ z : Y ~ g ֒ → Y ~ g [ z ]defined by (2.15) with c replaced by z .2.3. The Yangian double D Y ~ g . We now turn to the Yangian double associatedto g , as first considered in the work of Khoroshkin–Tolstoy [24] in the case where g is finite-dimensional. Let δ ( u ) = P r ∈ Z u r ∈ C [[ u ± ]] denote the formal deltafunction, so that u − δ ( v/u ) = X r ∈ Z v r u − r − ∈ C [[ u ± , v ± ]] . In what follows, we invoke the standard terminology for topological C [[ ~ ]]-algebras;see [22, Def. XVII.2.2], for instance. HE FORMAL SHIFT OPERATOR ON THE YANGIAN DOUBLE 9
Definition 2.5.
The Yangian double D Y ~ g is the unital, associative C [[ ~ ]]-algebratopologically generated by h ∈ h and the coefficients {X ± ir , H ir } i ∈ I ,r ∈ Z of the series X ± i ( u ) = X r ∈ Z X ± ir u − r − and H i ( u ) = X r ∈ Z H ir u − r − , subject to the following relations for all i, j ∈ I and h, h ′ ∈ h : H i = d i α ∨ i , (2.17) [ H i ( u ) , H j ( v )] = 0 , [ h, H j ( u )] = 0 , [ h, h ′ ] = 0 , (2.18) [ h, X ± j ( u )] = ± α j ( h ) X ± j ( u ) , (2.19) ( u − v ∓ ~ d ij ) H i ( u ) X ± j ( v ) = ( u − v ± ~ d ij ) X ± j ( v ) H i ( u ) , (2.20) ( u − v ∓ ~ d ij ) X ± i ( u ) X ± j ( v ) = ( u − v ± ~ d ij ) X ± j ( v ) X ± i ( u ) , (2.21) [ X + i ( u ) , X − j ( v )] = δ ij u − δ ( v/u ) H i ( v ) , (2.22) X π ∈ S m (cid:2) X ± i ( u π (1) ) , (cid:2) X ± i ( u π (2) ) , · · · , (cid:2) X ± i ( u π ( m ) ) , X ± j ( v ) (cid:3) · · · (cid:3)(cid:3) = 0 , (2.23)where in the last relation i = j and m = 1 − a ij .The C [ ~ ]-form D Y ~ g of D Y ~ g is defined to be the unital, associative C [ ~ ]-algebragenerated by h ∈ h and {X ± ir , H ir } i ∈ I ,r ∈ Z , subject to relations (2.17)–(2.23). Remark 2.6. (1) The relations (2.18)–(2.22) are understood to be expanded in the formalseries space D Y ~ g [[ u ± , v ± ]] to yield the corresponding relations for D Y ~ g .Similarly, (2.23) is to be expanded in D Y ~ g [[ u ± , . . . , u ± m , v ± ]].(2) The above relations are equivalent the relations (2.1)–(2.7) upon replacingall instances of x ± ik , x ± jk , h ik and h jk ( k ∈ N ) by X ± ik , X ± jk , H ik and H jk ,respectively, and allowing k to take arbitrary integer values.Let us now collect some facts about D Y ~ g and D Y ~ g which follow readily fromthe above definition. Let denote the natural C [ ~ ]-algebra homomorphism : D Y ~ g → D Y ~ g . Proposition 2.7. (1) induces an isomorphism of C [[ ~ ]] -algebras lim ←− n ( D Y ~ g / ~ n D Y ~ g ) ∼ −→ D Y ~ g . (2) For each i ∈ I , we have [ X + i , X − i ( u )] = H i ( u ) . Consequently, the set h ∪ {X ± ik } i ∈ I ,k ∈ Z generates D Y ~ g as a C [ ~ ] -algebra and D Y ~ g as a topological C [[ ~ ]] -algebra. (3) D Y ~ g is a Z -graded algebra with deg ~ = 1 , deg h = 0 , and deg X ± ir = deg H ir = r ∀ i ∈ I , r ∈ Z . (4) The assignment x ± ir
7→ X ± ir , h ir
7→ H ir , h h ∀ i ∈ I , r ∈ N and h ∈ h , extends to a homomorphism of Z -graded C [ ~ ] -algebras ı Y : Y ~ g → D Y ~ g . We shall set ı := ◦ ı Y : Y ~ g → D Y ~ g . It should be emphasized that, at this point, it is not clear that any of the maps , ı Y or ı are injective. As a consequence of (1) above, we haveKer( ) = \ n ∈ N ~ n D Y ~ g , and, as D Y ~ g is not necessarily separated, this ideal need not vanish. We will,however, see in Corollary 4.4 that both ı Y and ı are indeed injective.2.4. Translation automorphisms.
We now introduce the so-called translationautomorphisms of the Yangian double (see [24, (5.12)], for instance). These willplay a particularly important role in the proof of Theorem 5.5 in Section 5.
Proposition 2.8.
Fix i ∈ I . Then the assignment t i defined by t i ( h ) = h, t i ( X ± jr ) = X ± j,r ± δ ij , t i ( H jr ) = H jr ∀ j ∈ I , r ∈ Z and h ∈ h extends to an automorphism t i of D Y ~ g and of D Y ~ g .Proof. It suffices to prove the assertion for D Y ~ g . For each n ∈ Z , define anassignment t ni by t ni ( h ) = h, t ni ( H j ( u )) = H j ( u ) , t ni ( X ± j ( u )) = u ± nδ ij X ± j ( u ) ∀ j ∈ I , h ∈ h . It is straightforward to verify that t ni preserves the relations of Definition 2.5. Forinstance,[ t nk ( X + i ( u )) , t nk ( X − j ( v ))] = ( u/v ) nδ kj δ ij u − δ ( v/u ) H i ( v ) = δ ij u − δ ( v/u ) H i ( v ) , where we have used ( u/v ) n u − δ ( v/u ) = u − δ ( v/u ). It follows that t ni extends toa C [ ~ ]-algebra endomorphism of D Y ~ g , which satisfies t ni = ( t i ) n for all n ∈ Z . Inparticular, t i is an automorphism with inverse t − i . (cid:3) Derived subalgebras and classical limits
The algebras Y ~ ˙ g and D Y ~ ˙ g . In the current literature on Yangians of infinite-dimensional Kac–Moody algebras, both the full Yangian Y ~ g of Definition 2.1 andthe Yangian Y ~ ˙ g of the derived Lie subalgebra ˙ g ⊂ g , defined below, have indepen-dently been considered; see [14, 16, 37], for instance.The results of this paper, which are primarily stated for Y ~ g and D Y ~ g , areentirely valid for Y ~ ˙ g and D Y ~ ˙ g . In this subsection, we make this transparent byclarifying the precise relationship between Y ~ g and Y ~ ˙ g , and D Y ~ g and D Y ~ ˙ g . Definition 3.1.
The Yangian Y ~ ˙ g is the unital, associative C [ ~ ]-algebra generatedby { x ± ir , h ir } i ∈ I ,r ∈ N , subject to the relations (2.4) - (2.7) of Definition 2.1, in additionto [ h ir , h js ] = 0 , [ h i , x ± js ] = ± d ij x ± js ∀ i, j ∈ I , r, s ∈ N . We first observe that Y ~ ˙ g admits the structure of a Q -graded C [ ~ ]-algebra Y ~ ˙ g = M β ∈ Q Y ~ ˙ g β , HE FORMAL SHIFT OPERATOR ON THE YANGIAN DOUBLE 11 determined by assigning deg h ir = 0 and deg x ± ir = ± α i for all i ∈ I and r ∈ N .Next, let us fix a decomposition of (abelian) Lie algebras h = ˙ h ⊕ ¨ h , where ˙ h = M i ∈ I C α ∨ i . The Lie algebra ¨ h then acts on Y ~ ˙ g by the commuting C [ ~ ]-linear derivationsuniquely determined by(3.1) h · x β = β ( h ) x β ∀ h ∈ ¨ h , x β ∈ Y ~ ˙ g β . We can thus form the crossed product (or smash product) algebra Y ~ ˙ g ⋊ U (¨ h ) overthe complex numbers [29, Def. 4.1.3]. As a vector space, we have Y ~ ˙ g ⋊ U (¨ h ) = Y ~ ˙ g ⊗ C U (¨ h ) , with associative multiplication • defined on simple tensors by( x ⊗ h ) • ( y ⊗ h ′ ) = x ( h · y ) ⊗ h h ′ ∀ x, y ∈ Y ~ ˙ g and h, h ′ ∈ U (¨ h ) , where we have used the sumless Sweedler notation ∆( h ) = h ⊗ h for the standardcoproduct on U (¨ h ). As the underlying action of U (¨ h ) is C [ ~ ]-linear, this defines a C [ ~ ]-algebra structure on Y ~ ˙ g ⋊ U (¨ h ). We then have the following result. Proposition 3.2.
The assignment h ⊗ h, x ± ir x ± ir ⊗ , h ir h ir ⊗ , for h ∈ ¨ h , i ∈ I and r ∈ N , uniquely extends to an isomorphism of C [ ~ ] -algebras Y ~ g ∼ −→ Y ~ ˙ g ⋊ U (¨ h ) . The proof of the proposition is entirely analogous to the argument that U ( g )itself decomposes as U ( g ) ∼ = U ( ˙ g ) ⋊ U (¨ h ), and is therefore omitted.A nearly identical story unfolds if Y ~ g is replaced by D Y ~ g . The only subtletywhich arises is that the crossed product construction should be carried out in thecategory of topological C [[ ~ ]]-modules. We summarize these results below, beginningwith the definition of D Y ~ ˙ g . Definition 3.3.
The Yangian double D Y ~ ˙ g is the unital, associative C [[ ~ ]]-algebratopologically generated by {X ± ir , H ir } i ∈ I ,r ∈ Z , subject to the relations (2.20) - (2.23)of Definition 2.5, in addition to[ H ir , H js ] = 0 , [ H i , X ± js ] = ± d ij X ± js ∀ i, j ∈ I , r, s ∈ Z . The C [ ~ ]-form D Y ~ ˙ g of D Y ~ ˙ g is the unital, associative C [ ~ ]-algebra generated by {X ± ir , H ir } i ∈ I ,r ∈ Z , subject to the same set of relations.The algebra D Y ~ ˙ g is itself Q -graded with deg H ir = 0 and deg X ± ir = ± α i : D Y ~ ˙ g = M β ∈ Q D Y ~ ˙ g β . As in the Yangian case, we have an action of the Lie algebra ¨ h on D Y ~ ˙ g by deriva-tions, uniquely determined by (3.1), where x β now takes values in D Y ~ ˙ g β . Each such derivation is C [ ~ ]-linear, and therefore determines a C [[ ~ ]]-linear derivation ofthe algebra D Y ~ ˙ g ∼ = lim ←− n ( D Y ~ ˙ g / ~ n D Y ~ ˙ g ) . We thus have an action of ¨ h on D Y ~ ˙ g by derivations, and may form the spaces D Y ~ ˙ g ⋊ U (¨ h ) and D Y ~ ˙ g ⋊ U (¨ h ) , which are naturally algebras over C [ ~ ] and C [[ ~ ]], respectively. We then have thefollowing analogue of Proposition 3.2. Proposition 3.4.
The assignment h ⊗ h, X ± ir
7→ X ± ir ⊗ , H ir
7→ H ir ⊗ , for h ∈ ¨ h , i ∈ I and r ∈ Z , uniquely extends to yield algebra isomorphisms D Y ~ g ∼ −→ D Y ~ ˙ g ⋊ U (¨ h ) and D Y ~ g ∼ −→ D Y ~ ˙ g ⋊ ~ U (¨ h ) , where D Y ~ ˙ g ⋊ ~ U (¨ h ) is the ~ -adic completion of D Y ~ ˙ g ⋊ U (¨ h ) . The classical limits s and t . Modulo the ideal generated by ~ , the definingrelations of Y ~ ˙ g and D Y ~ ˙ g are of Lie type. It follows that Y ~ ˙ g and D Y ~ ˙ g deformthe enveloping algebras of certain infinite-dimensional complex Lie algebras s and t , respectively. In this section, we overview the abstract definitions of s and t ,together with their Y ~ g and D Y ~ g counterparts.Henceforth, the symbol a is understood to take value s or t , and we set Z s = N and Z t = Z . Definition 3.5.
The complex Lie algebra a is defined to be the quotient of the freeLie algebra on { X ± ir , H ir } i ∈ I ,r ∈ Z a by the ideal generated by the following relations,for i, j ∈ I and r, s ∈ Z a : [ H ir , H js ] = 0 , (3.2) [ H ir , X ± js ] = ± d ij X ± j,r + s , (3.3) [ X + ir , X − js ] = δ ij H i,r + s , (3.4) [ X ± i,r +1 , X ± js ] = [ X ± ir , X ± j,s +1 ] , (3.5) ad( X ± i ) − a ij ( X ± js ) = 0 for i = j. (3.6)We note that a is a Z a -graded Lie algebra with deg X ± ir = deg H ir = r for all i ∈ I and r ∈ Z a . Additionally, the assignment X ± ir x ± i ⊗ t r , H ir h i ⊗ t r ∀ i ∈ I , r ∈ Z a , extends to yield graded epimorphisms(3.7) π s : s ։ ˙ g [ t ] and π t : t ։ ˙ g [ t ± ] , which are isomorphisms when g is finite-dimensional. When g is an untwisted affineLie algebra, one has instead(3.8) s ∼ = uce ( ˙ g [ t ]) and t κ ∼ = uce ( ˙ g [ t ± ]) , HE FORMAL SHIFT OPERATOR ON THE YANGIAN DOUBLE 13 where uce ( p ) denotes the universal central extension of a perfect Lie algebra p , and t κ is a one-dimensional central extension of t , defined for g of any type, constructedas follows. Define a linear map ¯ κ : ˙ g [ t ± ] ⊗ ˙ g [ t ± ] → C by¯ κ ( f ( t ) , g ( t )) = Res t ( ∂ t ( f ( t )) , g ( t )) ∀ f ( t ) , g ( t ) ∈ ˙ g [ t ± ] , where the invariant form ( , ) | ˙ g × ˙ g has been naturally extended to a bilinear formon ˙ g [ t ± ] ⊗ ˙ g [ t ± ] with values in C [ t ± ], ∂ t : ˙ g [ t ± ] → ˙ g [ t ± ] is the formal derivativeoperator, and Res t : C [ t ± ] → C is the formal residue. One verifies as in [20, § κ is a C -valued 2-cocycle on ˙ g [ t ± ]. It follows that κ = ¯ κ ◦ π ⊗ t : t ⊗ t → C is a C -valued 2-cocycle on t . Definition 3.6.
The Lie algebra t κ is the central extension of t by the cocycle κ .That is, t κ = t ⊕ C K as a vector space, with Lie bracket given by [ t , K] = 0 and[ x, y ] = [ x, y ] t + κ ( x, y )K ∀ x, y ∈ t . The assertion of (3.8) is non-trivial, and has been established in the work ofMoody, Rao and Yokonuma [30]. These isomorphisms appear in the form statedabove in [16], where t κ is itself denoted t . A deeper analysis of these results will begiven in the course of the proof of Theorem 6.2 in Section 6.5.Returning to our general discussion of a , note that the assignment deg H ir = 0and deg X ± ir = ± α i , for all i ∈ I and r ∈ Z a , defines a Q -grading a = M β ∈ Q a β . The commutative Lie algebra ¨ h acts on a by the derivations uniquely determined by(3.1) with x β ∈ a β . We may therefore take the semidirect product of Lie algebras a ⋊ ¨ h . We then have the following result, where the notation x a , ± ir , h a ir is used to denote x ± ir , h ir ∈ Y ~ ˙ g if a = s , and X ± ir , H ir ∈ D Y ~ ˙ g if a = t . Proposition 3.7.
The assignment X ± ir x a , ± ir mod ~ , H ir h a ir mod ~ ∀ i ∈ I , r ∈ Z a uniquely extends to isomorphisms of graded algebras U ( s ) ∼ −→ Y ~ ˙ g / ~ Y ~ ˙ g and U ( t ) ∼ −→ D Y ~ ˙ g / ~ D Y ~ ˙ g . Tensoring with the identity on U (¨ h ) yields isomorphisms U ( s ⋊ ¨ h ) ∼ −→ Y ~ g / ~ Y ~ g and U ( t ⋊ ¨ h ) ∼ −→ D Y ~ g / ~ D Y ~ g . The first assertion of the proposition, for Y ~ ˙ g , is precisely [16, Prop. 2.6]. TheD Y ~ ˙ g analogue of this result follows from an identical argument (see also [16,Prop. 3.6]). The second part of the proposition is then a consequence of Proposi-tions 3.2 and 3.4, which imply there are algebra isomorphisms Y ~ g / ~ Y ~ g ∼ = Y ~ ˙ g / ~ Y ~ ˙ g ⋊ U (¨ h ) ∼ = U ( s ⋊ ¨ h ) , D Y ~ g / ~ D Y ~ g ∼ = D Y ~ ˙ g / ~ D Y ~ ˙ g ⋊ U (¨ h ) ∼ = U ( t ⋊ ¨ h ) , where we have employed the fact that U ( a ⋊ ¨ h ) ∼ = U ( a ) ⋊ U (¨ h ). Extending the shift automorphism
The primary goal of this section is to introduce the formal shift operator Φ z together with its evaluation Φ c at any invertible complex number c ∈ C × . This willbe achieved in Theorem 4.3, after first proving a collection of preliminary results oncompleted Yangians and formal series algebras. We will then conclude this sectionby spelling out a number of direct consequences to Theorem 4.3 in Sections 4.5 and4.6.4.1. Completed Yangian.
Let d Y ~ g denote the completion of Y ~ g with respect toits N -grading: d Y ~ g = Y k ∈ N Y ~ g k . Since ~ has degree one, d Y ~ g is a unital, associative C [[ ~ ]]-algebra. Consider now theideal Y ~ g + ⊂ Y ~ g generated by elements of strictly positive degree: Y ~ g + = M k> Y ~ g k . Lemma 4.1. d Y ~ g admits the following properties. (1) The canonical C [ ~ ] -algebra homomorphism Y ~ g → lim ←− n (cid:0) Y ~ g /Y ~ g n + (cid:1) extendsto an isomorphism of C [[ ~ ]] -algebras d Y ~ g ∼ −→ lim ←− n (cid:0) Y ~ g /Y ~ g n + (cid:1) . (2) d Y ~ g separated and complete as a C [[ ~ ]] -module, (3) d Y ~ g is a torsion free C [[ ~ ]] -module, provided Y ~ g is a torsion free C [ ~ ] -module.Proof. By Lemma 2.2, Y ~ g and Y ~ g generate Y ~ g as a C [ ~ ]-algebra, and conse-quently, we have Y ~ g n + = M k ≥ n Y ~ g k for all n ∈ N . Lemma 4.1 thus follows from Proposition A.1 of Appendix A with the algebra Ataken to be Y ~ g . Parts (2) and (3) may also be proven as in [7, Prop. 6.3]. (cid:3) Formal series spaces.
Let Y ~ g [ z ; z − ]] denote the algebra of formal Laurentseries in z − with ceofficients in Y ~ g : Y ~ g [ z ; z − ]] = [ n ∈ N z n Y ~ g [[ z − ]] ⊂ Y ~ g [[ z ± ]] . Define d Y ~ g z ⊂ Y ~ g [[ z − ]] by d Y ~ g z = Y k ∈ N Y ~ g k z − k . Let L d Y ~ g z denote the C [ z ± ]-submodule of Y ~ g [ z ; z − ]] generated by d Y ~ g z . Thefollowing proposition outputs a set of valuable properties characterizing this spaceand its ~ -adic completion. HE FORMAL SHIFT OPERATOR ON THE YANGIAN DOUBLE 15
Proposition 4.2.
Let v be an indeterminate and equip d Y v g [ z ± ] with the C [ ~ ] -algebra structure determined by ~ · vz . Then: (1) L d Y ~ g z is a Z -graded C [ ~ ] -algebra with L d Y ~ g z,k = z k d Y ~ g z . In particular, L d Y ~ g z = M n ∈ Z z n d Y ~ g z . (2) The graded linear map L d Y ~ g z → d Y v g [ z ± ] given by z n f ~ ( z ) z n f v (1) ∀ f ~ ( z ) ∈ d Y ~ g z , n ∈ Z , is an isomorphism of graded C [ ~ ] -algebras. (3) The ~ -adic completion L d Y ~ g z of L d Y ~ g z is the subspace of d Y ~ g [[ z ± ]] consist-ing of formal series X k ∈ Z z k f k ( z ) , f k ( z ) ∈ d Y ~ g z with the property that, for each m ∈ N , ∃ N m ∈ N such that f k ( z ) ∈ ( ~ /z ) m d Y ~ g z ∀ | k | ≥ N m . (4) For each c ∈ C × , the map E v c : L d Y ~ g z → d Y ~ g , f ( z ) f ( c ) , is an epimorphism of C [[ ~ ]] -algebras.Proof. As d Y ~ g z is a C -algebra, ~ ∈ z d Y ~ g z and z n d Y ~ g z · z m d Y ~ g z ⊂ z n + m d Y ~ g z ∀ n, m ∈ Z , L d Y ~ g z is a C [ ~ ]-algebra, which will be Z -graded provided the sum P n ∈ Z z n d Y ~ g z isdirect. This assertion is readily verified, and hence Part (1) holds.Part (2) is a consequence of Part (1), the definition of the C [ ~ ]-module structureon d Y v g [ z ± ], and the fact that d Y ~ g z → d Y ~ g , f ( z ) f (1) , is an isomorphism of C -algebras.Consider now Part (3). Since z ∈ d Y v g [ z ± ] is a unit, Part (2) yieldsL d Y ~ g z = lim ←− n (cid:16) L d Y ~ g z / ~ n L d Y ~ g z (cid:17) ∼ = lim ←− n (cid:16)d Y v g [ z ± ] /v n d Y v g [ z ± ] (cid:17) . Part (3) thus follows from the identification of Part (2), Lemma 4.1, and the fol-lowing straightforward general result:If A is a separated and complete C [[ ~ ]]-module, then the ~ -adic completion ofA[ z ± ] is equal to the subspace of A[[ z ± ]] consisting of all series X k ∈ Z x k z k ∈ A[[ z ± ]]satisfying the condition that, for each m ∈ N , ∃ N m ∈ N such that x k ∈ ~ m A ∀ | k | ≥ N m . Let us now turn to Part (4). Composing the algebra epimorphism d Y v g [ z ± ] ։ d Y ~ g , f v ( z ) f ~ ( c ) with the isomorphism of (2), we obtain an epimorphism of C [ ~ ]-algebras E v ′ c : L d Y ~ g z ։ d Y ~ g , given by evaluating z c . Since, by Lemma 4.1, d Y ~ g is separated and complete, E v ′ c induces E v c as in the statement of the proposition. (cid:3) The formal shift operator Φ z . Let τ c and τ z be the shift homomorphismsof (2.15) and (2.16), respectively, and recall that and ı are the natural homomor-phisms : D Y ~ g → D Y ~ g and ı : Y ~ g → D Y ~ g introduced in Section 2.3. In addition, we shall set ∂ ( n ) z = 1 n ! ( ∂ z ) n ∀ n ∈ N , where ∂ z is the formal derivative operator with respect to z . With the machineryof Sections 4.1 and 4.2 at our disposal, we are now prepared to state and prove ourfirst main result. Theorem 4.3. (1)
There is a unique homomorphism of C [[ ~ ]] -algebras Φ z : D Y ~ g → L d Y ~ g z with the property that Φ z ◦ ı = τ z . It is given by (4.1) Φ z ( h ) = h ∀ h ∈ h ,u Φ z ( H i ( u )) = X n ∈ N h i,n ∂ ( n ) z ( δ ( z/u )) , u Φ z ( X ± i ( u )) = X n ∈ N x ± i,n ∂ ( n ) z ( δ ( z/u )) , for all i ∈ I . (2) The compostion Φ z ◦ is a Z -graded C [ ~ ] -algebra homomorphism Φ z ◦ : D Y ~ g → L d Y ~ g z = M n ∈ Z z n d Y ~ g z ⊂ Y ~ g [ z ; z − ]] . (3) Fix c ∈ C × . Then Φ c = E v c ◦ Φ z is the unique homomorphism of C [[ ~ ]] -algebras Φ c : D Y ~ g → d Y ~ g satisfying Φ c ◦ ı = τ c . The proof of the theorem will be given in § u − δ ( z/u ) as u − δ ( z/u ) = exp( − z∂ u )(1 /u ) + exp( − u∂ z )(1 /z ) , we find that ∂ ( n ) z (cid:0) u − δ ( z/u ) (cid:1) = exp( − z∂ u )( u − n − ) + ( − n exp( − u∂ z )( z − n − ) . Hence, from the second line of (4.1), we obtain(4.2) (Φ z ◦ ı ) h i ( u ) = exp( − z∂ u ) h i ( u ) = h i ( u − z ) , (Φ z ◦ ı ) x ± i ( u ) = x ± i ( u − z ) , HE FORMAL SHIFT OPERATOR ON THE YANGIAN DOUBLE 17 for all i ∈ I . We may thus conclude that Φ z ◦ ı = τ z will hold, provided that Φ z , asgiven by (4.1), is an algebra homomorphism.The above expansion of ∂ ( n ) z (cid:0) u − δ ( z/u ) (cid:1) also implies that(4.3) Φ z ( X ± i, − n − ) = ( − n +1 ∂ ( n ) z x ± i ( − z ) ∀ i ∈ I , n ∈ N . In particular, since x ± i ( z ) ∈ z − d Y ~ g z and ∂ ( n ) z is a degree − n operator on L d Y ~ g z ,( − n +1 ∂ ( n ) z x ± i ( − z ) ∈ z − n − d Y ~ g z ⊂ L d Y ~ g z . Consequently, Part (2) of the theorem will follow automatically from Part (1) andthe second statement of Proposition 2.7.4.4.
Proof of Theorem 4.3.
Let us begin by establishing that there is at mostone homomorphism Φ z : D Y ~ g → L d Y ~ g z with the property that Φ z ◦ ı = τ z . Ourargument will also imply the uniqueness of Φ c , as in the statement of the theorem. Proof of uniqueness.
Let Φ z be such a homomorphism, and fix i ∈ I . Our startingpoint is the relation(4.4) [ ı ( t i ) , X ± is ] = ± d i X ± i,s +1 ∀ s ∈ Z , where t i = h i − ~ h i , as in Lemma 2.2. This relation is proven in the same wayas its Y ~ g -counterpart; see (2) of Remark 2.6 and Lemma 2.2. It implies thatad(T i ) k ( X ± is ) = ( ± k X ± i,s + k ∀ s ∈ Z , k ∈ N , where T i = ι ( t i )2 d i . Applying Φ z , and using that τ z ( t i ) = t i + zh i , we obtainad (cid:18) T i + z d i H i (cid:19) k Φ z ( X ± is ) = ( ± k Φ z ( X ± i,s + k ) ∀ s ∈ Z , k ∈ N . By (2.17) and (2.19), [ H i , Φ z ( X ± is )] = ± d i Φ z ( X ± is ) . It follows that the above isequivalent to ( z ± ad(T i )) k Φ z ( X ± is ) = Φ z ( X ± i,s + k ) ∀ s ∈ Z , k ∈ N . Fixing n ∈ N and taking k = n + 1 and s = − n −
1, we deduce that(4.5) ( z ± ad(T i )) n +1 Φ z ( X ± i, − n − ) = X ± i . As T i ∈ Y ~ g ,(4.6) ( z ± ad(T i )) − n − = X p ≥ ( − n ad( ∓ T i ) p ∂ ( n ) z ( z − p − )is a C [[ ~ ]]-linear endomorphism of L d Y ~ g z . Applying it to (4.5) and employing (4.4),we recover (4.3):Φ z ( X ± i, − n − ) = X p ≥ ( − n + p x ± ip ∂ ( n ) z ( z − p − ) = ( − n +1 ∂ ( n ) z x ± i ( − z ) . By Part (2) of Proposition 2.7, this identity, together with the requirement Φ z ◦ ı = τ z , completely determines Φ z . This proves the uniqueness of Φ z .Observe that, since the evaluation of (4.6) at z = c ∈ C × defines an honest C [[ ~ ]]-linear endormorphism of d Y ~ g , the above uniqueness argument is completely valid with z replaced by a fixed scalar c ∈ C × . It thus proves that there is at mostone C [[ ~ ]]-algebra homomorphism Φ c : D Y ~ g → d Y ~ g such that Φ c ◦ ı = τ c . Proof of (1) and (2) . Next, we prove that the assignment Φ z defined by (4.1)preserves the defining relations of D Y ~ g . Since L d Y ~ g z is separated and complete (by(3) of Proposition 4.2), this will imply that (4.1) indeed extends to a homomorphismof C [[ ~ ]]-algebras Φ z : D Y ~ g → L d Y ~ g z , which, by the remarks following the statement of the theorem, will complete theproof of both Parts (1) and (2) of the theorem.It is clear from (2.1) and (2.2) that Φ z preserves the relations (2.17) and (2.18). The relation (2.19) . By (2.3), for each h ∈ h and j ∈ I we have[Φ z ( h ) , Φ z ( X ± j ( u ))] = X n ∈ N [ h, x ± j,n ] ∂ ( n ) z (cid:0) u − δ ( z/u ) (cid:1) = ± α j ( h )Φ z ( X ± j ( u )) . The relations (2.20) and (2.21) . For each n ∈ N , we have(4.7) ( u − v ) ∂ ( n ) z (cid:0) u − δ ( z/u ) (cid:1) = ∂ ( n − z (cid:0) u − δ ( z/u ) (cid:1) + ( z − v ) ∂ ( n ) z (cid:0) u − δ ( z/u ) (cid:1) , ( u − z ) ∂ ( n ) z (cid:0) u − δ ( z/u ) (cid:1) = ∂ ( n − z (cid:0) u − δ ( z/u ) (cid:1) , where ∂ ( n − z (cid:0) u − δ ( z/u ) (cid:1) = 0 if n = 0. The second relation is obtained from thefirst by setting v = z , and the first relation is proven by induction on n .For each n, m ∈ N , set f n,m ( u, v, z ) = ∂ ( n ) z (cid:0) u − δ ( z/u ) (cid:1) ∂ ( m ) z (cid:0) v − δ ( z/v ) (cid:1) . Then (4.7) implies that f n,m ( u, v, z ) satisfies(4.8) ( u − v ) f n,m ( u, v, z ) = f n − ,m ( u, v, z ) − f n,m − ( u, v, z ) , where f − ,m ( u, v, z ) = f n, − ( u, v, z ) = 0.We now apply this to prove that Φ z preserves (2.20) and (2.21). Fix i, j ∈ I , andlet ( Y i ( u ) , y ir ) denote ( X ± i ( u ) , x ± ir ) or ( H i ( u ) , h ir ) for all r ∈ N . Then, by (4.8):( u − v ) (cid:2) Φ z ( Y i ( u )) , Φ z ( X ± j ( v )) (cid:3) = X n,m ∈ N [ y i,n , x ± j,m ]( u − v ) f n,m ( u, v, z )= X n,m ∈ N [ y i,n , x ± j,m ]( f n − ,m ( u, v, z ) − f n,m − ( u, v, z )) . Using (2.5) and (2.6), we can rewrite the right-hand side as X n,m ∈ N ([ y i,n +1 , x ± j,m ] − [ y i,n , x ± j,m +1 ]) f n,m ( u, v, z )= ± ~ d ij X n,m ∈ N { y i,n , x ± j,m } f n,m ( u, v, z ) = ± ~ d ij (cid:8) Φ z ( Y i ( u )) , Φ z ( X ± j ( v )) (cid:9) , where { x, y } = xy + yx . Thus, we have proven that( u − v ∓ ~ d ij )Φ z ( Y i ( u ))Φ z ( X ± j ( v )) = ( u − v ± ~ d ij )Φ z ( X ± j ( v ))Φ z ( Y i ( u )) , HE FORMAL SHIFT OPERATOR ON THE YANGIAN DOUBLE 19 which is precisely (2.20) if Y i ( u ) = H i ( u ), and (2.21) if Y i ( u ) = X ± i ( u ). The relation (2.22) . Fix i, j ∈ I . Then, by (2.4), we have[Φ z ( X + i ( u )) , Φ z ( X − j ( v ))] = δ ij X n ∈ N h i,n n X k =0 ∂ ( k ) z ( u − δ ( z/u )) ∂ ( n − k ) z ( v − δ ( z/v ))= δ ij X n ∈ N h i,n ∂ ( n ) z ( u − v − δ ( z/u ) δ ( z/v ))= δ ij u − δ ( v/u )Φ z ( H i ( v )) , where in the second equality we have used the generalized Leibniz identity, and inthe third equality we have used that u − v − δ ( z/u ) δ ( z/v ) = u − v − δ ( v/u ) δ ( z/v ). The Serre relations (2.23) . Fix i, j ∈ I with i = j and let m = 1 − a ij . For( n , . . . , n m , s ) ∈ N m +1 , set f n ,...,n m ,su ,...,u m ,v ( z ) = ∂ ( n ) z ( u − δ ( z/u )) · · · ∂ ( n m ) z ( u − m δ ( z/u m )) ∂ ( s ) z ( v − δ ( z/v )) . Then, since f n ,...,n m ,su ,...,u m ,v ( z ) is symmetric in { , . . . , m } , (4.1) gives X π ∈ S m (cid:2) Φ z ( X ± i ( u π (1) )) , (cid:2) Φ z ( X ± i ( u π (2) )) , · · · , (cid:2) Φ z ( X ± i ( u π ( m ) )) , Φ z ( X ± j ( v )) (cid:3) · · · (cid:3)(cid:3) = X n ,...,n m ,s ∈ N f n ,...,n m ,su ,...,u m ,v ( z ) X π ∈ S m h x ± i,n π (1) , h x ± i,n π (2) , · · · , h x ± i,n π ( m ) , x ± js i · · · ii = 0 , where the last equality holds by (2.7). Proof of (3) . Fix c ∈ C × . By Part (1) of the theorem and Part (4) of Proposition4.2, Φ c = E v c ◦ Φ z is a homomorphism of C [[ ~ ]]-algebras satisfying Φ c ◦ ı = τ c . Aswe have already established the uniqueness assertion, we are done. (cid:3) Consequences and formulas.
As a first, and rather immediate, corollary toTheorem 4.3 we obtain the injectivity of the natural homomorphisms from Y ~ g toboth D Y ~ g and D Y ~ g , and deduce the existence of injective translation endomor-phisms on the standard Borel subalgebras of Y ~ g . Corollary 4.4.
Define Y ~ ( b ± ) to be the subalgebra of Y ~ g generated by the Cartansubalgebra h and { x ± ir , h ir } i ∈ I ,r ∈ N . Then: (1) The algebra homomorphisms ı Y : Y ~ g → D Y ~ g and ı : Y ~ g → D Y ~ g are injective. (2) For each fixed i ∈ I , the assignment σ ± i : x ± jr x ± j,r + δ ij , h jr h jr , h h ∀ j ∈ I , r ∈ N and h ∈ h determines an injective C [ ~ ] -algebra endomorphism σ ± i of Y ~ ( b ± ) .Proof. Since Φ c ◦ ı = τ c is injective, ı is injective. As ı = ◦ ı Y , we can conclude ı Y is also injective. This proves Part (1).Consider now Part (2), and fix c ∈ C × and i ∈ I . Define σ ± i := τ − c ◦ Φ c ◦ t ± i ◦ ı | Y ~ ( b ± ) . This is an algebra endomorphism of Y ~ ( b ± ) which agrees with the assignment inthe statement of (2). It is injective since t ± i ( ı ( Y ~ ( b ± ))) ⊂ ı ( Y ~ ( b ± )), t i is anautomorphism, and Φ c ◦ ı , ı and τ − c are all injective. (cid:3) Remark 4.5.
When g is of finite type or of simply-laced affine type, one can deducethe existence of the algebra endomorphisms σ ± i of Y ~ ( b ± ) by appealing to the factthat Y ~ g is known to admit a triangular decomposition, as in [7, § g .The next result applies the endomorphisms σ ± i of Y ~ ( b ± ) to obtain a useful setof formulas re-expressing the definition of Φ z on each generating series X ± i ( u ). Corollary 4.6. (1)
For each i ∈ I , we have Φ z ( X ± i ( u )) = exp( σ ± i ∂ z )( u − δ ( z/u ) x ± i ) = u − δ (cid:18) z + σ ± i u (cid:19) ( x ± i ) . In particular, for each k ∈ N and ℓ ∈ Z , Φ z ◦ t ± ℓi ( X ± ik ) = exp( σ ± i ∂ z )( z k + ℓ x ± i ) = ( z + σ ± i ) ℓ τ z ( x ± ik ) , where t i ∈ Aut(D Y ~ g ) is as in Proposition 2.8. (2) For each i ∈ I , we have exp( − z∂ u ) x ± i ( u ) = Φ z ( X ± i ( u ) + ) = u − − u − ( z + σ ± i ) ( x ± i )exp( − u∂ z ) x ± i ( − z ) = Φ z ( X ± i ( u ) − ) = − z − − z − ( u − σ ± i ) ( x ± i ) , where X ± i ( u ) + = ı ( x ± i ( u )) and X ± i ( u ) − = X ± i ( u ) + − X ± i ( u ) . Proof.
Since ( σ ± i ) n ( x ± i ) = x ± in for all i ∈ I and n ∈ N , Part (1) follows directly from(4.1). As for Part (2), the leftmost equalities follow from (4.2) and (4.3), while therightmost equalites are readily deduced from Part (1). (cid:3) Similarity of Φ c and Φ a . We conclude Section 4 by establishing that, forany a, c ∈ C × , Φ c and Φ a are equal up to conjugation by a gradation automorphismgoverned by the ratio ca .For each a ∈ C × , introduce the C -algebra automorphism χ a of D Y ~ g by χ a = M k ∈ Z a k k ∈ Aut C ( D Y ~ g ) , where k is the identity map on the k -th graded component D Y ~ g k of D Y ~ g . Since χ a ( ~ n D Y ~ g ) = ~ n D Y ~ g for each n ∈ N , χ a extends to a C -algebra automorphismof D Y ~ g , which we again denote by χ a . These gives rise to an action of the multi-plicative group C × on D Y ~ g . That is, one has χ a ◦ χ c = χ a · c ∀ a, c ∈ C × . In addition, χ a restricts to a C -algebra automorphism of Y ~ g ∼ = ı ( Y ~ g ), whichextends by continuity to an automorphism χ ıa of the completed Yangian d Y ~ g . The HE FORMAL SHIFT OPERATOR ON THE YANGIAN DOUBLE 21 next proposition uses these automorphisms to illustrate the precise relation betweenΦ c and Φ a , for any a, c ∈ C × . Proposition 4.7.
For each pair of points a, c ∈ C × , one has the identity Φ c = χ ıa/c ◦ Φ a ◦ χ c/a . Proof.
The composition χ ıa/c ◦ Φ a ◦ χ c/a fixes ~ , and is thus a C [[ ~ ]]-algebra homo-morphism. Next, observe that, for each b ∈ C × , χ b satisfies χ b ( h ) = h, χ b ( A i ( u )) = A i ( u/b ) ∀ h ∈ h , i ∈ I , where A i ( u ) takes value u X ± i ( u ) or u H i ( u ). The assertion of the proposition there-fore follows from (4.1) together with the identity( a/c ) n ∂ ( n ) z ( δ ( zc/au )) (cid:12)(cid:12)(cid:12) z = a = ∂ ( n ) w ( δ ( w/u )) (cid:12)(cid:12)(cid:12) w = c , which is obtained by making the change of variables w = zc/a . (cid:3) Isomorphism with completed Yangian
The evaluation ideal J at t = 1 is defined to be the kernel of the compositeD Y ~ g ~ −−−→ U ( t ⋊ ¨ h ) ev g −−→ U ( g ) , where ~ ~ , under the identification of Proposition3.7, and ev g is the epimorphism of algebras induced by the composition(5.1) t ⋊ ¨ h π t ⊕ −−−→ ˙ g [ t ± ] ⋊ ¨ h ˙ev g ⊕ −−−−→ ˙ g ⋊ ¨ h ∼ = g , with π t as in (3.7) and ˙ev g : ˙ g [ t ± ] → ˙ g the evaluation morphism given by t
1. Inthis section, we will prove that the evaluation of Φ z at z = 1 induces an isomorphismof C [[ ~ ]]-algebras b Φ : \ D Y ~ g ∼ −→ d Y ~ g , where \ D Y ~ g is the completion of D Y ~ g with respect to the descending filtrationD Y ~ g = J ⊃ J ⊃ J ⊃ · · · ⊃ J n ⊃ · · · This will be achieved in Theorem 5.5 of Section 5.1. In Section 5.2, we will obtaina generalization of this result which holds for an arbitrary evaluation point c ∈ C × .We will then conclude this section with two applications of Theorem 5.5: In Section5.3, we will show that the natural inclusion ı extends to an isomorphism betweenthe evaluation completions of Y ~ g and D Y ~ g at t = 1. We will then demonstrate inSection 5.4 that Y ~ g can be realized as a degeneration of D Y ~ g , in the same waythat Y ~ g can be realized as a degeneration of the quantum loop algebra U ~ ( L g ).5.1. The isomorphism b Φ . In what follows, we shall set Φ = Φ , where Φ is themorphism Φ c from Theorem 4.3 with c taken to be 1. Since ∂ ( n ) z ( u − δ ( z/u )) = ( − n ∂ ( n ) u ( u − δ ( z/u )) ∀ n ∈ N , we deduce from (4.1) that Φ is given explicitly by the following data:(5.2) Φ : D Y ~ g → d Y ~ g , Φ( h ) = h, Φ( X ± i ( u )) = X n ∈ N ( − n x ± in ∂ ( n ) u ( δ ( u )) ∀ i ∈ I and h ∈ h . For each n ∈ N , introduce the ideal d Y ~ g ≥ n ⊂ d Y ~ g by d Y ~ g ≥ n = Y k ≥ n Y ~ g k . Equivalently, under the identification of Part (1) of Lemma 4.1, one has d Y ~ g ≥ n = lim ←− k>n (cid:0) Y ~ g n + /Y ~ g k + (cid:1) . Lemma 5.1.
We have Φ( J n ) ⊂ d Y ~ g ≥ n ∀ n ∈ N . Consequently, Φ induces a homomorphism of C [[ ~ ]] -algebras b Φ : \ D Y ~ g → d Y ~ g . Proof.
We proceed analogously to the proof of [7, Thm. 6.2 (1)]. To prove the firstassertion, it suffices to show that Φ( J ) ⊂ d Y ~ g + := d Y ~ g ≥ , as this will implyΦ( J n ) ⊂ ( d Y ~ g + ) n ⊂ d Y ~ g ≥ n ∀ n ∈ N . As the kernel of the evaluation homomorphism ev g is generated as an ideal by { X ± ir − X ± is } i ∈ I ,r,s ∈ Z , the ideal J ⊂ D Y ~ g is generated by(5.3) ~ D Y ~ g ∪ {X ± ir − X ± is } i ∈ I ,r,s, ∈ Z . Since ( u r − u s ) δ ( u ) = 0 for all r, s ∈ Z , (5.2) yields( u r − u s )Φ( X ± i ( u )) = X n> ( − n x ± in ( u r − u s ) ∂ ( n ) u ( δ ( u )) ∈ d Y ~ g + [[ u ± ]] ∀ i ∈ I . Applying the formal residue Res u : d Y ~ g [[ u ± ]] → d Y ~ g to this identity, we obtainΦ( X ± ir ) − Φ( X ± is ) = Res u (( u r − u s )Φ( X ± i ( u ))) ∈ d Y ~ g + . As ~ ∈ d Y ~ g + , this completes the proof of the first part of the lemma.We may thus conclude that Φ induces a family of C [[ ~ ]]-algebra homomorphisms(5.4) Φ n : D Y ~ g / J n → d Y ~ g / d Y ~ g ≥ n ∼ = Y ~ g /Y ~ g n + ∀ n ∈ N . Taking the inverse limit of this family, we obtain b Φ = lim ←− n Φ n : \ D Y ~ g → d Y ~ g . (cid:3) We will show that b Φ is an isomorphism by constructing its inverse explicitly. Set(5.5) Γ = ı ◦ τ − : Y ~ g ֒ → D Y ~ g , where we recall that τ − ∈ Aut( Y ~ g ) is defined in (2.15), and ı is the naturalhomomorphism Y ~ g → D Y ~ g , which by Corollary 4.4 is an embedding. Lemma 5.2.
We have Γ( Y ~ g + ) ⊂ J . Consequently, Γ induces a homomorphism of C [[ ~ ]] -algebras b Γ : d Y ~ g → \ D Y ~ g . HE FORMAL SHIFT OPERATOR ON THE YANGIAN DOUBLE 23
Proof.
On the generating set h ∪ { x ± i , h i } i ∈ I , Γ is given byΓ( h ) = h, Γ( x ± i ) = X ± i , Γ( h i ) = H i − H i ∈ J ∀ i ∈ I and h ∈ h . Since J = D Y ~ g , this implies that Γ( Y ~ g k ) ⊂ J k for all k ∈ N , and therefore thatΓ( Y ~ g + ) ⊂ J . Consequently, Γ induces a family of C [[ ~ ]]-algebra homomorphisms(5.6) Γ n : Y ~ g /Y ~ g n + → D Y ~ g / J n ∀ n ∈ N . Taking the inverse limit of this system, we obtain b Γ = lim ←− n Γ n : d Y ~ g → \ D Y ~ g . (cid:3) In order to prove that b Γ = b Φ − , we will first analyze how the family of automor-phisms { t i } ∈ I ⊂ Aut(D Y ~ g ) of Proposition 2.8 interact with the ideal J ⊂ D Y ~ g . Lemma 5.3.
Fix i ∈ I . Then (5.7) t i ( J ) = J , ( − t ± i ) J n ⊂ J n +1 ∀ n ∈ N . Consequently, t i induces t i,n ∈ Aut(D Y ~ g / J n ) satisfying (5.8) t ∓ i,n = n − X k =0 ( − t ± i,n ) k ∀ n ∈ N . Proof. As t i permutes the generating set (5.3) of J , we have t i ( J ) = J . It followsthat, for each n ∈ N , t i induces t i,n ∈ Aut(D Y ~ g / J n ) uniquely determined byq n ◦ t i = t i,n ◦ q n , where q n : D Y ~ g ։ D Y ~ g / J n is the natural quotient map.If the second relation of (5.7) holds, then ( − t ∓ i ) n D Y ~ g ⊂ J n for each n ∈ N .In particular, ( − t ∓ i,n ) n = 0, from which (5.8) follow readily.We are thus left to prove that ( − t ± i ) J n ⊂ J n +1 for all n ∈ N . Since( − t − i ) = ( t i − ) t − i and t − i ( J ) = J , we need only prove this for t i . Moreover, as t i ( J ) = J and(5.9) ( − t i )( x x · · · x n ) = n X j =1 x · · · x j − ( − t i )( x j ) t i ( x j +1 ) · · · t i ( x n )for any x , . . . , x n ∈ D Y ~ g and n >
0, it suffices to prove that( − t i )D Y ~ g ⊂ J and ( − t i ) J ⊂ J . As − t i is C [[ ~ ]]-linear and annihilates 1, (5.3) and (5.9) imply that these inclusionswill follow from( − t i ) X ± jr ∈ J and ( − t i )( X ± jr − X ± js ) ∈ J ∀ r, s ∈ Z , j ∈ I . By definition, we have ( − t i ) X ± jr = δ ij ( X ± ir − X ± i,r ± ) ∈ J . In addition, since ~ ∈ J and H i − H i ∈ J , we have( − t i )( X + jr − X + js ) = δ ij ( X + ir − X + i,r +1 − X + is + X + i,s +1 ) = δ ij d i [ ı ( t i − h i ) , X + is − X + ir ] ∈ J , where we have used (4.4) in the second equality. Similarly,( − t i )( X − jr − X − js ) = δ ij d i [ ı ( t i − h i ) , X − i,s − − X − i,r − ] ∈ J . (cid:3) Remark 5.4.
The lemma implies that, for each i ∈ I , t i extends to an automor-phism b t i of the C [[ ~ ]]-algebra \ D Y ~ g . More precisely, b t i := lim ←− n t i,n ∈ Aut( \ D Y ~ g ) . In addition, b t i satisfies the relation b t ∓ i = X k ∈ N ( − b t ± i ) k = lim ←− n n − X k =0 ( − t ± i,n ) k . With the above lemma at our disposal, we are now prepared to prove the mainresult of this section. Let b Φ and b Γ be as in Lemmas 5.1 and 5.2.
Theorem 5.5.
The C [[ ~ ]] -algebra homomorphisms b Φ : \ D Y ~ g → d Y ~ g and b Γ : d Y ~ g → \ D Y ~ g are mutual inverses. In particular, b Φ is an isomorphism of C [[ ~ ]] -algebras.Proof. Let { Φ n } n ∈ N and { Γ n } n ∈ N be as in (5.4) and (5.6), respectively. Since b Φ = lim ←− n Φ n and b Γ = lim ←− n Γ n , it suffices to prove that(5.10) Γ n = Φ − n ∀ n ∈ N . Fix n ∈ N . As Φ ◦ Γ = Φ ◦ ( ı ◦ τ − ) = τ ◦ τ − = Y ~ g , we haveΦ n ◦ Γ n = ∈ End( Y ~ g /Y ~ g n + ) . Hence, (5.10) will hold provided that Γ n is surjective, which we prove below.Since τ − is an automorphism we have ı ( Y ~ g ) ⊂ Im(Γ), and thus(5.11) (q n ◦ ı ) Y ~ g ⊂ Im(Γ n ) , where we recall that q n : D Y ~ g ։ D Y ~ g / J n is the natural quotient map. As t ± i ı ( Y ~ ( b ± )) ⊂ ı ( Y ~ ( b ± )), the identity (5.8) of Lemma 5.3 implies that( t ∓ i,n ◦ q n ◦ ı ) Y ~ ( b ± ) ⊂ (q n ◦ ı ) Y ~ ( b ± ) . Therefore,q n ( X ± i, − k − ) = ( t ∓ i,n ) k +1 q n ( X ± i ) ⊂ (q n ◦ ı ) Y ~ g ∀ k ∈ N , i ∈ I . As D Y ~ g is generated by {X ± i − k − } i ∈ I ,k ∈ N ∪ ı ( Y ~ g ), combining the above with (5.11)yields D Y ~ g / J n ⊂ (q n ◦ ı ) Y ~ g ⊂ Im(Γ n ) ⊂ D Y ~ g / J n . (cid:3) HE FORMAL SHIFT OPERATOR ON THE YANGIAN DOUBLE 25
Remark 5.6.
Fix n ∈ N and i ∈ I . The relation(Γ n ◦ Φ n ◦ q n ) X ± i, − k − = q n ( X ± i, − k − ) ∀ k ∈ N may also be proven directly as follows. Set X ± i ( u ) Φ n := − n − X k =0 ( − σ ± i ) k ∂ ku k ! (cid:18) x ± i − u (cid:19) ∈ Y ~ g [[ u ]] . By (2) of Corollary 4.6, Φ( X ± i ( u ) − ) ≡ X ± i ( u ) Φ n modulo d Y ~ g ≥ n [[ u ]]. By (1) of Corol-lary 4.6, τ c ◦ ( σ ± i ) k ( x ± i ) = ( c + σ ± i ) k x ± i ∀ k ∈ N , c ∈ C × . It follows thatΓ( X ± i ( u ) Φ n ) = n − X k =0 ( − t ± i ) k ∂ ku k ! (cid:18) X ± i u − (cid:19) = − X p ≥ u p n − X k =0 (cid:18) p + kp (cid:19) ( − t ± i ) k X ± i . Applying q n and using (5.8) together with ( − t ± i,n ) n = 0, we obtain(Γ n ◦ Φ n ◦ q n ) X ± i ( u ) − = q n ◦ Γ( X ± i ( u ) Φ n ) = − X p ≥ t ∓ ( p +1) i,n ( X ± i ) u p = q n ( X ± i ( u ) − ) . Change of evaluation point.
We now apply Proposition 4.7 to illustratethat the evaluation point c = 1 = t can be replaced by any c ∈ C × in the statementof Theorem 5.5. Let J c ⊂ D Y ~ g be the evaluation ideal at t = c . That is, J c is thekernel of the composite D Y ~ g ~ −−−→ U ( t ⋊ ¨ h ) ev c g −−→ U ( g ) , where ev c g is defined as in (5.1), but with ˙ev g replaced by the evaluation morphism˙ev c g : ˙ g [ t ± ] → ˙ g given by t c . Note that the above composition coincides withD Y ~ g χ c −→ D Y ~ g ~ −−−→ U ( t ⋊ ¨ h ) ev g −−→ U ( g ) , where χ c is as in Proposition 4.7. In particular, we have the equality χ c ( J c ) = J and may thus that χ c induces an isomorphism of C -algebras b χ c : \ D Y ~ g c ∼ −→ \ D Y ~ g , where \ D Y ~ g c is the completion of D Y ~ g with respect to its descending filtrationgiven by powers of J c . The following corollary then provides the desired general-ization of Theorem 5.5. Corollary 5.7.
Fix c ∈ C × . Then: (1) Φ c satisfies Φ c ( J c ) ⊂ d Y ~ g + and thus induces a C [[ ~ ]] -algebra homomorphism b Φ c : \ D Y ~ g c → d Y ~ g . (2) b Φ c is an isomorphism satisfying the relations b Φ c = χ ı /c ◦ b Φ ◦ b χ c and b Φ − c = b χ /c ◦ b Γ ◦ χ ıc . Proof.
Since χ c ( J c ) = J , Proposition 4.7 and Lemma 5.1 yieldΦ c ( J c ) = ( χ ı /c ◦ Φ)( J ) ⊂ d Y ~ g + . As in the proof of Lemma 5.1, this implies that Φ c gives rise to b Φ c as in the statementof Part (1). As for Part (2), it follows by continuity that b Φ c and χ ı /c ◦ b Φ ◦ b χ c areboth determined by their values on the image of D Y ~ g , and therefore coincide byProposition 4.7. The assertion that b Φ c is invertible with inverse b χ /c ◦ b Γ ◦ χ ıc nowfollows immediately from Theorem 5.5. (cid:3) The evaluation completion of Y ~ g . A simplification of the classical resultunderlying Theorem 5.5 is provided by the observation that the translation w t + 1 induces an isomorphism \C [ w ± ] ∼ −→ C [[ t ]] , where \C [ w ± ] = lim ←− n ( C [ w ± ] / J n )and J = ( w −
1) is the evaluation ideal of C [ w ± ] at w = 1. On the other hand, thenatural inclusion C [ t ] ֒ → C [ w ± ] sending t to w extends to an isomorphism d C [ t ] ∼ −→ \C [ w ± ] , where C [ t ] is completed with respect to its evaluation ideal J + = ( t − J + ⊂ Y ~ g at t = 1 to be the kernel of the composite Y ~ g ~ −−−→ U ( s ⋊ ¨ h ) → U ( g ) , where the second arrow is the epimorphism induced by the composition s ⋊ ¨ h π s ⊕ −−−→ ˙ g [ t ] ⋊ ¨ h ˙ev g ⊕ −−−−→ ˙ g ⋊ ¨ h ∼ = g , with π s as in (3.7) and ˙ev g as defined at the beginning of Section 5. Since the abovecoincides with the composite Y ~ g ı −→ D Y ~ g ~ −−−→ U ( t ⋊ ¨ h ) ev g −−→ U ( g ) , one has the equality ı ( J + ) = J ∩ ı ( Y ~ g ) ⊂ J . Consequently, ı induces a homomor-phism of C [[ ~ ]]-algebras b ı : d Y ~ g × → \ D Y ~ g , where d Y ~ g × is the completion of Y ~ g with respect to the descending filtration Y ~ g = J + ⊃ J + ⊃ · · · ⊃ J n + ⊃ · · · Our goal is to prove that b ı is an isomorphism using our work in the previoussection. We begin with the following analogue of Lemma 5.2, which asserts thatthe gradation and evaluation completions of Y ~ g are isomorphic. Lemma 5.8.
Setting Γ = τ − , we have the equality Γ ( Y ~ g + ) = J + . Consequently, Γ induces an isomorphism of C [[ ~ ]] -algebras b Γ : d Y ~ g ∼ −→ d Y ~ g × . HE FORMAL SHIFT OPERATOR ON THE YANGIAN DOUBLE 27
Proof.
The proof that Γ = ı ◦ Γ satisfies Γ( Y ~ g + ) ⊂ J , given in Lemma 5.2, showsthat Γ ( Y ~ g + ) ⊂ J + . Conversely, by Lemma 5.1, we have Φ( J ) ⊂ d Y ~ g + , and thus Γ − ( J + ) = τ ( J + ) = Φ( ı ( J + )) ⊂ Y ~ g ∩ d Y ~ g + = Y ~ g + . (cid:3) Combining this lemma with Theorem 5.5 yields an isomorphism of C [[ ~ ]]-algebras b Γ ◦ b Φ : \ D Y ~ g ∼ −→ d Y ~ g × . It follows from the identity Γ ◦ Φ ◦ ı = τ − ◦ τ = Y ~ g that b ı is a right inverse,and therefore the unique inverse, of this isomorphism. We have thus proven thefollowing corollary, which realizes our current goal. Corollary 5.9.
The C [[ ~ ]] -algebra homomorphisms b ı : d Y ~ g × → \ D Y ~ g and b Γ ◦ b Φ : \ D Y ~ g ∼ −→ d Y ~ g × are mutual inverses. In particular, b ı is an isomorphism of C [[ ~ ]] -algebras. This corollary affords the evaluation completion of D Y ~ g a rather explicit de-scription. Namely, it coincides with the gradation completion of Y ~ g with respectto the shifted N -grading Y ~ g = M n ∈ N Γ ( Y ~ g n ) , where Γ = τ − , as above. More precisely, one has the equality \ D Y ~ g = Y n ∈ N ( ı ◦ Γ )( Y ~ g n ) = Y n ∈ N Γ( Y ~ g n ) . The natural homomorphism D Y ~ g → \ D Y ~ g can be expressed in terms of thesecoordinates using Corollary 5.9. Alternatively, using Remark 5.4 we find that, foreach i ∈ I , one has the identity(5.12) u − δ (cid:16)b t ± i /u (cid:17) = δ ( u + z ) | z =1 − b t ± i in End( \ D Y ~ g )[[ u ± ]]. The right-hand side may be rewritten as δ ( u + z ) | z =1 − b t ± i = exp((1 − b t ± i ) ∂ u ) δ ( u ) = X n ∈ N (1 − b t ± i ) n ∂ ( n ) u ( δ ( u )) . Applying both sides of (5.12) to X ± i = ı ( x ± i ) therefore yields X ± i ( u ) = u − δ (cid:16)b t ± i /u (cid:17) X ± i = X n ∈ N (1 − b t ± i ) n ( X ± i ) ∂ ( n ) u ( δ ( u ))= X n ∈ N ( − n Γ( x ± in ) ∂ ( n ) u ( δ ( u ))in \ D Y ~ g , where in the last equality we have applied the second identity of Part (1)of Corollary 4.6 with z = − ℓ = n and k = 0. Note that, by (5.2), the abovecomputation recovers the identity X ± i ( u ) = ( b ı ◦ b Γ ◦ b Φ)( X ± i ( u )) of Corollary 5.9. Degeneration.
It was observed by Drinfeld [3, §
6] and later proven by Guayand Ma [13] that the Yangian of a finite-dimensional simple Lie algebra g may beviewed as a degeneration of the corresponding quantum loop algebra U ~ ( L g ). Inthe form presented in [13] and [7] this result can be stated as follows. The quantumloop algebra U ~ ( L g ) admits a descending filtration given by powers of its evaluationideal J at t = 1, and there is an isomorphism of graded C [ ~ ]-algebrasgr J ( U ~ ( L g )) ∼ −→ D Y ~ g . It was shown in [7, Prop. 6.5] that this isomorphism can be realized as the associatedgraded map gr(Φ
GTL ) of the algebra homomorphismΦ
GTL : U ~ ( L g ) → d Y ~ g of geometric type constructed in [7]. In this section, we present a D Y ~ g -analogueof this result in which U ~ ( L g ) and Φ GTL are replaced by D Y ~ g and Φ, and g is takento be an arbitrary symmetrizable Kac–Moody algebra.In what follows, we view D Y ~ g , Y ~ g and d Y ~ g as N -filtered algebras, with descend-ing filtrations {J n } n ∈ N , { Y ~ g n + } n ∈ N and { d Y ~ g ≥ n } n ∈ N , respectively. Note thatgr( d Y ~ g ) ∼ = gr Y ~ g = M n ∈ N Y ~ g n + (cid:14) Y ~ g n +1 + ∼ = M n ∈ N Y ~ g n = Y ~ g , as graded C [ ~ ]-algebras. Let Φ and Γ be as in (5.2) and (5.5), respectively. Corollary 5.10.
Γ : Y ~ g → D Y ~ g is filtered and the induced homomorphism gr(Γ) : Y ~ g → gr(D Y ~ g ) . is an isomorphism of graded C [ ~ ] -algebras with inverse given by gr(Φ) : gr(D Y ~ g ) → gr( d Y ~ g ) ∼ = Y ~ g . Proof.
By Lemma 5.2, Γ( Y ~ g + ) ⊂ J , and hence Γ is filtered. Similarly, Lemma 5.1implies that Φ is a filtered morphism.In the proof of Theorem 5.5 we showed thatΓ n : Y ~ g /Y ~ g n + → D Y ~ g / J n and Φ n : D Y ~ g / J n → Y ~ g /Y ~ g n + are mutual inverses for each n ∈ N . By Lemmas 5.1 and 5.2, we haveΓ n +1 ( Y ~ g n + /Y ~ g n +1 + ) = J n / J n +1 ∀ n ∈ N . Letting Γ ( n +1) and Φ ( n +1) denote the restrictions of Γ n and Φ n to Y ~ g n + /Y ~ g n +1 + and J n / J n +1 , respectively, we find thatgr(Γ) = ⊕ n ∈ N Γ ( n +1) : Y ~ g ∼ = M n ∈ N Y ~ g n + /Y ~ g n +1 + → gr(D Y ~ g ) = M n ∈ N J n / J n +1 is an isomorphism with inverse gr(Φ) = ⊕ n ∈ N Φ ( n +1) . (cid:3) HE FORMAL SHIFT OPERATOR ON THE YANGIAN DOUBLE 29
The above result can be rephrased in the language of one-parameter deformationsas follows. Let Y ~ ,v ( g ) be the C [ ~ , v ]-subalgebra of D Y ~ g [ v ± ] generated by v − J and D Y ~ g . Equivalently, Y ~ ,v ( g ) is the extended Rees algebra Y ~ ,v ( g ) = M n ∈ Z v − n J n ⊂ D Y ~ g [ v ± ] , where J − n = D Y ~ g for all n ∈ N . Then, by Corollary 5.10, Y ~ ,v ( g ) is a flatdeformation of the Yangian Y ~ g over C [ v ]. Indeed, Y ~ ,v ( g ) ⊂ D Y ~ g [ v ± ] is a torsionfree C [ v ]-module and one has Y ~ ,v ( g ) /vY ~ ,v ( g ) ∼ = M n ∈ Z J n / J n +1 = gr(D Y ~ g ) ∼ = gr(Φ) Y ~ g .
6. D Y ~ g as a flat deformation We now apply our construction to prove a Poincar´e–Birkhoff–Witt theorem forD Y ~ g , applicable when g is of finite type or of simply-laced affine type. The pre-cise statement of this result, given in Theorem 6.2, simultaneously establishes theinjectivity of both Φ z and Φ c for all such g , and therefore that D Y ~ g can be viewedas both a subalgebra of L d Y ~ g z ⊂ d Y ~ g [[ z ± ]] and of d Y ~ g . Here we note that theinjectivity of Φ is not an immediate consequence of Theorem 5.5, since the naturalhomomorphism D Y ~ g → \ D Y ~ g has kernel equal to the intersection of all powers J n of J , which need not vanish.Theorem 6.2 nevertheless implies that this intersection is indeed trivial, at least inthe finite and simply-laced affine cases.6.1. The classical limit of Φ . Since the quotient map Y ~ g → Y ~ g / ~ Y ~ g ∼ = U ( s ⋊ ¨ h )is N -graded, it induces an isomorphism d Y ~ g / ~ d Y ~ g ∼ −→ \ U ( s h ) , where s h := s ⋊ ¨ h and \ U ( s h ) is the formal completion of U ( s h ) with respect to its N -grading: \ U ( s h ) = Y n ∈ N U ( s h ) n . The classical limit ¯Φ of Φ is then the homomorphism U ( t ⋊ ¨ h ) → \ U ( s h ) uniquelydetermined by the requirement that the following diagram commute:(6.1) D Y ~ g d Y ~ g U ( t ⋊ ¨ h ) \ U ( s h ) Φ¯Φ where the vertical arrows are given by reducing modulo ~ . By (5.2), ¯Φ is givenexplicitly by the formulas¯Φ( h ) = h, ¯Φ( X ± i ( u )) = X n ∈ N ( − n X ± in ∂ ( n ) u ( δ ( u )) ∀ i ∈ I and h ∈ h , where X ± i ( u ) = X r ∈ Z X ± ir u − r − ∈ t [[ u ± ]] . Remark 6.1.
Since Φ ◦ ı = τ admits an invertible classical limit, it follows thatthe classical limit of ı , which coincides with the natural homomorphism U ( s ⋊ ¨ h ) → U ( t ⋊ ¨ h ) , is injective. This justifies our use of the same notation for generators of s and t .The above formulas for ¯Φ imply that ¯Φ( t ) ⊂ b s , where b s is the Lie algebra b s = Y n ∈ N s n ⊂ [ U ( s ) . We may therefore define φ to be the homomorphism of Lie algebras φ := ¯Φ | t : t → b s . Consider now the injection C [ w ] ֒ → C [[ t ]] given by w t +1. As t +1 is invertiblein C [[ t ]] with inverse ( t + 1) − = X k ≥ ( − k t k , this morphism uniquely extends to γ : C [ w ± ] ֒ → C [[ t ]]. We thus obtain an injectivehomomorphism of Lie algebras ⊗ γ : ˙ g [ w ± ] = ˙ g ⊗ C [ w ± ] ֒ → ˙ g [[ t ]] = ˙ g ⊗ C [[ t ]] . The homomorphism φ then satisfies the commutative diagram(6.2) t b s ˙ g [ w ± ] ˙ g [[ t ]] π t φ b π s ⊗ γ where π s and π t are the graded homomorphisms defined in (3.7), and b π s is obtainedfrom π s by extending by continuity. We end this subsection by noting that, when g is of finite type, the vertical arrows in (6.2) are isomorphisms and, consequently, φ is injective. We will see below that this holds in a much more general context.6.2. D Y ~ g as a flat deformation. The following theorem is the main result ofthis section.
Theorem 6.2.
Suppose that φ : t → b s is injective and that Y ~ ˙ g is a torsion free C [ ~ ] -module. Then: (1) For any fixed c ∈ C × , the algebra homomorphisms Φ z : D Y ~ g → L d Y ~ g z and Φ c : D Y ~ g → d Y ~ g are injective. (2) D Y ~ ˙ g and D Y ~ g are flat deformations of U ( t ) and U ( t ⋊ ¨ h ) , respectively,over C [[ ~ ]] . In particular, there are isomorphisms of C [[ ~ ]] -modules D Y ~ ˙ g ∼ = U ( t )[[ ~ ]] and D Y ~ g ∼ = U ( t ⋊ ¨ h )[[ ~ ]] . Moreover, the hypotheses on φ and Y ~ ˙ g are satisfied whenever g is of finite type orsimply-laced affine type. HE FORMAL SHIFT OPERATOR ON THE YANGIAN DOUBLE 31
Remark 6.3.
In fact, we will show that φ is injective whenever g is of untwistedaffine type with underlying finite-dimensional simple Lie algebra ¯ g ≇ sl . Although Y ~ ˙ g is expected to be torsion free for all such g , this remains a conjecture. Proof.
Let us first clarify why the hypotheses on φ and Y ~ ˙ g are satisfied in theclaimed cases. That Y ~ ˙ g is torsion free when g is of finite type is due to Levendorskii[28] (see also [6, Thm. B.6] and [15, Prop. 2.2]). We have seen that φ is injectivein this case at the end of Section 6.1.It has recently been proven independently in [16] and [37] that Y ~ ˙ g is a torsionfree C [ ~ ]-module when g is of simply-laced affine type. We will prove that φ isinjective when g is of untwisted affine type in Section 6.6 using the identifications s ∼ = uce ( ˙ g [ t ]) and t κ ∼ = uce ( ˙ g [ t ± ]), which are made concrete in Sections 6.3–6.5.Now let us turn to proving (1) and (2) under the assumption that g is such thatthe hypotheses on Y ~ ˙ g and φ hold. Proof of (1) . By Part (3) of Theorem 4.3 and Proposition 4.7, we have
E v ◦ Φ z = Φ and Φ c = χ ı /c ◦ Φ ◦ χ c ∀ c ∈ C × . Therefore, it suffices to show that Φ is injective. Taking the direct sum of φ withthe identity map on ¨ h , we obtain an injective homomorphism of Lie algebras φ ⊕ : t ⋊ ¨ h → b s ⋊ ¨ h , where the action of ¨ h on b s is obtained from that of ¨ h on s by extending by continuity.By the Poincar´e–Birkhoff–Witt theorem for enveloping algebras, the above mapinduces an injective homomorphism of algebras U ( t ⋊ ¨ h ) → U ( b s ⋊ ¨ h ) ⊂ \ U ( s h ) , which is precisely the classical limit ¯Φ of Φ introduced in Section 6.1. In particular,¯Φ is injective.Suppose now that x ∈ D Y ~ g is nonzero. We will employ a standard argumentto show that x / ∈ Ker(Φ). Since D Y ~ g is a separated C [[ ~ ]]-module, there is k ∈ N such that x = ~ k y, where y / ∈ ~ D Y ~ g . Since the image of y in D Y ~ g / ~ D Y ~ g is nonzero and ¯Φ is injective, the commutativ-ity of the diagram (6.1) implies that y / ∈ Ker(Φ). Moreover, as Y ~ g = Y ~ ˙ g ⋊ U (¨ h ) istorsion free, Part (3) of Lemma 4.1 implies that d Y ~ g is torsion free. We may thusconclude that Φ( x ) = ~ k Φ( y ) = 0 , and therefore that Φ is injective, as desired. Proof of (2) . By Proposition 3.7, D Y ~ ˙ g and D Y ~ g are deformations of U ( t ) and U ( t ⋊ ¨ h ), respectively, over C [[ ~ ]].To prove that they are flat deformations, it suffices to show that they are sep-arated, complete and torsion free C [[ ~ ]]-modules (see [22, Prop. XVI.2.4], for in-stance). They are separated and complete by definition, having been defined topo-logically in terms of generators and relations. They are torsion free since Φ isinjective and d Y ~ g is torsion free, as explained in the Proof of (1). (cid:3) Kassel’s realization.
For the remainder of Section 6, we assume that g is anuntwisted affine Kac–Moody algebra with underlying simple Lie algebra ¯ g ≇ sl .We then have ˙ g ∼ = ¯ g [ v ± ] ⊕ C c as a vector space, with Lie bracket determined by [ c, ˙ g ] = 0 and[ x ⊗ v r , y ⊗ v s ] = [ x, y ] ⊗ v r + s + rδ r, − s ( x, y ) c for all x, y ∈ ¯ g and r, s ∈ Z . By [16, Prop. 4.7], there are isomorphisms(6.3) uce ( ˙ g [ t ]) ∼ = uce (¯ g [ v ± , t ]) and uce ( ˙ g [ t ± ]) ∼ = uce (¯ g [ v ± , t ± ]) . To prove that φ is injective, we shall make use of isomorphisms t κ ∼ −→ uce (¯ g [ v ± , t ± ]) and s ∼ −→ uce (¯ g [ v ± , t ])obtained in the work of Moody–Rao–Yokonuma [30], which coupled with (6.3) yield(3.8). Their construction is partly based on a general result due to Kassel [21], whichprovides an explicit realization of uce (¯ g ⊗ A ), where A is an arbitrary commutative,associative algebra over the complex numbers. For the sake of completeness, werecall some of the relevant general theory below, beginning with Kassel’s realization.After briefly discussing some auxiliary properties of this realization in Section 6.4,we will review the relevant results from [30] in Section 6.5.Let (Ω( A ) , d ) be the module of K¨ahler differentials associated to A . That is,Ω( A ) is the A -module Ω( A ) = ( A ⊗ A ) /M, where A acts on A ⊗ A by left multiplication in the first tensor factor, and M isthe submodule of A ⊗ A generated by 1 ⊗ ab − a ⊗ b − b ⊗ a for all a, b ∈ A . Thedifferential map d is then the derivation d : A → Ω( A ) , d ( a ) = 1 ⊗ a mod M ∀ a ∈ A. In particular, we can (and will) write ad ( b ) for the equivalence class of the tensor a ⊗ b in Ω( A ). We shall use the same notation for generators of the quotient z ( A ) := Ω( A ) /d ( A ) . Consider the alternating bilinear map ε : (¯ g ⊗ A ) × (¯ g ⊗ A ) → z ( A ) determined by ε ( x ⊗ a, y ⊗ b ) = ( x, y ) bd ( a ) ∀ x, y ∈ ¯ g , a, b ∈ A. Using the invariance of ( · , · ) and the fact that in z ( A ) we have d ( abe ) = 0 = ab · d ( e ) + ae · d ( b ) + be · d ( a ) ∀ a, b, e ∈ A, one readily concludes that ε satisfies the cocycle equation ε ( x ⊗ a, [ y ⊗ b, z ⊗ e ]) + ε ( y ⊗ b, [ z ⊗ e, x ⊗ a ]) + ε ( z ⊗ e, [ x ⊗ a, y ⊗ b ]) = 0 , for all x, y, z ∈ ¯ g and a, b, e ∈ A . It follows that the vector space u ( A ) := (¯ g ⊗ A ) ⊕ z ( A )admits the structure of a Lie algebra with bracket given by [ u ( A ) , z ( A )] = 0 and(6.4) [ x ⊗ a, y ⊗ b ] = [ x, y ] ⊗ ab + ( x, y ) bd ( a ) ∀ x, y ∈ ¯ g , a, b ∈ A. It is clear that u ( A ) is a central extension of ¯ g ⊗ A . In fact, we have the followingremarkable result due to Kassel [21, Thm. 3.3] (see also [30, Prop. 2.2]). HE FORMAL SHIFT OPERATOR ON THE YANGIAN DOUBLE 33
Proposition 6.4. u ( A ) is isomorphic to the universal central extension of ¯ g ⊗ A : u ( A ) ∼ = uce (¯ g ⊗ A ) . Gradings on u ( A ) . If in addition A = L k ∈ Z A k is a Z -graded algebra, thenthe Lie algebra ¯ g ⊗ A is naturally Z -graded, with k -th graded component(¯ g ⊗ A ) k = ¯ g ⊗ A k ∀ k ∈ Z . The grading on A also naturally induces a Z -graded A -module structure on Ω( A ),compatible with that on A ⊗ A . As the subspace d ( A ) is itself graded, z ( A ) is agraded C -vector space. By (6.4), it follows that u ( A ) inherits the structure of a Z -graded Lie algebra, with k -th graded component u ( A ) k = (¯ g ⊗ A ) k ⊕ z ( A ) k , where z ( A ) k is the k -th graded component of z ( A ). If A k = { } for all k < A is N -graded), then we may complete u ( A ) with respect to its induced N -grading to obtain a Lie algebra [ u ( A ) = (¯ g ⊗ Y k ∈ N A k ) ⊕ Y k ∈ N z ( A ) k , with Lie bracket determined by (6.4), for a, b ∈ b A = Q k ∈ N A k , together with therequirement that Q k ∈ N z ( A ) k be central.The next result we will need concerns the functorial nature of u ( A ) and itscompatibility with the above completion process. In what follows, A and B areassociative, commutative C -algebras, with A taken to be N -graded, as above. Ad-ditionally, let us assume we are given an algebra homomorphism γ : B → b A. Lemma 6.5.
There is a unique Lie algebra homomorphism φ γ : u ( B ) → [ u ( A ) with the property that φ γ | ¯ g ⊗ B = ⊗ γ . Explicitly, φ γ | z ( B ) is given by (6.5) φ γ ( bd ( e )) = γ ( b ) d ( γ ( e )) ∀ b, e ∈ B. Proof.
The bracket relation (6.4) implies that, if φ γ exists, then it must satisfy(6.5). Conversely, if (6.5) determines a well-defined map z ( B ) → d z ( A ) := Y k ∈ N z ( A ) k , then (6.4) implies that φ γ , uniquely determined by φ γ | ¯ g ⊗ B = ⊗ γ and (6.5), willbe a Lie algebra homomorphism.Since the natural quotient map A ⊗ A → z ( A ) is N -graded, it induces a linearmap A b ⊗ A → d z ( A ), where A b ⊗ A is the completion of A ⊗ A with respect to its N -grading. The composition B ⊗ γ ⊗ γ −−−→ b A ⊗ ֒ → A b ⊗ A → d z ( A )then sends b ⊗ e to γ ( b ) d ( γ ( e )) and factors through Ω( B ) and z ( B ) = Ω( B ) /d ( B ),as desired. (cid:3) The Moody–Rao–Yokonuma isomorphism.
We now narrow our focus tothe special case where A is the Z -graded algebra C [ v ± , t ± ] or the N -graded algebra C [ v ± , t ], where deg t = 1 and deg v = 0.As in [30, § A -module Ω( A ) is freely generated by d ( v ) and d ( t ).Using that, in z ( A ), we have0 = d ( v r t s ) = sv r t s − d ( t ) + rt s v r − d ( v ) , one deduces ( cf . [4, 30]) that z ( A ) admits the vector space decomposition z ( C [ v ± , t ± ]) = M ( r,s ) ∈ Z × Z C K r,s ⊕ C c v ⊕ C c t , z ( C [ v ± , t ]) = M ( r,s ) ∈ Z × N + C K r,s ⊕ C c v , where K , := 0, and for ( r, s ) ∈ Z × Z × we have(6.6) K r,s = 1 s v r − t s d ( v ) , K s, = − s v s t − d ( t ) , c v = v − d ( v ) , c t = t − d ( t ) . By (6.3) and Proposition 6.4, we can (and will) identify uce ( ˙ g [ t ± ]) and uce ( ˙ g [ t ])with the Lie algebras u ( C [ v ± , t ± ]) and u ( C [ v ± , t ]), respectively. In particular, uce ( ˙ g [ t ± ]) = ¯ g [ v ± , t ± ] ⊕ z ( C [ v ± , t ± ])as a vector space, with Lie structure such that z ( C [ v ± , t ± ]) is central and[ x ⊗ v r t s , y ⊗ v k t ℓ ] = [ x, y ] ⊗ v r + k t s + ℓ + ( x, y ) v k t ℓ d ( v r t s )for all x, y ∈ ¯ g and r, s, k, ℓ ∈ Z . Moreover, in terms of the basis (6.6), we have v k t ℓ d ( v r t s ) = δ r, − k δ s, − ℓ ( rc v + sc t ) + ( rℓ − sk )K r + k,s + ℓ , By definition, this is equivalent to v k t ℓ d ( v r t s ) = ((cid:16) rℓ − sks + ℓ (cid:17) v r + k − t s + ℓ d ( v ) if s = − ℓ,δ r, − k rv − d ( v ) + sv r + k t − d ( t ) if s = − ℓ. The Lie algebra uce ( ˙ g [ t ]) may then be characterized as the Lie subalgebra uce ( ˙ g [ t ]) = ¯ g [ v ± , t ] ⊕ M ( r,s ) ∈ Z × N + C K r,s ⊕ C c v ⊂ uce ( ˙ g [ t ± ]) . As a consequence of the general discussion in § uce ( ˙ g [ t ± ]) and uce ( ˙ g [ t ]) are Z and N -graded Lie algebras, respectively, with gradings determined by deg t = 1.In order to make precise the isomorphisms of (3.8), let us specify g to be rank ℓ + 1, with I taken to be { , . . . , ℓ } so that ¯ I = { , . . . , ℓ } labels the simple roots of¯ g . Let x ± θ ∈ ¯ g ± θ be such that ( x + θ , x − θ ) = 1, where θ is the highest root of ¯ g .The following result is a translation of [30, Prop. 3.5]. It appears in the formbelow in [16]; see Propositions 4.4 and 4.7 therein. Recall that t κ is the one-dimensional central extension of t introduced in Definition 3.6. Proposition 6.6.
The assignment K c t , X ± ir x ± i ⊗ t r , X ± r x ∓ θ ⊗ v ± t r ∀ i ∈ ¯ I , r ∈ Z uniquely extends to an isomorphism of Z -graded Lie algebras ψ : t κ ∼ −→ uce ( ˙ g [ t ± ]) . Moreover, ψ induces isomorphisms of graded Lie algebras ψ | s : s ∼ −→ uce ( ˙ g [ t ]) and ψ t : t ∼ −→ uce ( ˙ g [ t ± ]) / C c t . HE FORMAL SHIFT OPERATOR ON THE YANGIAN DOUBLE 35
Injectivity of φ . We now combine the results collected in Sections 6.3–6.5 toprove that φ : t → b s is injective when g is of untwisted affine type.Since ψ | s from Proposition 6.6 is graded, it extends to an isomorpism c ψ | s : b s ∼ −→ uce [ ( ˙ g [ t ]) = \ u ( C [ v ± , t ]) . As illustrated in Section 6.4, the right-hand side above may be identified with(¯ g ⊗ C [ v ± ][[ t ]]) ⊕ Y s ∈ N z ( C [ v ± , t ]) s , where z ( C [ v ± , t ]) s = ( C c v if s = 0 , L r ∈ Z C K r,s if s ∈ N + . Next, let us us extend γ : C [ w ± ] ֒ → C [[ t ]] of (6.2) to a homomorphism γ : C [ v ± , w ± ] ֒ → \C [ v ± , t ] = C [ v ± ][[ t ]]by setting γ ( v ) = v . By Lemma 6.5, ⊗ γ : ¯ g [ v ± , w ± ] → ¯ g ⊗ C [ v ± ][[ t ]] extendsuniquely to a homomorphism of Lie algebras φ γ : uce ( ˙ g [ w ± ]) → uce [ ( ˙ g [ t ])with φ γ ( f dg ) = γ ( f ) d ( γ ( g )) ∀ f, g ∈ C [ v ± , w ± ] . The following proposition completes our proof that φ : t → b s is injective when g isof untwisted affine type, and therefore completes the proof of Theorem 6.2. Proposition 6.7.
Let c ψ | s and φ γ be as above. Then: (1) Ker( φ γ ) = C c w . Consequently, φ γ induces an injection ¯ φ γ : uce ( ˙ g [ w ± ]) / C c w ֒ → uce [ ( ˙ g [ t ]) . (2) φ : t → b s is injective, and satisfies φ = ( c ψ | s ) − ◦ ¯ φ γ ◦ ψ t .Proof. Let us begin by establishing that the kernel of φ γ coincides with C c w . Proof of (1) . Since φ γ | ¯ g [ v ± ,w ± ] = ⊗ γ is injective and φ γ ( z ( C [ v ± , w ± ])) ⊂ \ z ( C [ v ± , t ]) , it suffices to show that the restriction of φ γ to z ( C [ v ± , w ± ]) has kernel C c w .In \ z ( C [ v ± , t ]), we have the relations γ ( v r w − ) d ( γ ( w )) = X k ≥ ( − k v r t k d ( t ) , v r t k d ( t ) = − rk + 1 v r − t k +1 d ( v ) . Hence, φ γ is determined on z ( C [ v ± , w ± ]) by φ γ ( v r w s d ( v )) = v r ( t + 1) s d ( v ) ,φ γ ( v r w − d ( w )) = − rv r − X k ≥ ( − k t k +1 k + 1 d ( v ) = − rv r − log( t + 1) d ( v ) , for all r, s ∈ Z . In particular, φ γ ( c w ) = φ γ ( w − d ( w )) = 0. Since z ( C [ v ± , w ± ]) has basis { v r w s d ( v ) } ( r,s ) ∈ Z × Z × ∪ { v r w − d ( w ) } r ∈ Z ∪ { v − d ( v ) } , to show that Ker( φ γ ) = C c w , it suffices to prove that the set { v r ( t + 1) s d ( v ) } ( r,s ) ∈ Z × Z × ∪ { v r − log( t + 1) d ( v ) } r ∈ Z × ∪ { c v } is linearly independent in \ z ( C [ v ± , t ]) = Y s ∈ N + z ( C [ v ± , t ]) s ⊕ C c v ∼ = t C [ v ± ][[ t ]] ⊕ C v − ⊂ C [ v ± ][[ t ]] , where the embedding of vector spaces \ z ( C [ v ± , t ]) ⊂ C [ v ± ][[ t ]] alluded to above isdetermined by identifying v r t s d ( v ) with v r t s .This follows readily from the observation that { ( t + 1) s } s ∈ Z × ∪ { log( t + 1) } is alinearly independent set in C [[ t ]], which can be deduced using the injectivity of γ and the observation that γ ( w s − ) = ( s ∂ t ( t + 1) s if s = 0 ,∂ t log( t + 1) if s = 0 . Proof of (2) . It suffices to verify the identity φ = ( c ψ | s ) − ◦ ¯ φ γ ◦ ψ t on the generatingset { X ± ir } i ∈ I ,r ∈ Z of t . This is easily done directly, using the explicit formulas for ψ t and ψ s given in Proposition 6.6 (see also (6.2)). (cid:3) Appendix A. Grading completions
In this appendix we prove Proposition A.1 which serves to clarify a number ofproperties satisfied by the grading completion of any N -graded C [ ~ ]-algebra. Thisproposition has been applied to prove Lemma 4.1 and, though it is elementary, hasbeen included for sake of completeness. Proposition A.1.
Let
A = L n ∈ N A n be a N -graded C [ ~ ] -algebra satisfying: (a) ~ A ⊂ A + = L n> A n , (b) A n + = L k ≥ n A k for each n ∈ N .Then the formal completion of A with respect to its N -grading, b A = Y n ∈ N A n , is a unital, associative C [[ ~ ]] -algebra. Moreover: (1) The canonical C [ ~ ] -algebra homomorphism Υ : A → lim ←− n (cid:0) A / A n + (cid:1) extendsto an isomorphism of C [[ ~ ]] -algebras b Υ : b A = Y n ∈ N A n ∼ −→ lim ←− n (cid:0) A / A n + (cid:1) . (2) b A is separated and complete as a C [[ ~ ]] -module. (3) b A is a torsion free C [[ ~ ]] -module, provided A is a torsion free C [ ~ ] -module. HE FORMAL SHIFT OPERATOR ON THE YANGIAN DOUBLE 37
Proof.
First note that the condition (a) guarantees that both completions of Aappearing above are unital, associative C [[ ~ ]]-algebras.To prove (1), note that (b) implies that, for each n ∈ N , we have b A / b A ≥ n ∼ = A / A n + , where b A ≥ n = Y k ≥ n A k . We thus have a canonical homomorphism of C [[ ~ ]]-algebras b Υ : b A → lim ←− n (cid:16) b A / b A ≥ n (cid:17) ∼ = lim ←− n (cid:0) A / A n + (cid:1) . Moreover, the composite of b Υ with the inclusion A ֒ → b A coincides with Υ.Under the identifying A / A n + ∼ = L n − k =0 A k , the projection p n +1 : A / A n +1+ → A / A n + coincides with the truncation operator x + . . . + x n x + . . . + x n − . Wemay therefore identifylim ←− n (cid:0) A / A n + (cid:1) = { ( x + . . . + x n − ) n ∈ N : x k ∈ A k } ⊂ Y n ∈ N n − M k =0 A k ! Under this identification, we have b Υ : X k ≥ x k ( x + . . . + x n − ) n ∈ N , from which the bijectivity of b Υ follows immediately.Consider Part (3). If x = P k x k ∈ b A is such that ~ x = 0, then we must have ~ x k = 0 for each k . As A is assumed to be torsion free, we can conclude that each x k , and thus x itself, vanishes.It remains to prove (2). By (a), ~ n b A ⊂ b A ≥ n , and thus \ n ∈ N ~ n b A ⊂ \ n ∈ N b A ≥ n = { } . Therefore, b A is a separated C [[ ~ ]]-module. To show it is also complete, we mustargue that the natural homomorphismΘ : b A → lim ←− n (cid:16)b A / ~ n b A (cid:17) is surjective. To this end, note that any x ∈ lim ←− n (cid:16)b A / ~ n b A (cid:17) may be represented as x = (cid:16)P n ≥ q k ( x k,n ) (cid:17) k ∈ N , where(i) x k,n ∈ A n satisfy x k,n − x ℓ,n ∈ ~ min( k,ℓ ) A for all k, ℓ, n ∈ N . (ii) q k : A → A / ~ k A is the natural quotient map.Set x n = x n +1 ,n ∈ A n for all n ∈ N . We claim that x is equal to the image of P n x n under Θ. To prove this, it suffices to show that x n − x k,n ∈ ~ k A ∀ k, n ∈ N . By (i), x n − x k,n ∈ ~ min( n +1 ,k ) A for all k and n , hence the assertion is true for k < n . If k ≥ n then, by (a) and (b), we have x n − x k,n ∈ ~ n +1 A ⊂ M k>n A k , which implies that x n − x k,n = 0 ∈ ~ k A, as desired. (cid:3)
References [1] M. Bershtein and A. Tsymbaliuk,
Homomorphisms between different quantum toroidal andaffine Yangian algebras , J. Pure Appl. Algebra (2019), no. 2, 867–899.[2] V. Drinfel’d,
Hopf algebras and the quantum Yang-Baxter equation , Soviet Math. Dokl. (1985), no. 1, 254–258.[3] , Quantum groups , Proceedings of the International Congress of Mathematicians, Vol.1, 2 (Berkeley, Calif., 1986), 1987, pp. 798–820.[4] B. Enriquez,
PBW and duality theorems for quantum groups and quantum current algebras ,J. Lie Theory (2003), no. 1, 21–64.[5] , Quasi-Hopf algebras associated with semisimple Lie algebras and complex curves ,Selecta Math. (N.S.) (2003), no. 1, 1–61.[6] M. Finkelberg and A. Tsymbaliuk, Shifted quantum affine algebras: integral forms in type A ,Arnold Math. J. (2019), no. 2-3, 197–283.[7] S. Gautam and V. Toledano Laredo, Yangians and quantum loop algebras , Selecta Math.(N.S.) (2013), no. 2, 271–336.[8] , Yangians, quantum loop algebras, and abelian difference equations , J. Amer. Math.Soc. (2016), no. 3, 775–824.[9] S. Gautam, V. Toledano Laredo, and C. Wendlandt, The meromorphic R-matrix of the Yan-gian , 2019. arXiv:1907.03525 .[10] N. Guay,
Cherednik algebras and Yangians , Int. Math. Res. Not. (2005), 3551–3593.[11] , Affine Yangians and deformed double current algebras in type A , Adv. Math. (2007), no. 2, 436–484.[12] ,
Quantum algebras and quivers , Selecta Math. (N.S.) (2009), no. 3-4, 667–700.[13] N. Guay and X. Ma, From quantum loop algebras to Yangians , J. Lond. Math. Soc. (2) (2012), no. 3, 683–700.[14] N. Guay, H. Nakajima, and C. Wendlandt, Coproduct for Yangians of affine Kac-Moodyalgebras , Adv. Math. (2018), 865–911.[15] N. Guay, V. Regelskis, and C. Wendlandt,
Equivalences between three presentations of or-thogonal and symplectic Yangians , Lett. Math. Phys. (2019), no. 2, 327–379.[16] ,
Vertex representations for Yangians of Kac-Moody algebras , J. ´Ec. polytech. Math. (2019), 665–706.[17] K. Iohara, Bosonic representations of Yangian double D Y ℏ ( g ) with g = gl N , sl N , J. Phys. A (1996), no. 15, 4593–4621.[18] N. Jing, S. Koˇzi´c, A. Molev, and F. Yang, Center of the quantum affine vertex algebra intype A , J. Algebra (2018), 138–186.[19] N. Jing, F. Yang, and M. Liu, Yangian doubles of classical types and their vertex represen-tations , J. Math. Phys. (2020), no. 5, 051704, 39.[20] V. Kac, Infinite-dimensional Lie algebras , Third edition, Cambridge University Press, Cam-bridge, 1990.[21] C. Kassel,
K¨ahler differentials and coverings of complex simple Lie algebras extended over acommutative algebra , Proceedings of the Luminy conference on algebraic K -theory (Luminy,1983), 1984, pp. 265–275.[22] , Quantum groups , Graduate Texts in Mathematics, vol. 155, Springer-Verlag, NewYork, 1995.[23] S. Khoroshkin,
Central extension of the Yangian double , Alg`ebre non commutative, groupesquantiques et invariants (Reims, 1995), 1997, pp. 119–135.[24] S. Khoroshkin and V. Tolstoy,
Yangian double , Lett. Math. Phys. (1996), no. 4, 373–402.[25] R. Kodera, Higher level Fock spaces and affine Yangian , Transform. Groups (2018), no. 4,939–962. HE FORMAL SHIFT OPERATOR ON THE YANGIAN DOUBLE 39 [26] ,
Affine Yangian action on the Fock space , Publ. Res. Inst. Math. Sci. (2019),no. 1, 189–234.[27] , Braid group action on affine Yangian , SIGMA Symmetry Integrability Geom. Meth-ods Appl. (2019), Paper No. 020, 28.[28] S. Levendorski˘ı, On PBW bases for Yangians , Lett. Math. Phys. (1993), no. 1, 37–42.[29] S. Montgomery, Hopf algebras and their actions on rings , CBMS Regional Conference Seriesin Mathematics, vol. 82, Published for the Conference Board of the Mathematical Sciences,Washington, DC; by the American Mathematical Society, Providence, RI, 1993.[30] R. V. Moody, S. E. Rao, and T. Yokonuma,
Toroidal Lie algebras and vertex representations ,Geom. Dedicata (1990), no. 1-3, 283–307.[31] M. Nazarov, Double Yangian and the universal R -matrix , Jpn. J. Math. (2020), no. 1,169–221.[32] A. Tsymbaliuk, The affine Yangian of gl revisited , Adv. Math. (2017), 583–645.[33] , Classical limits of quantum toroidal and affine Yangian algebras , J. Pure Appl.Algebra (2017), no. 10, 2633–2646.[34] C. Wendlandt,
The restricted quantum double of the Yangian . In preparation.[35] Y. Yang and G. Zhao,
The cohomological Hall algebra of a preprojective algebra , Proc. Lond.Math. Soc. (3) (2018), no. 5, 1029–1074.[36] ,
Cohomological Hall algebras and affine quantum groups , Selecta Math. (N.S.) (2018), no. 2, 1093–1119.[37] , The PBW theorem for affine Yangians , Transform. Groups (2020).doi:10.1007/s00031-020-09572-6.
Department of Mathematics, The Ohio State University.
E-mail address ::