Yangians and Quantum Loop Algebras – II, in Preparation

Home , Quantum group

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1.2. Since their construction by Drinfeld [9, 10], these aﬃne quantum groups have been extensively studied from several perspectives (see, e.g., [5, chap. 12], [2] and references therein), and are widely believed to share the same ﬁnite–dimensional representation theory. This belief is corroborated in part by the following facts

(1) The quantum loop algebra U~(Lg) degenerates to the Yangian Y~(g). Specif- ically, if U~(Lg) is filtered by the powers of the evaluation ideal at z = 1, its associated graded is isomorphic to Y~(g) [10, 17]. (2) Finite–dimensional simple modules over U~(Lg) are parametrised by I–tuples of (Drinfeld) polynomials Pi(u) i I satisfying Pi(0) = 1, where I is the set of vertices of the Dynkin{ diagram} ∈ of g [4]. Similarly, finite–dimensional simple modules over Y~(g) are classified by I–tuples of monic polynomials [11, 28, 29, 3]. (3) If g is simply laced, there exists, for every w NI, a Steinberg variety Z(w) ∈ endowed with an action of GL(w) C× (here GL(w) = i I GLwi ), and algebra homomorphisms × ∈ Q GL(w) C× Ψ : U~(Lg) K × (Z(w)) U → GL(w) C× Ψ : Y~(g) H × (Z(w)) Y → The variety Z(w) and the homomorphism ΨU were constructed by Nakajima [26], while ΨY was constructed by Varagnolo [33].

1.3. The above results go some way towards relating the categories of finite– dimensional representations of U~(Lg) and Y~(g). For example, exponentiating the roots of Drinfeld polynomials yields, via (2), a surjective map exp∗ between the set of isomorphism classes of irreducible finite–dimensional modules of Y~(g) and those of U~(Lg)–modules. If g is simply laced, the geometric realisations (3) imply further that exp∗ preserves the dimensions of these representations [33]. Despite these results however, and to the best of our knowledge, no natural relation between the categories of finite–dimensional representations of U~(Lg) and Y~(g) is known. Part of the difficulty in exploiting, say, the geometric realisations to pursue this question lies in the fact that the homomorphisms ΨU , ΨY are neither injective nor surjective [26]. Moreover, although these realisations yield all irreducible

representations, the categories Repfd(U~(Lg)) and Repfd(Y~(g)) are not semisimple.

1.4. The aim of the present paper is to clarify the relation between U~(Lg) and Y~(g). We do so by constructing an explicit algebra homomorphism [ Φ : U~(Lg) Y~(g) −→ [ where Y~(g) is the completion of Y~(g) with respect to its N–grading, and show that it induces an isomorphism of completed algebras. We also show that Φ exponentiates YANGIANSANDQUANTUMLOOPALGEBRAS 3

the roots of Drinfeld polynomials, though we defer the study of the corresponding pull–back functor

F =Φ∗ : Rep (Y~(g)) Rep (U~(Lg)) fd → fd to the sequel of this paper [13]. To state out results more precisely, recall that U~(Lg) and Y~(g) are deformations 1 of the loop and current algebras U(g[z, z− ]) and U(g[s]) respectively. Denote by

U~(Lh),U~(Lb ) U~(Lg) and Y~(h),Y~(b ) Y~(g) ± ⊂ ± ⊂

1 1 the subalgebras deforming U(h[z, z− ]), U(b [z, z− ]) and U(h[s]), U(b [s]) respectively, where h g is the Lie algebra of H± and b g are the opposite± Borel ⊂ ± ⊂ subalgebras corresponding to a choice αi i I of simple roots of g. For any i I, { } ∈ ∈ let sli g be the corresponding 3–dimensional subalgebra and denote by 2 ⊂

i i U~(Lsl ) U~(Lg) and Y~(sl ) Y~(g) 2 ⊂ 2 ⊂

i 1 i the subalgebras which deform U(sl2[z, z− ]) and U(sl2[s]) respectively. Then, the main result of this paper is the following

[ Theorem. There exists an explicit algebra homomorphism Φ : U~(Lg) Y~(g) with the following properties → (1) Φ is deﬁned over Q[[~]]. \ [ \ (2) Φ induces an isomorphism U~(Lg) Y~(g), where U~(Lg) is the completion → of U~(Lg) with respect to the ideal of z = 1. (3) Φ induces Drinfeld’s degeneration of U~(Lg) to Y~(g). [ (4) Φ restricts to a homomorphism U~(Lh) Y~(h) which induces the exponen- tiation of roots on Drinfeld polynomials.→ \ (5) Φ restricts to a homomorphism U~(Lb ) Y~(b ). ± → ± i \i (6) Φ restricts to a homomorphism U~(Lsl ) Y~(sl ) for any i I. 2 → 2 ∈ It is interesting to note that Theorem 1.4 stands in stark contrast with the analogous ﬁnite–dimensional situation. Indeed, if g ≇ sl2, no explicit isomorphisms are known between the quantum group U~g and the undeformed enveloping algebra Ug[[~]] [5, i 6.4]. Moreover, if g ≇ sl , no algebra isomorphism U~g Ug[[~]] maps U~sl to § 2 → 2 Usli [[~]] for every i I [31, Prop. 3.2]. 2 ∈

1.5. The homomorphism Φ has the following form. Let Ei,k, Fi,k,Hi,k i I,k Z be { } ∈ ∈ the loop generators of U~(Lg) and xi,m± ,ξi,m i I,m N those of Y~(g) (see [11] and { } ∈ ∈ 4 S.GAUTAMANDV.TOLEDANOLAREDO

Section 2 for definitions). Then, 1 Φ(Hi,0)= di− ti,0 ~ rm Φ(H )= t i,r 1 i,m m! qi qi− m 0 − X≥ + kσi + + Φ(Ei,k)= e gi,m xi,m m 0 ≥ − X kσi Φ(Fi,k)= e gi,m− xi,m− m 0 X≥ In the formulae above, r Z , k Z, q = e~/2 and q = qdi , where the d are ∈ ∗ ∈ i i the symmetrising integers for the Cartan matrix of g. The ti,m i I,m N are an { } ∈ ∈ alternative set of generators of the commutative subalgebra Y~(h) Y~(g) generated ⊂ by the elements ξi,m i I,m N. They are defined in [22] by equating the generating functions { } ∈ ∈ m 1 m 1 ~ ti,mu− − = log(1 + ~ ξi,mu− − ) m 0 m 0 X≥ X≥ The elements gi,m± i I,m N lie in the completion of Y~(h), and are constructed as { } ∈ ∈ follows. Consider the formal power series v G(v) = log vQ[[v]] ev/2 e v/2 ∈ − − and define γ (v) Y 0[[v]] by i ∈ t d r+1 c γ (v)= ~ i,r G(v) i r! −dv r 0 X≥ Then, 1/2 m ~ γi(v) g± v = exp (1.1) i,m q q 1 2 m 0 i i− X≥ − Finally, σi± are the homomorphisms of the subalgebras Y~(b ) Y~(g) generated ± ⊂ by ξj,r,xj,r± j I,r N, which fix the ξj,r and act on the remaining generators as the { } ∈ ∈ shifts x± x± . j,r → j,r+δij Note that the formulae connecting the generators Hi,k of U~(Lg) and ti,m of { } 1 { } Y~(h) essentially coincide with those connecting the generators of h[z, z− ] and h[s].

1.6. The above formulae apply equally well when g is a symmetrisable Kac–Moody algebra. Our proof of Theorem 1.4 shows that they deﬁne a homomorphism from the quantum aﬃnization U~g of the quantum group U~g [20, 26], to the completion of the Yangian Y~(g), provided the following holds (1) the entries of the Cartand matrix of g satisfy a a 3 for i = j I. ij ji ≤ 6 ∈ (2) the PBW theorem holds for Y~(g). YANGIANSANDQUANTUMLOOPALGEBRAS 5

The first assumption is equivalent to requiring that all rank 2 subalgebras of g be finite–dimensional, and is needed in our proof of the q–Serre relations. The second is required for the construction of certain straightening homomorphisms on Y~(g) which are needed in the proof of Theorem 1.4. We note that, for the Yangians associated to affine Kac–Moody algebras, the PBW theorem was proved by Guay in type An, for n 4 [16] and, more recently, by Guay–Nakajima for all simply laced cases [18]. In≥ particular, the above formulae define a homomorphism from the tor quantum toro¨ıdal algebra U~ (g) associated to a simply laced, simple Lie algebra g ≇ sl2, to the completion of the affine Yangian Y~g.

1.7. We also construct in this paper a homomorphismb similar to the one described in 1.5 for g = gln, by relying on the geometric realisation of U~(Lgln) obtained by Ginzburg and Vasserot [15, 34]. More precisely, fix integers 1 n d, and let ≤ ≤ = 0= V V V = Cd F { 0 ⊆ 1 ⊆···⊆ n } be the variety of n–step flags in Cd. The cotangent bundle T may be realised as ∗F d T ∗ = (V ,x) End(C ) x(Vi) Vi 1 F { • ∈F× | ⊂ − } and therefore admits a morphism T ∗ via the second projection, where = x End(Cd) xn = 0 is the cone ofF→Nn–step nilpotent endomorphisms. DefineN the { ∈ | } Steinberg variety Z = T ∗ T ∗ . The group GLd C× acts on T ∗ and Z and there are surjective algebraF× homomorphismsN F × F

× GLd C ΨU : U~(Lgln) K × (Z) → × GLd C Ψ : Y~(gl ) H × (Z) Y n →

see [15, 34] for the deﬁnition of ΨU . To understand these more explicitly, one can use the convolution actions of × × × GLd C GLd C GLd C × K × (Z) on K × (T ∗ ) and of H × (Z) on HGLd C (T ∗ ), which are faithful. By using the equivariantF Chern character, we construct× anF algebra homomorphism \ Φ : U~(Lgl ) Y~(gl ) n → n which intertwines these two actions.

1.8. In the sequel to this paper [13], we shall prove that, for g semisimple, a modi- fication of the pull–back functor Φ∗ converges for numerical values of ~, and defines a functor Rep (Y (g)) Rep (U (Lg)) fd a → fd ǫ where Y (g) is the specialisation of Y~(g) at ~ = a C R and ǫ = exp(πia), and a ∈ \ defines an equivalence of an explicit subcategory of Repfd(Yag) with Repfd(Uǫ(Lg)). 6 S.GAUTAMANDV.TOLEDANOLAREDO

1.9. It is worth pointing out that most of our results relating U~(Lg) and Y~(g) have analogues for the affine and degenerate affine Hecke algebras and ′ associated to an affine Weyl group W [24], and were in fact inspired by theseH andH their further study in [6]. Indeed, in [24], Lusztig constructs an explicit isomorphism between appropriate completions of and ′. In this context, the isomorphism can be understood in terms of, andH in fact obtainedH from, the geometric realisations G C× Ξ : ∼ K × (Z) H −→ G C× Ξ′ : ′ ∼ H × (Z) H −→ where Z is the Steinberg variety corresponding to W [7, 14].

1.10. Outline of the paper. In Section 2, we review the definition of the quantum loop algebra U~(Lg) and Yangian Y~(g) of a semisimple Lie algebra g. We also introduce shift homomorphisms of the subalgebras Y~(b ), and straightening ± homomorphisms of the subalgebra Y~(h). [ In Section 3, we consider assignments mapping the generators of U~(Lg) to Y~(g). [ These have the form described in 1.5, where the elements g± Y~(h) are however i,m ∈ not necessarily given by formula (1.1). Our main result, Theorem 3.4, gives necessary and sufficient conditions for these elements to give rise to an algebra homomorphism. We call such homomorphisms of geometric type since, for g simply laced, they are related to the Chern character in the geometric realisation described in 1.2. The proof that the elements given by (1.1) satisfy the conditions of Theorem 3.4, and therefore give rise to an algebra homomorphism Φ, is given in Section 4 (Theorem 4.7). We also prove that the action of Φ on Drinfeld polynomials exponentiates their roots (Corollary 4.5). In Section 5, we prove the essential uniqueness of homomorphisms of geometric type by showing that any two differ by conjugation by an element of the torus H [ and an invertible element of Y~(h) (Theorem 5.11). In Section 6, we show that any homomorphism of geometric type Φ induces an \ [ \ isomorphism U~(Lg) Y~(g), where U~(Lg) is the completion with respect to the evaluation ideal at z →= 1 (Theorem 6.2). We show moreover that the associated graded map coincides with Drinfeld’s degeneration of U~(Lg) to Y~(g) (Proposition 6.5). Section 7 contains similar results for g = gln. In addition to constructing an \ explicit homomorphism Φ : U~(Lgln) Y~(gln) (Theorem 7.6), we review the geometric realisations of these algebras and→ show that Φ intertwines them (Theorem 7.19). Appendix A contains a proof of the Serre relations which is required to complete the proof of Theorem 3.4.

1.11. Acknowledgments. We are very grateful to Ian Grojnowski from whom we learned that the quantum loop algebra and Yangian should be isomorphic after appropriate completions. His explanations and friendly insistence helped us overcome our initial doubts. We are also grateful to V. Drinfeld for showing us a proof that YANGIANSANDQUANTUMLOOPALGEBRAS 7

ﬁnite–dimensional representations separate elements of the Yangian and allowing us to reproduce it in Appendix A. We would also like to thank N. Guay for sharing a preliminary version of the preprint [17] and E. Vasserot for useful discussions.

2. Quantum loop algebras and Yangians 2.1. Let g be a complex, semisimple Lie algebra and ( , ) a non–degenerate, invari- · · ant bilinear form on g. Let h g be a Cartan subalgebra of g, αi i I h∗ a basis ⊂ { } ∈ ⊂ of simple roots of g relative to h and aij = 2(αi, αj)/(αi, αi) the entries of the corresponding Cartan matrix A. Set d = (α , α )/2, so that d a = d a for any i, j I. i i i i ij j ji ∈ Let ν : h h∗ be the isomorphism determined by the inner product ( , ) and set 1 → · · hi = ν− (αi)/di. Choose root vectors ei gαi ,fi g αi such that [ei,fi] = hi. Recall that g is presented on generators e∈,f , h subject∈ − to the relations { i i i}

[hi, hj ] = 0 [h , e ]= a e [h ,f ]= a f i j ij j i j − ij j [ei,fj]= δijhi for any i, j I and, for any i = j I ∈ 6 ∈

1 aij ad(ei) − ej = 0

1 aij ad(fi) − fj = 0

A closely related, but slightly less standard presentation may be obtained by 1 setting ti = ν− (αi)= dihi and choosing, for any i I, root vectors xi± g αi such + ∈ ∈ ± that [xi ,xi−]= ti. Then g is presented on xi±,ti i I subject to the relations { } ∈

[ti,tj] = 0

[t ,x±]= d a x± i j ± i ij j + [xi ,xj−]= δijti 1 aij ad(xi±) − xj± = 0

2.2. Throughout this paper, q and ~ are formal variables related by q2 = e~. For di ~di/2 any i I, we set qi = q = e . We use the standard notation for Gaussian integers∈

qn q n [n] = − − q q q 1 − − n [n] ! [n] ! = [n] [n 1] [1] = q q q − q · · · q k [k] ![n k] ! q q − q 8 S.GAUTAMANDV.TOLEDANOLAREDO

2.3. The quantum loop algebra [11]. Let U~(Lg) be the unital, associative algebra over C[[~]] topologically generated by elements Ei,k, Fi,k,Hi,k i I,k Z subject to the following relations { } ∈ ∈ (QL1) For i, j I and r, s Z ∈ ∈

[Hi,r,Hj,s] = 0

(QL2) For any i, j I and k Z, ∈ ∈ [H , E ]= a E [H , F ]= a F i,0 j,k ij j,k i,0 j,k − ij j,k (QL3) For any i, j I and r Z , ∈ ∈ × [ra ] [ra ] [H , E ]= ij qi E [H , F ]= ij qi F i,r j,k r j,r+k i,r j,k − r j,r+k (QL4) For i, j I and k, l Z ∈ ∈ E E qaij E E = qaij E E E E i,k+1 j,l − i j,l i,k+1 i i,k j,l+1 − j,l+1 i,k aij aij F F q− F F = q− F F F F i,k+1 j,l − i j,l i,k+1 i i,k j,l+1 − j,l+1 i,k (QL5) For i, j I and k, l Z ∈ ∈ ψi,k+l φi,k+l [Ei,k, Fj,l]= δij − 1 q q− i − i (QL6) Let i = j I and set m = 1 a . For every k ,...,k Z and l Z 6 ∈ − ij 1 m ∈ ∈ m s m ( 1) Ei,kπ(1) Ei,kπ(s) Ej,lEi,kπ(s+1) Ei,kπ(m) = 0 − s q · · · · · · π Sm s=0 i X∈ X m s m ( 1) Fi,kπ(1) Fi,kπ(s) Fj,lFi,kπ(s+1) Fi,kπ(m) = 0 − s q · · · · · · π Sm s=0 i X∈ X where the elements ψi,r, φi,r are deﬁned by

r ~di 1 s ψ (z)= ψ z− = exp H exp (q q− ) H z− i i,r 2 i,0  i − i i,s  r 0 s 1 X≥ X≥   r ~di 1 s φi(z)= φi, rz = exp Hi,0 exp (qi qi− ) Hi, sz − − 2 − − −  r 0 s 1 X≥ X≥   with ψi, k = φi,k = 0 for every k 1. − 0 ≥ We shall denote by U U~(Lg) the commutative subalgebra generated by the ⊂ elements Hi,r i I,r Z. { } ∈ ∈ YANGIANSANDQUANTUMLOOPALGEBRAS 9

2.4. The Yangian [11]. Let Y~(g) be the unital, associative C[~]–algebra generated by elements x± ,ξi,r i I,r N, subject to the following relations { i,r } ∈ ∈ (Y1) For any i, j I and r, s N ∈ ∈ [ξi,r,ξj,s] = 0 (Y2) For i, j I and s N ∈ ∈ [ξ ,x± ]= d a x± i,0 j,s ± i ij j,s (Y3) For i, j I and r, s N ∈ ∈ diaij~ [ξ ,x± ] [ξ ,x± ]= (ξ x± + x± ξ ) i,r+1 j,s − i,r j,s+1 ± 2 i,r j,s j,s i,r (Y4) For i, j I and r, s N ∈ ∈ diaij~ [x± ,x± ] [x± ,x± ]= (x± x± + x± x± ) i,r+1 j,s − i,r j,s+1 ± 2 i,r j,s j,s i,r (Y5) For i, j I and r, s N ∈ ∈ + [xi,r,xj,s− ]= δijξi,r+s (Y6) Let i = j I and set m = 1 a . For any r , , r N and s N 6 ∈ − ij 1 · · · m ∈ ∈ x± , x± , , x± ,x± = 0 i,rπ(1) i,rπ(2) · · · i,rπ(m) j,s · · · π Sm X∈ h h h i ii Y~(g) is an N–graded algebra by deg(ξi,r) = deg(xi,r± )= r and deg(~) = 1.

2.5. PBW theorem for Y~(g). For any positive root β of g, choose a sequence of simple roots α ,...,α such that β = α + + α and i1 ik i1 · · · ik

[xi±, [xi± , , [xi±− ,xi± ] ]] g β 1 2 · · · k 1 k · · · ∈ ± are non–zero vectors. For any r N, deﬁne x± Y~(g) by choosing a partition ∈ β,r ∈ r = r + + r of length k and setting 1 · · · k x± = [x± , [x± , , [x± ,x± ] ]] β,r i1,r1 i2,r2 · · · ik−1,rk−1 ik,rk · · ·

Theorem ([23]). Fix a total order on the set = ξi,r,x± i I,r N,β Σ+ . Then, the G { β,r} ∈ ∈ ∈ ordered monomials in the elements of form a basis of Y~(g). G 0 Let Y ,Y ± Y~(g) be the subalgebras generated by the elements ξi,r i I,r N ⊂ 0 0 { } ∈0 ∈+ (resp. xi,r± i I,r N) and Y ≥ ,Y ≤ Y~(g) the subalgebras generated by Y ,Y 0{ } ∈ ∈ ⊂ and Y ,Y − respectively. The following is a direct consequence of Theorem 2.5. Corollary. 0 (1) Y is a polynomial algebra in the generators ξi,r i I,r N. { } ∈ ∈ (2) Y ± is the algebra generated by elements xi,r± i I,r N subject to the relations { } ∈ ∈ (Y4) and (Y6). 0 0 (3) Y ≥ (resp. Y ≤ ) is the algebra generated by elements ξi,r,xi,r± i I,r N sub- { } ∈ ∈ ject to the relations (Y1)–(Y4) and (Y6). 10 S.GAUTAMANDV.TOLEDANOLAREDO

(4) Multiplication induces an isomorphism of vector spaces 0 + Y − Y Y Y~(g) ⊗ ⊗ → 2.6. The shift operators σ±. Fix i I. By Corollary 2.5 (3), the assignment i ∈ x± x± ξj,r ξj,r j,r → j,r+δij → extends to an algebra homomorphism Y 0 Y 0 (resp. Y 0 Y 0) which we ≥ → ≥ ≤ → ≤ shall denote by σi±. 2.7. The relations (Y2)–(Y3). We rewrite below the deﬁning relations (Y2)– (Y3) of Y~(g) in terms of the shift operators σj± and the generating series

r 1 1 ξ (u)=1+ ~ ξ u− − Y~(g)[[u− ]] (2.1) i i,r ∈ r 0 X≥ Lemma. The relations (Y2)–(Y3) are equivalent to

~diaij [ξi(u),xj,s± ]= ± ξi(u)xj,s± u σ± ~d a /2 − j ± i ij 1 where the rational function on the right–hand side is expanded in powers of u− . r 1 Proof. Set a = diaij/2. Multiplying (Y3) by ~u− − and summing over r 0 yields ≥ 1 u[ξ (u) 1 ~u− ξ ,x± ] [ξ (u) 1,x± ]= ~a x± ,ξ (u) 1 i − − i,0 j,s − i − j,s+1 ± { j,s i − } where x,ξ = xξ + ξx. Using (Y2) and x,ξ = [x,ξ] + 2ξx, yields { } { } (u σ± ~a)[ξ (u),x± ]= 2~aξ (u)x± (2.2) − j ± i j,s ± i j,s 0 r 1 as claimed. Conversely, taking the coeﬃcients of u and u− − in (2.2) yields (Y2) and (Y3) respectively.

2.8. The relations (Y4) and (Y6). We shall use the following notation for an operator T End(V ), T End(V m) is deﬁned as • ∈ (i) ∈ ⊗ i 1 m i T = 1⊗ − T 1⊗ − (i) ⊗ ⊗ for an algebra A, ad(m) : A m End(A) is deﬁned as • ⊗ → ad(m) (a a )= ad(a ) ad(a ) 1 ⊗···⊗ m 1 ◦···◦ m Proposition. (1) The relation (Y4) for i = j is equivalent to the requirement that the following holds for any A(v , v ) 6 C[[v , v ]] 1 2 ∈ 1 2 A(σ±,σ±)(σ± σ± a~)x± x± = A(σ±,σ±)(σ± σ± a~)x± x± i j i − j ∓ i,0 j,0 i j i − j ± j,0 i,0 where a = diaij/2. YANGIANSANDQUANTUMLOOPALGEBRAS 11

(2) The relation (Y4) for i = j is equivalent to the requirement that the following holds for any B(v , v ) C[[v , v ]] such that B(v , v )= B(v , v ) 1 2 ∈ 1 2 1 2 2 1 µ B(σ± ,σ± )(σ± σ± d ~)x± x± = 0 (2.3) i,(1) i,(2) i,(1) − i,(2) ∓ i i,0 ⊗ i,0 2 where µ : Y~(g)⊗ Y~(g) is the multiplication. (3) The relation (Y6)→ is equivalent to the requirement that the following holds for any i = j and A C[v , . . . , v ]Sm with m = 1 a 6 ∈ 1 m − ij m (m) ⊗ ad A(σi,±(1),...,σi,±(m)) xi,±0 xj,l± = 0 Proof. (1) The relation (Y4)

[x± ,x± ] [x± ,x± ]= a~(x± x± + x± x± ) i,r+1 j,s − i,r j,s+1 ± i,r j,s j,s i,r may be rewritten as

r s r s σ± σ± σ± σ± a~ x± x± = σ± σ± σ± σ± a~ x± x± i j i − j ∓ i,0 j,0 i j i − j ± j,0 i,0 (2) If i = j, a = di and the above may be written as

r s µ σ± σ± σ± σ± d ~ x± x± = i,(1) i,(2) i,(1) − i,(2) ∓ i i,0 ⊗ i,0 s r µ σ± σ± σ± σ± d ~ x± x± i,(1) i,(2) i,(2) − i,(1) ± i i,0 ⊗ i,0 which is equivalent to

r s s r µ σ± σ± + σ± σ± σ± σ± d ~ x± x± = 0 i,(1) i,(2) i,(1) i,(2) i,(1) − i,(2) ∓ i i,0 ⊗ i,0 (3) is just the reformulation of (Y6).

Corollary. If (2.3) holds for some B C[[v , v ]], then B(v , v )= B(v , v ). ∈ 1 2 1 2 2 1 Proof. By (2) of Proposition 2.8, we may assume that B(v , v )= B(v , v ) and 1 2 − 2 1 therefore that B = (v1 v2)B where B is symmetric in v1 v2. Using the grading − ↔ r s s r on Y~(g), we may further assume that B is proportional to v1v2 + v1v2 for some r s N. An application of (Y4) then yields ≥ ∈ r s s r µ (σ± σ± )(σ± σ± + σ± σ± )(σ± σ± d ~)x± x± i,(1) − i,(2) i,(1) i,(2) i,(1) i,(2) i,(1) − i,(2) ∓ i i,0 ⊗ i,0 = 2 (x± x± x± x± d ~x± x± ) (x± x± x± x± d ~x± x± ) i,r+2 i,s− i,r+1 i,s+1∓ i i,r+1 i,s − i,r+1 i,s+1− i,r i,s+2∓ i i,r i,s+1 If r s + 2, the above is not zero by the PBW Theorem 2.5 and B = 0. If r = s≥+ 1, a further application of (Y4) shows that the second of the above two parenthesized summands is zero and again B = 0 by Theorem 2.5. Finally, if r = s, (Y4) implies that the two parenthesized summands are opposites of each other and again B = 0. 12 S.GAUTAMANDV.TOLEDANOLAREDO

2.9. An alternative system of generators for Y 0. The following generators of Y 0 were introduced in [22]. For any i I, deﬁne the formal power series ∈ r 1 0 1 t (u)= ~ t u− − Y [[u− ]] i i,r ∈ r 0 X≥ by

r 1 t (u) = log(ξ (u)) = log 1+ ~ ξ u− − (2.4) i i  i,r  r 0 X≥   0 Since (2.4) can be inverted, ti,r i I,r N is another system of generators of Y . These { } ∈ ∈ are homogeneous, with deg(t )= r, since ζ ξ (u)= ξ (ζ 1u), where is the action i,r · i i − · of ζ C∗ on Y~(g) given by the grading. Moreover, ti,0 = ξi,0 and ti,r = ξi,r mod ~ for any∈ r 1 since ≥ r 1 2 ti(u)= ~ ξi,ru− − mod ~ r 0 X≥ To compute the commutation relations between ti,r and xj,s± , we introduce the following formal power series (inverse Borel transform of ti(u)) vr B (v)= B(t (u)) = ~ t Y 0[[v]] (2.5) i i i,r r! ∈ r 0 X≥ Lemma. For any i, j I we have ∈ aij v aij v ± qi qi− σ v B (v),x± = − e j x± (2.6) i j,s ± v j,s h i aij Proof. To simplify notations set a = ~d a /2, so that ea = q± . By Lemma 2.7 ± i ij i

1 u σj± + a ξi(u)xj,s± ξi(u)− = − xj,s± u σ± a − j − so that 1 1 (σ± a)u− [t (u),x ] = log − j − x i j,s± 1 j,s± 1 (σ± + a)u − j − ! Using pv 1 1 e B log(1 pu− ) = − (2.7) − v this yields

± ± (σ a)v (σ +a)v av av 1 e j − 1 e j e e ± − σj v [Bi(v),xj,s± ]= − − xj,s± = − e xj,s± v − v ! v as claimed. YANGIANSANDQUANTUMLOOPALGEBRAS 13

Remark. Expanding the right–hand side of (2.6) as a power series in v yields the commutation relations r/2 2l ⌊ ⌋ r (~diaij/2) [ti,r,xj,s± ]= diaij xj,r± +s 2l ± 2l 2l + 1 − Xl=0 These relations were obtained in this form in [22, Lemma 1.4].

2.10. The operators λi±(v). We introduce below operators which straighten mono- 0 0 mials of the form x± ξ, ξ Y , into elements of Y Y . i,m ∈ · ± 0 Proposition. There are operators λi±;s i I,s N on Y such that the following holds { } ∈ ∈ 0 0 (1) For any ξ Y , the elements λ± (ξ) Y are uniquely determined by the ∈ i;s ∈ requirement that, for any m N, ∈ xi,m± ξ = λi±;s(ξ)xi,m± +s (2.8) s 0 X≥ (2) For any ξ, η Y 0, ∈ λi±;s(ξη)= λi±;k(ξ)λi±;l(η) (2.9) kX+l=s 0 0 (3) The operator λi;s : Y Y is homogeneous of degree s. 0 0 → − (4) Let λ±(v) : Y Y [v] be given by i → s λi±(v)(ξ)= λi±;s(ξ) v s 0 X≥ 0 0 and extend the N–grading on Y to Y [v] by deg(v) = 1. Then λi±(v) is an algebra homomorphism of degree 0. ǫ1 ǫ2 (5) λ (v1) and λ (v2) commute for any i1, i2 I and ǫ1,ǫ2 . i1 i2 ∈ ∈ {±} (6) For any i I, ∈ + + λi (v)λi−(v)=id= λi−(v)λi (v) (7) For any i, j I, ∈ aij v2 aij v2 − qi qi v1v2 λj±(v1) (Bi(v2)) = Bi(v2) − e ∓ v2 (8) For any i I and r N, ∈ ∈ r λ±(v)(t )= t d a v mod ~ j i,r i,r ∓ i ij Proof. (1)–(2) by Lemma 2.7, (2.8) holds when ξ is one of the generators ξj,r of 0 0 Y . Since (2.8) holds for ξη if it holds for ξ, η Y , with λi;s(ξη) given by (2.9), the 0 ∈ λi;s can be deﬁned as operators on Y . The fact they are uniquely characterised by (2.8) and satisfy (2.9) follows from Corollary 2.5. (3) the linear independence of the elements on the right–hand side of (2.8) implies 0 that deg(λi;s(ξ)) = deg(ξ) s for any homogeneous ξ Y . (4) is a rephrasing of − ∈ 0 (2) and (3). (5) and (6) follow from (7) since the elements ti,n generate Y . (7) follows from Lemma 2.9. (8) is a direct consequence of (7). { } 14 S.GAUTAMANDV.TOLEDANOLAREDO

Remark. Using the shift operators, the relation (2.8) can be rewritten as

xi,m± ξ = λi±(σi±)(ξ) xi,m±

3. Homomorphisms of geometric type

Let Y~g be the completion of Y~(g) with respect to its N–grading. In this section, we define an assignment d Φ : Hi,r, Ei,r, Fi,r i I,r Z Y~g { } ∈ ∈ −→ and find necessary and sufficient conditions for Φ to extend to a homomorphism d U~(Lg) Y~g. → 3.1. Definitiond of Φ. Define 1 Φ(Hi,0)= di− ti,0 and, for r Z ∈ × k ~ r Bi(r) Φ(Hi,r)= ti,k = q q 1 k! q q 1 i i− k 0 i i− − X≥ − 0 where Bi(v) is the formal power series (2.5). Let U be the polynomial ring on 1 the generators Hi,r i I,r Z . The above assignment extends to an homomorphism { } ∈ ∈ Φ0 : U 0 Y 0. e → 0 Let now g± i I,m N be elements of Y , and define further { i,m} ∈ ∈ e c + + Φ(Ei,0)= c gi,mxi,m m 0 X≥ Φ(Fi,0)= gi,m− xi,m− m 0 X≥

In terms of the shift operators σi±, the above may be written as + + + Φ(Ei,0)= gi (σi )xi,0 (3.1)

Φ(Fi,0)= gi−(σi−)xi,−0 (3.2) where m 0 g±(v)= g± v Y[[v] i i,m ∈ m 0 X≥ with the completion of Y 0[v] taken with respect to the N–grading which extends that on Y 0 by deg(v) = 1.

1 e0 0 U is isomorphic to the subalgebra U of U~(Lg) generated by {Hi,r}i∈I,r∈Z by the PBW Theorem for U~(Lg) [1], but we shall not need this fact. YANGIANSANDQUANTUMLOOPALGEBRAS 15

[ 3.2. Homomorphisms of geometric type. If Φ : U~(Lg) Y~(g) is a homomorphism of the above form, we shall say that it is of geometric→ type since its form is related to the Chern character in the geometric realisations of U~(Lg) and Y~(g) discussed in 1.2. Such a homomorphism has the following properties i \i i (1) It restricts to a homomorphism U~(Lsl2) Y~(sl2) for any I, where U~(Lsl2) i → U~(Lg) and Y~(sl2) Y~(g) are the subalgebras generated by Ei,r, Fi,r,Hi,r r Z ⊂ ⊂ { } ∈ and x± ,ξi,k k N respectively. { i,k } ∈ \ (2) It restricts to a homomorphism U~(Lb ) Y~(b ), where U~(Lb+),U~(Lb ) ± → ± − U~(Lg) and Y~(b ) Y~(g) are the subalgebras generated respectively by ⊂ ± ⊂ Ei,r,Hi,r i I,r Z, Fi,r,Hi,r i I,r Z and x± ,ξi,k i I,k N . { } ∈ ∈ { } ∈ ∈ { i,k } ∈ ∈ Note however that, unless g± = 0 for any i I and m 1, Φ does not map the i,m ∈ ≥ [ quantum group U~g = Ei,0, Fi,0,Hi,0 i I U~(Lg) to Ug[[~]] Y~(g). h i ∈ ⊂ ⊂ 3.3. The following result shows that the requirement that Φ extends to an algebra homomorphism determines its value on generators Ei,k, Fi,k. Proposition. The assignment Φ is compatible with relations (QL2)–(QL3) if, and only if

+ kσi + + + Φ(Ei,k)= e gi (σi )xi,0 (3.3) − kσi Φ(Fi,k)= e gi−(σi−)xi,−0 (3.4) Proof. We only consider the case of the E s. Let i, j I and k Z. By (Y2), ′ ∈ ∈ + 1 kσj + + + [Φ(Hi,0), Φ(Ej,k)] = [di− ξi,0, e gj (σj )xj,0] + kσj + + 1 + = e gj (σj )[di− ξi,0,xj,0] = aijΦ(Ej,k) so that Φ is compatible with (QL2). Next, if r Z , Lemma 2.9 yields ∈ × 1 + kσj + + + [Φ(Hi,r), Φ(Ej,k)] = 1 [Bi(r), e gj (σj )xj,0] q q− i − i raij raij q q− + + i i rσj kσj + + + = − 1 e e gj (σj )xj,0 r(q q− ) i − i [ra ] = ij qi Φ(E ) r j,r+k and Φ is compatible with (QL3).

Conversely, if Φ is compatible with (QL3) then Φ(Ei,r)= r/[2r]qi [Φ(Hi,r), Φ(Ei,0)] + for r = 0 and the computation above shows that this is equal to erσi Φ(E ). 6 i,0 16 S.GAUTAMANDV.TOLEDANOLAREDO

0 0 3.4. Necessary and suﬃcient conditions. Let λi±(v) : Y Y [v] be the homomorphism deﬁned in Proposition 2.10. → [ Theorem. The assignment Φ extends to an algebra homomorphism U~(Lg) Y~(g) if, and only if the following conditions hold → (A) For any i, j I ∈ + + + gi (u)λi (u)(gj−(v)) = gj−(v)λj−(v)(gi (u)) (B) For any i I and k Z ∈ ∈ ku + + 0 ψi,k φi,k e g (u)λ (u)(g−(u)) =Φ − i i i m 1 u =ξi,m qi q− − i

(C) For any i, j I and a = d a /2 ∈ i ij u v a~ v u a~ e e ± e e ± g±(u)λ±(u)(g±(v)) − = g±(v)λ±(v)(g±(u)) − i i j u v a~ j j i v u a~ − ∓ − ∓ Proof. By construction and Proposition 3.3, Φ is compatible with the relations (QL1)–(QL3). The result then follows from Lemmas 3.5 and 3.6 below and the proof of the q–Serre relations (Proposition A.1 in the Appendix).

3.5.

Lemma. Φ is compatible with the relation (QL5) if, and only if (A) and (B) hold.

Proof. Compatibility with (QL5) reads

0 ψi,k+l φi,k+l [Φ(Ei,k), Φ(Fj,l)] = δijΦ − 1 q q− i − i for i, j I and k, l Z. We begin by computing the left–hand side. To this end, it will be∈ convenient to∈ write formulae (3.3)–(3.4) as

+,(k) + ,(k) Φ(Ei,k)= gi,m xi,m and Φ(Fi,k)= gi,m− xi,m− m 0 m 0 X≥ X≥ ,(k) 0 ,(k) m kv where gi,m± Y are deﬁned by m 0 gi,m± v = e gi±(v). This yields ∈ ≥ +,(Pk) + ,(l) Φ(Ei,k)Φ(cFj,l)= gi,m xi,mgj,n− xj,n− m,n 0 X≥ +,(k) + ,(l) + = gi,m λi,s gj,n− xi,m+sxj,n− m,n,s 0 X≥ +,(k) + ,(l) + = gi,m λi,s gj,n− xj,n− xi,m+s + δijξi,m+n+s m,n,s 0 X≥ YANGIANSANDQUANTUMLOOPALGEBRAS 17

,(l) +,(k) + where we used (Y5). Similarly, Φ(Fj,l)Φ(Ei,k)= m,n,s gj,m− λj,s− gi,n xj,m− +sxi,n. Deﬁne R(k,l),L(k,l) Y 0[[u, v]] by P ∈

(k,l) +,(kc) m + s ,(l) n ku lv + + R = g u λ u g− v = e e g (u)λ (u)(g−(v)) i,m i,s  j,n  i i j m 0 s 0 n 0 X≥ X≥ X≥   (k,l) ,(l) m s +,(k) n ku lv + L = g− v λ− v g u = e e g−(v)λ−(v)(g (u)) j,m j,s  i,n  j j i m 0 s 0 n 0 X≥ X≥ X≥   By the PBW Theorem 2.5, Φ is compatible with (QL5) if, and only if R(k,l) = L(k,l) and, for i = j, ψ φ R(k,l) =Φ0 i,k+l − i,k+l m n 1 u v =ξi,m+n qi q− − i The ﬁrst equation is clearly equivalent to (A) and the second to (B).

3.6. Lemma. Φ is compatible with the relation (QL4) if, and only if (C) holds. Proof. We prove the claim for the E’s only. Compatibility with (QL4) reads Φ(E )Φ(E ) qaij Φ(E )Φ(E )= qaij Φ(E )Φ(E ) Φ(E )Φ(E ) i,k+1 j,l − i i,k j,l+1 i j,l i,k+1 − j,l+1 i,k for any i, j I and k, l Z. Assume ﬁrst i = j and set a = diaij/2, so that aij a~ ∈ ∈ 6 qi = e . Since + + rσi sσj + + + + + + + + Φ(Ei,r)Φ(Ej,s)= e e gi (σi )λi (σi ) gj (σj ) xi,0xj,0 the above reduces to

+ lσ+ + σ++a~ ekσi e j g+(σ+)λ+(σ+)(g+(σ+)) eσi e j x+ x+ i i i i j j − i,0 j,0 + lσ+ + ~ σ+ = ekσi e j g+(σ+)λ+(σ+)(g+(σ+)) eσi +a e j x+ x+ j j j j i i − j,0 i,0 Using (1) of Proposition 2.8, we get

+ + σ σj +a~ + σ++a~ e i e eσi e j x+ x+ = − σ+ σ+ a~ x+ x+ − i,0 j,0 σ+ σ+ a~ i − j − i,0 j,0 i − j − + σ++a~ eσi e j = − σ+ σ+ + a~ x+ x+ σ+ σ+ a~ i − j j,0 i,0 i − j − The PBW Theorem 2.5 then shows that the above is equivalent to (C). Assume now that i = j, then + + rσi + + + sσi + + + Φ(Ei,r)Φ(Ei,s)= e gi (σi )xi,0 e gi (σi )xi,0 rσ+ sσ+ = µ e i,(1) e i,(2) g+(σ+ )λ+(σ+ ) g+(σ+ ) x+ x+ i i,(1) i i,(1) i i,(2) i,0 ⊗ i,0 18 S.GAUTAMANDV.TOLEDANOLAREDO

The compatibility with (QL4) therefore reduces to

kσ+ lσ+ σ+ σ+ +d ~ µ e i,(1) e i,(2) g+(σ+ )λ+(σ+ )(g+(σ+ )) e i,(1) e i,(2) i x+ x+ i i,(1) i i,(1) i i,(2) − i,0 ⊗ i,0 lσ+ kσ+ σ+ +d ~ σ+ = µ e i,(1) e i,(2) g+(σ+ )λ+(σ+ )(g+(σ+ )) e i,(2) i e i,(1) x+ x+ i i,(1) i i,(1) i i,(2) − i,0 ⊗ i,0 that is, to

kσ+ lσ+ lσ+ kσ+ µ e i,(1) e i,(2) + e i,(1) e i,(2)

σ+ σ+ +d ~ g+(σ+ )λ+(σ+ )(g+(σ+ )) e i,(1) e i,(2) i x+ x+ = 0 i i,(1) i i,(1) i i,(2) − i,0 ⊗ i,0 By (2) of Proposition 2.8 and Corollary 2.8, this equation is equivalent to the requirement that eu ev+di~ g+(u)λ+(u)(g+(v)) − i i i u v d ~ − − i be symmetric under u v, which is precisely condition (C) for i = j. ↔

3.7. For later use, we shall need the following

\0 Lemma. Let gi±(u) i I Y [u] be elements satisfying condition (B) of Theorem 3.4. Then, { } ∈ ⊂

1 \0 gi±(u)= mod Y [u]+ di± + where di± i I C× satisfy di di− = di for each i I. In particular, each gi±(u) is invertible.{ } ∈ ⊂ ∈ Proof. Condition (B) for k = 0 yields

~ ~ di di Hi,0 Hi,0 + + 0 e 2 e− 2 g (u)λ (u) g−(u) m =Φ − i i i u =ξi,m 1 qi qi− ! − \0 Computing mod ~, and a fortiori mod Y [u]+, yields ~ ~ di di Hi,0 Hi,0 0 e 2 e− 2 0 1 Φ − 1 =Φ (Hi,0)= di− ti,0 q q− i − i ! \0 \0 Write gi±(u)= pi± mod Y [u]+, where pi± C[tj,0]j I. Computing mod Y [u]+, we get ∈ ∈ + + + + + g (u)λ (u) g−(u) m = p (tj,0)λ (p−(tj,0))ξi,0 = p (tj,0)p−(tj,0 diaij)ξi,0 i i i u =ξi,m i i;0 i i i − where we used (8) of Proposition 2.10. Comparing both sides and using ξi,0 = ti,0 yields the claim. YANGIANSANDQUANTUMLOOPALGEBRAS 19

4. Existence of homomorphisms [ In this section, we construct an explicit homomorphism U~(Lg) Y~(g) by ex- hibiting a joint solution to equations (A)–(C) of Theorem 3.4. We→ begin by giving an intrinsic expression for the right–hand side of equation (B) (Proposition 4.2). Until 4.7, we ﬁx i I, and consider the subalgebras U 0 U 0, Y 0 Y 0 generated ∈ i ⊂ i ⊂ by Hi,r r Z and ξi,r r N respectively. { } ∈ { } ∈ e e 4.1. The functions G(v) and γi(v). Consider the formal power series v G(v) = log vQ[[v]] (4.1) ev/2 e v/2 ∈ − − Deﬁne γ (v) Y[0[v] by i ∈ i + t γ (v)= ~ i,r ( ∂ )r+1 G(v) i r! − v r 0 X≥ vr Recall that Bi(v)= ~ r 0 ti,r r! is the inverse Borel transform of ti(u). This allows us to write γ (u) more compactly≥ as i P γ (v)= B ( ∂ )G′(v) (4.2) i − i − v 4.2. Proposition. The following holds in Y 0 for any k Z i ∈ ψ φ ~ Φ0 i,k − i,k = c ekv exp (γ (v)) 1 1 i n qi q− qi q− v =ξi,n − i − i

The above identity will be proved in 4.3–4.6 by injectively mapping both sides to a family of polynomial rings, and verifying their equality there.

4.3. Universal Drinfeld polynomials. Fix an integer m 1. Following [26, 33], consider the rings ≥

1 1 1 Sm S(m)= C[q± , A1± ,...,Am± ]

Sm R(m)= C[~, a1, . . . , am] Define a homomorphism U : U 0 S(m) by D i → m 1 q z q− A U (ψ (z))e = i − i p = U (φ (z)) (4.3) D i z A D i p=1 p Y − where the first and second equalities are obtained by expanding the middle term in powers of z 1 and z respectively. Similarly, define Y : Y 0 R(m) by − D i → m Y u + di~ ap (ξi(u)) = − (4.4) D u ap pY=1 − 20 S.GAUTAMANDV.TOLEDANOLAREDO

The homomorphism U (resp. Y ) gives the action of ψ (z), φ (z) (resp. ξ (u)) on D D i i i the highest weight vector of the indecomposable U~(Lsl2) (resp. Y~(sl2)) module 1 1 with Drinfeld polynomial (1 A− z) (1 A z) (resp. (u a ) (u a )). − 1 · · · − m− − 1 · · · − m 0 0 4.4. The following result spells out the image of the generators of Ui and Yi under U and Y respectively. D D e Proposition. (1) The following holds: U (ψ )= qm = U (φ ) 1 and, for any r N D i,0 i D i,0 − ∈ ∗ m 1 ′ U 1 r qiAp qi− Ap (ψi,r) = (qi qi− ) Ap − (4.5) D − Ap Ap′ p=1 p′=p X Y6 − m 1 ′ U 1 r qiAp qi− Ap (φi, r)= (qi qi− ) Ap− − (4.6) D − − − Ap Ap′ p=1 p′=p X Y6 − Moreover, U maps H to m and, for any r Z , D i,0 ∈ ∗ 2r m U 1 1 qi− r ((q q− )H )= − A (4.7) D i − i i,r r p p=1 X (2) The homomorphism Y maps ξ to d m and, for any r N D i,0 i ∈ m Y r ap ap′ + di~ (ξr)= di ap − (4.8) D ap ap′ p=1 p′=p X Y6 − Moreover, for any r N, ∈ m r+1 r+1 1 a (ap di~) Y (t )= p − − (4.9) D r r + 1 ~ p=1 X (3) If B (v) Y 0[[v]] is the series deﬁned by (2.5), then i ∈ i m 1 e di~v Y (B (v)) = − − eapv (4.10) D i v p=1 X U m U 1 Proof. (1) The fact that (ψi,0)= qi = (φi,0)− follows by taking the values of the middle term P (z) inD (4.3) at z = andD z = 0 respectively. Next, the partial fraction decomposition of P (z) is readily∞ seen to be

m m 1 1 ′ qiz qi− Ap m 1 qiAp qi− Ap 1 − = qi + (qi qi− ) Ap − (4.11) z Ap −  Ap Ap′  z Ap p=1 p=1 p′=p Y − X Y6 − −   1 The relations (4.5)–(4.6) follow by expanding this into powers of z− and z respectively. For later use, note that the evaluation of (4.11) at z = 0 yields the identity m 1 ′ U m m 1 qiAp qi− Ap (ψi,0 φi,0)= qi qi− = (qi qi− ) − (4.12) D − − − Ap Ap′ p=1 p′=p X Y6 − YANGIANSANDQUANTUMLOOPALGEBRAS 21

Since U (ψ )= qm, it follows that U (H )= m, and D i,0 i D i,0 m 2 U 1 s U 1 z qi− Ap exp (q q− ) H z− = (ψ− ψ (z)) = − D   i − i i,s  D i,0 i z A s 1 p=1 p X≥ Y −    1 taking the log of both sides and expanding in powers of z− (resp. z) yields (4.7) for r> 0 (resp. r< 0). (2) The fact that Y (ξ )= d m follows by taking the coeﬃcient of u 1 in (4.4). D i,0 i − The partial fraction decomposition of Y (ξ (u)) is D i m m u + di~ ap ap ap′ + di~ 1 − =1+ di~ − u ap  ap ap′  u ap p=1 p=1 p′=p Y − X Y6 − −  r 1  and (4.8) follows by taking the coeﬃcient of u− − . Taking the log of both sides of (4.4) yields Y 1 1 (t (u)) = log 1 a u− + log 1 (a d ~)u− (4.13) D i − − p − p − i p X and therefore (4.9). (3) Follows by applying (2.7) to (4.4).

Y 0 Corollary. The homomorphism : Yi m 1 R(m) is injective. D → ≥ r Proof. This follows from (4.9) and the factL that the power sums p ap are alge- braically independent. P 4.5. Let R\(m) be the completion of R(m) with respect to the N–grading deﬁned by deg(~) = deg(a ) = 1. Since the map Y : Y 0 R(m) preserves the grading, it p D i → extends to a homomorphism Y 0 R\(m). i → Corollary. Let ch : S(m) R(m) be the homomorphism deﬁned by → c q e~/2 and A eap 7−→ p 7−→ Then, the following diagram commutes

U 0 / Ui D S(m)

Φ0 ch e Y 0 / \ Yi D R(m) 0 where Φ is deﬁned in Section 3.1c. 0 Proof. It suﬃces to check the commutativity on the generators Hi,r r Z of Ui . { } ∈0 The statement now follows from (4.7), (4.10) and the fact that, for r =0,Φ (Hi,r)= 1 6 B (v)/(q q− ) . e i i − i v=r

22 S.GAUTAMANDV.TOLEDANOLAREDO

4.6. Proof of Proposition 4.2. By Corollaries 4.4 and 4.5, it suﬃces to prove that, for any k Z, ∈ ψ φ ~ ch U i,k − i,k = Y ekv exp (γ (v)) 1 1 i n D qi q− qi q− D v =ξn − i − i

By (4.5)–(4.6) and (4.12), the left–hand side is equal to

1 ′ k qiAp qi− Ap ch Ap −  Ap Ap′  p p′=p X Y6 − We now compute the right–hand side. By (4.2) and (4.10),

di~∂v Y ap∂v 1 e (γ (v)) = e− − ∂ G(v)= G(v a ) G(v a + d ~) D i ∂ v − p − − p i p v p X X where we used ea∂v G(v)= G(v + a). Thus,

v 1 ap Y kv kv v ap qie qi− e e exp (γi(v)) = e − − D v a + d ~ ev eap p p i Y − − By (4.8), the substitution vn = ξ in a formal power series F Y[0[v] gives i,n ∈ i ′ Y Y ap ap + di~ n (F (v) v =ξn )= di (F (ap)) − D | D ap ap′ p p′=p X Y6 − v ap ap v 1 ap Since (v ap)/(e e ) = e− and (qie q− e )/(v ap + di~) = − − |v=ap − i − v=ap ap 1 e (qi qi− )/di~, this implies that − Y kv e exp (γ (v)) n D i |v =ξi,n ′ 1 ′ ap 1 ap ′ qi qi− kap ap ap qie qi− e ap ap + di~ = − e − a − a ′ − ~ ap ap′ + di~ e p e p ap ap′ p p′=p X Y6 − − − 1 a 1 a ′ q q− q e p q− e p = i − i ekap i − i ~ eap eap′ p p′=p X Y6 − as claimed.

4.7. A joint solution to equations (A)–(C). Proposition 4.2 suggests replacing equation (B) of Theorem 3.4 by the stronger requirement that, for any i I ∈ + + ~ gi (v)λi (v)(gi−(v)) = 1 exp (γi(v)) (B) q q− i − i Using equation (A) for j = i and u = v then shows that e

+ ~ gi−(v)λi−(v)(gi (v)) = 1 exp (γi(v)) q q− i − i YANGIANSANDQUANTUMLOOPALGEBRAS 23

+ Applying the operator λi−(v) to the ﬁrst of these equations, and using λi (v)λi−(v)= id (Proposition 2.10) and the second equation, yields λi−(v)(γi(v)) = γi(v). Simi- + + larly, applying λi (v) to the second of these equations yields λi (v)(γi(v)) = γi(v). This suggests in turn that a solution to the above equations may be given by gi±(v)= gi(v), where 1/2 ~ γi(v) gi(v)= 1 exp (4.14) q q− 2 i − i We now show that this is indeed the case.

Theorem. The series gi±(v) = gi(v) satisfy the conditions (A),(B) and (C) of Theorem 3.4, and therefore give rise to a homomorphism Φ : U~(Lg) Y~g. e→ 4.8. We shall need the following d Lemma. Let i, j I, and set a = d a /2. Then, ∈ i ij G(v u + a~) G(v u a~) λ±(u) (g (v)) = g (v)exp − − − − i j j ± 2 where G(v) is given by (4.1).

Proof. By Proposition 2.10,

a~v a~v e e− uv λ±(u)(B (v)) = B (v) − e i j j ∓ v Since γ (v)= B ( ∂ )∂ G(v), we get j − j − v v a~∂v a~∂v e e− u∂v λi±(u)γj(v)= γj(v) − e− ∂vG(v) ± ∂v = γ (v) (G(v u + a~) G(v u a~)) j ± − − − − The claim follows by exponentiating.

4.9. Proof of condition (A). We need to prove that for every i, j I, we have ∈ + gi(u)λi (u)(gj (v)) = gj(v)λj−(v)(gi(u)) By Lemma 4.8, this is equivalent to

G(v u + a~) G(v u a~) g (u)g (v)exp − − − − i j 2 G(u v a~) G(u v + a~) = g (u)g (v)exp − − − − i j 2 The result now follows since G is an even function. 24 S.GAUTAMANDV.TOLEDANOLAREDO

4.10. Proof of condition (B). Lemma 4.8 implies that G(d ~) G( d ~) g (u)λ+(u)(g (eu)) = g (u)2 exp i − − i i i i i 2 2 = gi(u) ~ = 1 exp (γi(u)) q q− i − i where the second equality holds because G is even.

4.11. Proof of condition (C). Let i, j I and set a = diaij/2. We need to prove that ∈ u v a~ v u a~ e e ± e e ± g (u)λ±(u)(g (v)) − = g (v)λ±(v)(g (u)) − i i j u v a~ j j i v u a~ − ∓ − ∓ By Lemma 4.8 and the fact that G is even, we get the following equivalent assertion eu ev a~ ev eu a~ exp (G(v u a~) G(v u a~)) − ± = − ± − ± − − ∓ u v a~ v u a~ − ∓ − ∓ Using the deﬁnition of G, the above becomes the equality v u a~ ev eu a~ eu ev a~ ev eu a~ − ± − ± − ± = − ± ev a~ eu v u a~ u v a~ v u a~ ± − − ∓ − ∓ − ∓

5. Uniqueness of homomorphisms The aim of this section is to prove that homomorphisms of geometric type are unique up to conjugation and scaling.

5.1. Let be the set of solutions g = gi±(u) i I of equations (A)–(C) of Theorem G { } ∈ \0 3.4. Given a collection r = ri±(u) i I of invertible elements of Y [u], set { } ∈

r g = ri±(u) gi±(u) i I · { · } ∈ Lemma. Let g . Then, r g if and only if the following holds ∈ G · ∈ G (A ) For any i, j I, 0 ∈ + + + ri (u)λi (u)(rj−(v)) = rj−(v)λj−(v)(ri (u)) (B ) For any i I, 0 ∈ + + + ri (u)λi (u)(ri−(u))=1= ri−(u)λi−(u)(ri (u))

(C±) For any i, j I, 0 ∈

ri±(u)λi±(u)(rj±(v)) = rj±(v)λj±(v)(ri±(u)) YANGIANSANDQUANTUMLOOPALGEBRAS 25

Proof. Let h = r g. The following assertions are straightforward to check · h satisfies (A) if and only if r satisfies (A0). • h satisfies (B) if r satisfies (B ). • 0 h satisfies (C) if and only if r satisfies (C±). • 0 There remains to prove that if h lies in , then r satisfies (B0). G + + We claim that (A0) and (C0±) imply that ci(u) = ri (u)λi (u)(ri−(u)) lies in C[[~, u]]. Assuming this, write c (u)= c(n)un, where c(n) C[[~]]. Then, i n i i ∈ + + + + h (u)λ (u)(h−(u)) m = Pci(u)g (u)λ (u)(g−(u)) m i i i u =ξi,m i i i u =ξi,m

(n) + + = c g (u)λ (u)(g−(u )) m i i i i u =ξi,m+n n 0 X≥ 0 + + = ci(σ ) g (u)λ (u)(g−(u)) m i i i i u =ξi,m where σ0 : Y 0 Y 0 is the algebra homomorphism deﬁned by σ0( ξ ) = ξ . i → i j,m j,m+δij Since both h and g satisfy (B) with k = 0, this yields

~di ~di ~di ~di Hi,0 Hi,0 Hi,0 Hi,0 0 e 2 e− 2 0 0 e 2 e− 2 Φ − 1 = ci(σi )Φ − 1 q q− q q− i − i ! i − i ! 0 An inductive argument using the C[~]–linear N–grading on Y given by deg(ξj,m)= m and deg(~) = 0 then shows that c(0) = 1 and c(n) = 0 for any n 1, so that r i i ≥ satisﬁes (B0). To prove our claim, set + + + ci(u)= ri (u)λi (u)(ri−(u)) = ri−(u)λi−(u)(ri (u)) + 1 so that r−(u)= c (u)λ−(u)(r (u)) . By (A ), the following holds for every i, j I i i i i − 0 ∈ + + + 1 + 1 + ri (u)λi (u) cj(v)λj−(v)(rj (v))− = cj(v)λj−(v)(rj (v)− )λj−(v)(ri (u))

+ Since λi (u) and λj−(v) commute, we get

+ + + + + + λi (u)(cj (v)) ri (u)λj−(v)(rj (v)) = cj(v)λj−(v) ri (u)λi (u)(rj (v))

+ + + = cj(v)λj−(v) rj (v)λj (v)(ri (u))

+ + = cj(v) ri (u)λj−(v)(rj (v))

+ + where the second equality uses (C0 ) and the third one λj−(v)λj (v) = 1. We have therefore proved that λ+(u)(c (v)) = c (v) for every i, j I i j j ∈ By deﬁnition of the operators λi±, this implies that the coeﬃcients of cj(v) lie in the centre of Y~(g), which is trivial. 26 S.GAUTAMANDV.TOLEDANOLAREDO

5.2. The uniqueness of homomorphisms of geometric type relies on the following

+ \0 Proposition. Let r (u) i I 1+ Y [u]+ be a collection of invertible elements { i } ∈ ⊂ + 0 satisfying condition (C0 ) of Lemma 5.1. Then, there exists an element ξ 1+ Y + such that, for any i I ∈ ∈ + + 1 c r (u)= ξ λ (u)(ξ)− i · i Moreover, if ζ Y 0× is any element such that r+(u)= ζ λ+(u)(ζ) 1, then ζ = cξ ∈ i · i − for some c C[[~]]×. ∈ c The proof of Proposition 5.2 is given in 5.3– 5.9. § §

5.3. We begin by linearising the problem. Set

r (u) = log(r (u)) Y\0[u] i i ∈ + By condition (C+), the following holds for any i, j I 0 ∈ (λ+(u) 1)(r (v)) = (λ+(v) 1)(r (u)) (5.1) i − j j − i and we need to show that r (u) = (λ+(u) 1)η (5.2) i i − for some η Y 0 . ∈ + c 5.4. Rank 1 case. We assume ﬁrst that I = 1 and accordingly drop the subscript i from our formulae. We shall prove (5.2|)| by working with an adapted system of generators of Y 0. Recall that, by Proposition 2.10,

e~v e ~v (λ+(u) 1)B(v)= − − euv − − v vk Deﬁne B′(v)= t′ by equating the coeﬃcients of v in k! k k 0 X≥ v ~v vn B′(v)= B(v)= t −e~v e ~v −e~v e ~v n! n − − n 0 − − X≥ 0 The elements tk′ k N give another system of generators of Y which are homogeneous, with deg({ t})=∈ k = deg(t ) for any k N, and satisfy k′ k ∈ + k λ (u)(tk′ )= tk′ + u (5.3) YANGIANSANDQUANTUMLOOPALGEBRAS 27

5.5. Since the operator λ+(u) : Y 0 Y 0[u] is homogeneous with respect to the N–grading extending that on Y 0 by deg(→ u) = 1, it suﬃces to prove (5.2) when r(u) is homogeneous of degree n N. Moreover, since λ+(u) is C[~]–linear and the formulae (5.3) do not involve ~∈, we may further assume that the coeﬃcients of r(u) 0 0 lie in the C–subalgebra Y Y generated by the tk′ . 0 ⊂ { } An element of Y [u]n has the form

n µ r¯(u)= aµtµ′ u −| | (5.4) µ n |X|≤ where a C[t ] and, for a partition µ of length l, we deﬁne t = t t . The µ ∈ 0′ µ′ µ′ 1 · · · µ′ l 0 + proof of the existence of η Y n such that (λ (u) 1)(η) = r(u) proceeds in two steps: ∈ − (1) Show that, modulo elements of the form (λ+(u) 1)(η), − n µ r¯(u)= aµtµ′ u −| | (5.5) µ

where aµ C do not depend on t0′ . (2) Show that∈ any r(u) of the form (5.5) is equal to (λ+(u) 1)(η) for some − η Y 0 . ∈ n 0 5.6. Proof of (1). Forr ¯(u) Y n of the form (5.4), choose bν C[t0′ ] for every ν n such that ∈ ∈ ⊢ b (t′ + 1) b (t′ )= a (t′ ) ν 0 − ν 0 ν 0 Then

+ n µ r¯(u) (λ (u) 1) b t′ = a′ t′ u −| | − − ν ν µ µ ν n ! µ

(λ+(v) 1)r(u) −

n µ (a (t + 1) a (t )) t′ u −| | = 0 µ 0 − µ 0 µ µ

5.7. Proof of (2). Letr ¯(u) be of the form (5.5). For any 0 l n, write ≤ ≤ n µ rl(u)= aµtµ′ u −| | µ 0 and let D(u) : Y 0 Y 0[u] be the diﬀerential operator m ′ + →k D(u)= m 1 u ∂tm . Since (λ (u) 1)(tk′ )= u we get, for any partition µ ≥ − P + (λ (u) 1)(t′ )= D(u)(t′ ) + terms of smaller length − µ µ Thus, (5.1) implies that

D(u)¯rk(v)= D(v)¯rk(u)

0 This cross–derivative condition implies the existence of η Y n such that rk(u) = D(u)η. This implies thatr ¯(u) (λ+(u) 1)(η) has smaller∈ k. This completes the proof of− the existence− part of Proposition 5.2 when g is of rank 1.

5.8. Arbitrary rank. The argument for arbitrary g rests on the following

0 Lemma. There exist generators ̟i,r i I,r N of Y which are homogeneous, with { } ∈ ∈ deg(̟i,r)= r and such that

r λ±(u)̟ = ̟ δ u i j,r j,r ± i,j r Proof. By Proposition 2.10, the generating series Bj(v)= ~ r 0 tj,rv /r! satisfy ≥ 1 uv P (λ±(u) 1)~− B (v)= Q (v)e i − j ∓ ij

where Qij(v) = 2sinh(~diaijv/2)/~v. Since Qij = diaij mod ~, the matrix Q = 1 1 (Q ) is invertible. Set B (v) = ~ Q− B (v). Then (λ±(u) 1)B (v) = ij i′ − − j ij j i − j′ δ euv which, in terms of the expansion B (v) = ̟ vr/r! yields the required ij P i′ i,r transformation± property. P Since deg(v) = 1, the stated homogeneity of the ̟i,r is equivalent to Ad(ζ)(Bi′(v)) = 2 Bi′(ζ v) where Ad(ζ) denotes the action of ζ C× on Y~(g)[[v]] corresponding to ∈ 1 2 the N–grading. This in turn follows from the fact that Ad(ζ)(~− Bj(v)) = Bj(ζ v) and Ad(ζ)Q(v)= Q(ζ2v).

Using the generators ̟i,r, the proof of the existence part of Proposition 5.2 in higher rank follows the same argument as the one used for proving the suﬃciency of the cross derivative condition (here the existence of a primitive for any i I is guaranteed by the rank 1 case). ∈ YANGIANSANDQUANTUMLOOPALGEBRAS 29

0× + + 1 5.9. Uniqueness of ξ. Let ζ Y be an element such that ri (u)= ζ λi (u)(ζ)− for each i I. Then ∈ · ∈ + c1 + + 1 λi (u)(ζξ− )= λi (u)(ζ)λi (u)(ξ)− + 1 + 1 = ri (u)− ζri (u)ξ− 1 = ζξ− 1 1 By Proposition 2.10 (6), we get that λi±(u)(ζξ− ) = ζξ− for each i I. By 1 ∈ deﬁnition of the operators λi±(u), this implies that the coeﬃcients of ζξ− lie in the centre of Y~(g), which is trivial. This completes the proof of the last assertion of Proposition 5.2.

5.10. Torus action. The adjoint action of h on Y~(g) exponentiates to one of the algebraic torus H = HomZ(Q, C×) where Q h∗ is the root lattice. This action preserves homomorphisms of geometric type and⊂ acts on the corresponding formal 1 power series by ζ gi±(u) = ζi± gi±(u) where H ζ ζi = ζ(αi) is the ith coordinate function· { on H. } { } ∋ → 5.11. Uniqueness of homomorphisms of geometric type. [ Theorem. Let Φ, Φ : U~(Lg) Y~(g) be two homomorphisms of geometric type. ′ → Then, there exists ζ H and ξ 1+ Y 0 such that ∈ ∈ + Φ′ = Ad(ξ) (ζ Φ) c ◦ · Moreover, ζ is unique and ξ is unique up to multiplication by c C[[~]] . ∈ × 0 Proof. Let g±(u) , h±(u) Y\[[u]] be elements of corresponding to Φ and { i } { i } ⊂ G Φ′ respectively. By Lemma 3.7, we may use the action of H to assume that gi±(u)= 0 1 h±(u) mod Y\[u] . By Lemma 5.1, the elements r±(u) = h±(u) g±(u) 1+ i + i i · i − ∈ \0 Y [u]+ satisfy conditions (A0)–(C0±). By Proposition 5.2, we may ﬁnd an element ξ 1+ Y 0 such that r+(u)= ξ λ+(u)(ξ 1). It follows that for any i I ∈ + i · i − ∈ + + + Φ′(E )= h (σ )x c i,0 i i i,0 + + + + + = ri (σi )gi (σi )xi,0 + + 1 + + + = ξλi (σi )(ξ− )gi (σi )xi,0 + + + 1 = ξgi (σi )xi,0ξ− Moreover, for any r Z, ∈ rσ+ rσ+ 1 1 Φ′(Ei,r)= e i Φ′(Ei,0)= e i ξΦ(Ei,0)ξ− = ξΦ(Ei,r)ξ− + 1 1 By (B0), ri−(u) = λi−(u)(ri (u)− ) = ξλi−(u)(ξ− ) and it follows similarly that 1 0 Φ′(Fi,r)= ξΦ(Fi,r)ξ− for any i I and r Z. Since Φ and Φ′ coincide on U and 0 ∈ ∈ Ad(ξ)(η) = η for any η Y it follows that Φ′ = Ad(ξ) Φ. The last assertion of Proposition 5.2 implies∈ the uniqueness of ξ up to multiplication◦ by an element of C[[~]]×. 30 S.GAUTAMANDV.TOLEDANOLAREDO

6. Isomorphisms of geometric type

We prove in this section that any homomorphism of geometric type Φ : U~(Lg) [ → Y~(g) extends to an isomorphism of completed algebras and induces Drinfeld’s degeneration of U~(Lg) to Y~(g).

6.1. Classical limit. The specialisations of the quantum loop algebra U~(Lg) and 1 Yangian Y~(g) at ~ = 0 are the enveloping algebras U(g[z, z− ]) and U(g[s]) respectively. Speciﬁcally, if ei,fi, hi i I are the generators of g given in Section 2.1, the assignments { } ∈

e zk E , f zk F , h zr H i ⊗ → i,k i ⊗ → i,k i ⊗ → i,r and r 1 + r 1 r 1 ei s xi,r, fi s xi,r− , hi s ξi,r ⊗ → √di ⊗ → √di ⊗ → di extend respectively to isomorphisms

1 U(g[z, z− ]) ∼ U~(Lg)/~U~(Lg) and U(g[s]) ∼ Y~(g)/~Y~(g) → → [ Proposition. Let Φ : U~(Lg) Y~(g) be the homomorphism given by Theorem 4.7. Then, the specialisation of Φ at→~ = 0 is the homomorphism

1 exp∗ : U(g[z, z− ]) U(g[[s]]) U\(g[s]) −→ ⊂ given on g[z, z 1] by exp (X zk)= X eks. − ∗ ⊗ ⊗ 1 Proof. Since Φ(H )= d− t and, for r = 0, i,0 i i,0 6 ~ rk Φ(Hi,r)= ti,k q q 1 k! i i− k 0 − X≥ setting ~ = 0 yields Φ (h z0)= h s0, and |~=0 i ⊗ i ⊗ k k r 1 s r rs Φ ~=0 (hi z )= dihi = hi e | ⊗ di ⊗ k! ⊗ k 0 X≥ Further, since g+(u)= 1 mod ~ by (4.14), we get i √di

k k r 1 rσ+ 0 s r rs Φ (ei z )= e i diei s = ei = ei e |~=0 ⊗ √d ⊗ ⊗ k! ⊗ i k 0 p X≥ + where we used the fact that, in the classical limit, the operator σi corresponds to multiplication by s. Similarly, Φ (f zr)= f ers. |~=0 i ⊗ i ⊗ YANGIANSANDQUANTUMLOOPALGEBRAS 31

6.2. Let U~Lg be the kernel of the composition J ⊂ ~ 0 z 1 U~Lg → U(Lg) → Ug −−−→ −−−→ and let \ n U~(Lg) = lim U~(Lg)/ ←− J be the completion of U~(Lg) with respect to the ideal . J

Theorem. Let Φ : U~(Lg) Y~g be a homomorphism of geometric type. Then, → [ (1) Φ maps to the ideal Y~(g) = Y~(g) . J d + n 1 n (2) The corresponding homomorphism ≥ Q \ [ Φ : U~(Lg) Y~(g) → is an isomorphism. b

Proof. (1) Note ﬁrst that is generated by ~U~(Lg) and the elements Hi,r J 1 { − Hi,s, Ei,r Ei,s, Fi,r Fi,s i I,r,s Z since its image in U(g[z, z− ]) is generated by the classes− of these elements.− } ∈ Note∈ next that, for r, s = 0 6 ~ rk sk Φ(Hi,r Hi,s)= − ti,k − q q 1 k! i i− k 1 − X≥ while

k ~ r ~ 1 Φ(Hi,r Hi,0)= ti,k + d− ti,0 − q q 1 k! q q 1 − i i i− k 1 i i− − X≥ − 1 1 which lies in n 1 Y~(g)n since ~/(qi qi− )= di− mod ~. Finally, for r, s Z, ≥ − ∈ + + Q Φ(E E ) = (erσi esσi )g+(σ+)e i,r − i,s − i i i,0 ∈J and similarly Φ(Fi,r Fi,s) . (2) By Theorem 5.11− , it suﬃces∈J to prove this for the explicit homomorphism given by Theorem 4.7. The result then follows Proposition 6.3 below and the fact that, by Proposition 6.1, the specialisation of Φ at ~ = 0 is an isomorphism U(g\[z, z 1]) − → U\(g[s]). b

1 \ 6.3. Let J U(g[z, z− ]) be the kernel of evaluation at z = 1 and U(Lg) the completion of⊂U(Lg) with respect to J. Proposition. \ \ 1 (1) U~(Lg) is a ﬂat deformation of U(g[z, z− ]). [ \ (2) Y~(g) is a ﬂat deformation of U(g[s]) over C[[~]]. 32 S.GAUTAMANDV.TOLEDANOLAREDO

1 Proof. (1) Set, for brevity = U~(Lg) and U = U(g[z, z ]). We claim that is U − U a flat deformation of /~ , and that /~ = U. U U U U ∼ To prove the first assertion it suffices to show, by [21, Prop XVI.2.4], that bis a separated, complete andb torsion–freeb Cb[[~]]–module.b b To show that it is separated,U ~ ~k k n b note that , so that lim←− / and ∈J U ⊂ n>kJ J b ~k = 0 U { } k 0 \≥ To show completeness, note that b / n if n k /~k = lim ( / n)/(~k /~k n) = lim U J ≤ U U ←− U J U U∩J ←− /~k + n if n > k n n U U J from whichb b it readily follows that the map ~k lim←− / U −→ k U U is surjective. Finally, to prove thatb is torsion–free,b b note that the kernel of multipli- n 1 Un n n n 1 cation by ~ on / is ~− (~ )/ . We claim that ~ = ~ − , which U J U∩J J n 1 n U∩J J implies that the kernel of ~ on isb limn − / = 0 . To prove the claim, use the flatness of to identify it withU the C[[~J]]–moduleJ U{[[~}]], so that = J ~U[[~]]. U J ⊕ Let a , . . . , a and write a b= a0 + ~a , where a0 J and a U[[~]]. Then 1 n ∈J i i i i ∈ i ∈ 0 n 1 0 0 n 1 a a = a a a mod ~ − = = a a mod ~ − 1 · · · n 1 2 · · · n J · · · 1 · · · n J from which the claim follows. The fact that /~ = U follows by taking limits in the sequence U U ∼ n 1 n n 0 ~ / − / U/J 0 b b →b U J →U J → → The latter is exact since, under the natural surjection U, the ideal n is mapped to J n with kernel ~ n = ~ n 1. U → J U∩J J − [ (2) Since Y~(g) is the completion of Y~(g) with respect to the ideal Y~(g)+ of elements of positive degree, it follows as in (1) that it is a separated and complete n n 1 C[[~]]–module. The lack of torsion of Y~(g) implies that ~Y~(g) Y~(g) = ~Y~(g) − ∩ + + [ [ and therefore that Y~(g) is torsion–free. Thus, Y~(g) is a flat deformation of [ [ \ \ Y~(g)/~Y~(g) ∼= Y~(g)/~Y~(g) ∼= U(g[s]) as claimed.

6.4. Drinfeld’s degeneration. Consider the descending ﬁltration 0 2 U~(Lg)= (6.1) J ⊃J ⊃J ⊃··· n n+1 deﬁned by the powers of and let gr (U~(Lg)) = n 0 / be its associated graded. J J ≥ J J L YANGIANSANDQUANTUMLOOPALGEBRAS 33

+ Theorem ([10, 17]). Let di± i I C× be such that di di− = di. Then, the fol- { } ∈ ⊂ lowing assignment extends uniquely to an isomorphism of graded algebras Y~(g) ∼ → gr (U~(Lg)) J ξ d H U~(Lg)/ i,0 7−→ i i,0 ∈ J + + x d E U~(Lg)/ , x− d−F U~(Lg)/ i,0 7−→ i i,0 ∈ J i,0 7−→ i i,0 ∈ J + + 2 2 x d (E E ) / , x− d− (F F ) / i,1 7−→ i i,1 − i,0 ∈J J i,1 7−→ i i,1 − i,0 ∈J J Remark. The fact that U~(Lg) degenerates to Y~(g) is stated, without proof, in [10, 6]. The formulae above and the proof that they deﬁne an isomorphism Y~(g) = § ∼ gr (U~g) are given in [17]. J 6.5. Relation to Drinfeld’s degeneration. By Theorem 6.2, a homomorphism of geometric type Φ induces a homomorphism [ gr(Φ) : gr (U~(Lg)) Y~(g)=gr\ Y~(g) J −→ Y~(g)+

0 × Let g±(v) Y[[v] be the elements deﬁning Φ. By Lemma 3.7, { i }⊂ 1 [0 gi±(v)= mod Y [v]+ (6.2) di± + for some d± C such that d d− = d . i ∈ × i i i

Proposition. gr(Φ) is the inverse of the degeneration isomorphism ı : Y~(g) ∼ → U~(Lg) given by Theorem 6.4.

Proof. It suﬃces to verify the claim on the generators ξi,0,xi,±0,xi,±1 i I of Y~(g). Now, { } ∈ gr(Φ) ı(ξ ) = gr(Φ)(d H )= ξ ◦ i,0 i i,0 i,0 and + + [ gr(Φ) ı(x )= d Φ(E ) mod Y~(g) ◦ i,0 i i,0 + + + + + [ = di gi (σi )xi,0 mod Y~(g)+ + = xi,0 by (6.2). Moreover, + + [ gr(Φ) ı(xi,1)= di Φ(Ei,1 Ei,0) mod Y~(g) 2 ◦ − ≥ + + σi + + + [ = di (e 1)gi (σi )xi,0 mod Y~(g) 2 − ≥ + = xi,1

And similarly gr(Φ) ı(x− )= x− for r = 0, 1. ◦ i,r i,r 34 S.GAUTAMANDV.TOLEDANOLAREDO

7. Geometric solution for gln

In this section, we construct a homomorphism of geometric type for gln, and show that it intertwines the geometric realisations of the corresponding loop algebra and Yangian constructed by Ginzburg–Vasserot [15, 34].

7.1. The quantum loop algebra [8]. Throughout this section, we ﬁx n 2 and mostly follow the notation of [34]. Set I = 1,...,n 1 and J = 1,...,n ≥. Then, { − } { } U~(Lgln) is topologically generated over C[[~]] by elements Ei,r, Fi,r, Dj,r i I,j J,r Z. To describe the relations, introduce the formal power serie{s } ∈ ∈ ∈

r r Ei(z)= Ei,rz− Fi(z)= Fi,rz− r Z r Z X∈ X∈ and

s ~Dj,0 1 s Θj±(z)= Θj,± sz∓ = exp exp (q q− ) Dj, sz∓ ± ± 2 ± − ±  s 0 s 1 X≥ X≥ The relations are   (QL1-gl) For any j, j J and r, s Z, ′ ∈ ∈ [Dj,r, Dj′,s] = 0 (QL2-gl) For any i I and j J, ∈ ∈ 1 cji Θj±(z)Ei(w)Θj±(z)− = ϑcji (q z/w)Ei(w)

1 cji 1 Θj±(z)Fi(w)Θj±(z)− = ϑcji (q z/w)− Fi(w) qmζ 1 where c = δ + δ , ϑ (ζ) = − , and the right–hand side is ji − ji ji+1 m ζ qm 1 2 − expanded in powers of z∓ . (QL3-gl) For any i, i I, ′ ∈ ′ ′ i i ′ Ei(z)Ei (w)= ϑaii′ (q − z/w)Ei (w)Ei(z) ′ ′ i i 1 ′ Fi(z)Fi (w)= ϑaii′ (q − z/w)− Fi (w)Fi(z)

where aii′ = 2δii′ δ i i′ ,1 are the entries of the Cartan matrix of sln and the equalities are understood− | − | as holding after both side have been multiplied by the denominator of the function ϑm. (QL4-gl) For any i, i I, ′ ∈ + Θi+1(z) Θi−+1(z) 1 ′ ′ (q q− )[Ei(z), Fi (w)] = δi,i δ(z/w) + − Θi (z) − Θi−(z) ! r where δ(ζ)= r Z ζ is the formal delta function. ∈

2 P ±1 −1 note that the expansions in z are related by the symmetry ϑm(ζ )= ϑ−m(ζ). YANGIANSANDQUANTUMLOOPALGEBRAS 35

(QL5-gl) For any i, i I such that i i = 1, ′ ∈ | − ′| 1 E (z )E (z )E ′ (w) (q + q− )E (z )E ′ (w)E (z )+ i 1 i 2 i − i 1 i i 2 E ′ (w)E (z )E (z ) + (z z ) = 0 i i 1 i 2 1 ↔ 2 1 F (z )F (z )F ′ (w) (q + q− )F (z )F ′ (w)F (z )+ i 1 i 2 i − i 1 i i 2 F ′ (w)F (z )F (z ) + (z z ) = 0 i i 1 i 2 1 ↔ 2 For any i, i I such that i i 2, ′ ∈ | − ′|≥ Ei(z)Ei′ (w)= Ei′ (w)Ei(z)

Fi(z)Fi′ (w)= Fi′ (w)Fi(z) In terms of the generators E , F , D , the relations (QL1-gl)–(QL4-gl) read { i,r i,r j,r}

cjir [cjir] [D , E ]= c E [D , E ]= q− E j,0 i,k ji i,k j,r i,k r i,k+r

cjir [cjir] [D , F ]= c F [D , F ]= q− F j,0 i,k − ji i,k j,r i,k − r i,k+r ′ ′ i a ′ +i a ′ +i i q Ei,k+1Ei′,l q ii Ei,kEi′,l+1 = q ii Ei′,lEi,k+1 q Ei′,l+1Ei,k − ′ − ′ a ′ +i i i a +i q ii F F ′ q F F ′ = q F ′ F q ij F ′ F i,k+1 i ,l − i,k i ,l+1 i ,l i,k+1 − i ,l+1 i,k + Pi,k+l Pi,k− +l [E , F ′ ]= δ ′ − i,k i ,l ii q q 1 − − s where Pi±(z)= s 0 Pi,± sz∓ =Θi±+1(z)/Θi±(z) ≥0 ± We denote by U U~(Lgl ) the commutative subalgebra generated by the P ⊂ n elements Dj,r.

7.2. The Yangian Y~(gl ). The following deﬁnition can be found in [25, 3.1]. n § Y~(gln) is the algebra over C[~] generated by elements ei,r,fi,r,θj,r i I,j J,r N sub- { } ∈ ∈ ∈ ject to the following relations3 (Y1-gl) For any j, j J and r, s N, ′ ∈ ∈ [θj,r,θj′,s] = 0 (Y2-gl) For any j J and i I, ∈ ∈ [θj,0, ei,s]= cjiei,s [θ ,f ]= c f j,0 i,s − ji i,s

[θ , e ] [θ , e ]= ~c e θ j,r+1 i,s − j,r i,s+1 ji i,s j,r [θ ,f ] [θ ,f ]= ~c θ f j,r+1 i,s − j,r i,s+1 − ji j,r i,s where c = (δ δ ). ji − ji − j,i+1 3our conventions are adapted to [15, 34]. They diﬀer from those of [25] by the permutation ei,r ↔ fi,r and the relabelling θi,r ↔ hi,r. 36 S.GAUTAMANDV.TOLEDANOLAREDO

(Y3-gl) For any i I, ∈ [e , e ] [e , e ]= ~(e e + e e ) i,r+1 i,s − i,r i,s+1 i,r i,s i,s i,r [f ,f ] [f ,f ]= ~(f f + f f ) i,r+1 i,s − i,r i,s+1 − i,r i,s i,s i,r For any i I n 1 and r, s N, ∈ \ { − } ∈ [e , e ] [e , e ]= ~e e i,r+1 i+1,s − i,r i+1,s+1 − i+1,s i,r [f ,f ] [f ,f ]= ~f f i,r+1 i+1,s − i,r i+1,s+1 i,r i+1,s (Y4-gl) For any i, i I, ′ ∈ [ei,r,fi′,s]= δi,i′ pi,r+s r 1 1 where pi(v)=1+ ~ r 0 prv− − = θi+1(v)θi(v)− . (Y5-gl) For any i, i I such that≥ i i = 1, and r , r ,s N, ′ ∈ P | − ′| 1 2 ∈ ′ ′ [ei,r1 , [ei,r2 , ei ,s]] + [ei,r2 , [ei,r1 , ei ,s]] = 0

′ ′ [fi,r1 , [fi,r2 ,fi ,s]] + [fi,r2 , [fi,r1 ,fi ,s]] = 0 For i, i I such that i i > 1, and r, s N ′ ∈ | − ′| ∈ [ei,r, ei′,s]=0=[fi,r,fi′,s]

The Yangian Y~(gln) is N–graded by deg(ei,r) = deg(fi,r) = deg(θj,r) = r and deg(~) = 1.

0 7.3. Shift homomorphisms. Let Y Y~(gln) be the commutative subalgebra +⊂ generated by the elements θj,r and Y ,Y − Y~(gln) the subalgebras generated 0 { } ⊂ by Y and the elements ei,r (resp. fi,r ), i I, r N. { } { } 0∈ ∈ For any i I, deﬁne, as in Section 2.6, a Y –linear homomorphism σ± of Y by ∈ i ± e ′ e ′ (resp. f ′ f ′ ). The deﬁnition of σ± relies on the PBW i ,r → i ,r+δii′ i ,r → i ,r+δii′ i theorem for Y~(gln), which is proved in [27].

7.4. Alternative generators for Y 0. Deﬁne an alternative family of generators 0 dj,r j J,r N of Y by { } ∈ ∈ r 1 dj(u)= ~ dj,ru− − = log(θj(u)) r 0 X≥ vr Set B (v)= ~ d Y 0[[v]]. The following commutation relations are proved j j,r r! ∈ r 0 exactly as theirX≥ counterparts in Lemma 2.9. Lemma. The following holds for any j J and i I, ∈ ∈ cji~v 1 e− + [B (v), e ]= − eσi v e j i,s v i,s

cji~v 1 e− − [B (v),f ]= − eσi v f j i,s − v i,s YANGIANSANDQUANTUMLOOPALGEBRAS 37

7.5. The operators λi±(v). The following result is analogous to, and proved in the same way as, Proposition 2.10. 0 Proposition. There are operators λi±;s i I,s N on Y such that the following holds. { } ∈ ∈ (1) For any ξ Y 0, ∈ + ei,rξ = λi;s(ξ)ei,r+s s 0 X≥ fi,rξ = λi−;s(ξ)fi,r+s s 0 X≥ 0 0 (2) The operator λ±(v) : Y Y [v] given by i → s λi±(v)(ξ)= λi±;s(ξ)v s N X∈ is an algebra homomorphisms of degree 0 with respect to the N–grading on Y 0[v] extending that on Y 0 by deg(v) = 1. ǫ ǫ′ (3) The operators λi (v) and λi′ (v′) commute for any i, i′ I and ǫ,ǫ′ . Moreover, ∈ ∈ {±} + λi (v)λi−(v)= Id (4) For any i I and j J, ∈ ∈ cji~v2 e− 1 v1v2 (λi±(v1) 1)Bj(v2)= − e (7.1) − ± v2

\0 7.6. Let gi±(u) i I be a collection of elements in Y [u]. Deﬁne, as in Section 3.1, { } ∈ \ an assignment Φ : E , F , D Y~(gl ) by { i,r i,r j,r}→ n Φ(Dj,0)= θj,0

Bj(r) Φ(Dj,r)= 1 for r = 0 q q− 6 −+ kσi + + Φ(Ei,k)= e gi (σi ) ei,0 − kσi Φ(Fi,k)= e gi−(σi−) fi,0 and denote the restriction of Φ to U 0 by Φ0. For any i I, set ∈ r 1 1 0 1 ξ (u)=1+ ~ ξ u− − = θ (u)θ (u)− Y [[u− ]] i i,r i+1 i ∈ r 0 X≥ s 1 0 1 Pi±(z)= Pi,± sz∓ =Θi±+1(z)Θi±(z)− U [[z∓ ]] ± ∈ s 0 X≥ Theorem. The assignment Φ extends to an algebra homomorphism if and only if the following conditions hold. (A) For any i, i I, ′ ∈ + + + gi (u)λi (u)(gi−′ (v)) = gi−′ (v)λi−′ (v)(gi (u)) 38 S.GAUTAMANDV.TOLEDANOLAREDO

(B) For any i I and k Z, ∈ ∈ P + P ku + + 0 i,k i,k− e g (v)λ (v)(g−(v)) m =Φ − i i i |v =ξi,m q q 1 − − ! (C0) For any i, i I such that i i > 1, ′ ∈ | − ′| gi±(u)λi±(u)(gi±′ (v)) = gi±′ (v)λi±′ (v)(gi±(u)) (C1) For any i I ∈ u v ~ v u ~ e e ± e e ± g±(u)λ±(u)(g±(v)) − = g±(v)λ±(v)(g±(u)) − i i i u v ~ i i i v u ~ − ∓ − ∓ (C2) For any i I n 1 , ∈ \ { − } 1 u v 1 u ~/2 v+~/2 ± e e ± e − e g±(u)λ±(u)(g± (v)) − = g± (v)λ± (v)(g±(u)) − i i i+1 u v i+1 i+1 i u v ~ − − − ! The proof of Theorem 7.6 is given in 7.7–7.11. It follows the same lines as that of Theorem 3.4, with the exception of the q–Serre relations (QL5-gl) which are proved directly. 7.7. A proof similar to that of Lemmas 3.5 and 3.6 yields the following (1) Φ is compatible with the relation (QL4-gl) if, and only if (A) and (B) hold. (2) Φ is compatible with the relation (QL3-gl) if, and only if (C0)–(C2) hold. By virtue of condition (C0), Φ is compatible with the the q–Serre relations (QL5-gl) whenever i i′ > 1. We therefore need only consider the case i i′ = 1. We shall in fact restrict| − to| i = i + 1 since the case i = i 1 is dealt with| − similarly.| ′ ′ − 7.8. The essential ingredient is the following analogue of Lemma A.4. We leave it to the reader to carry out the construction of the auxiliary algebra Y (see A.2), § the operators σ , σ and σ ′ on Y ( A.3) and the map p ′ : Y i,(1) i,(2) i 2αi+αi′ § ii 2αi+αi′ → Y~(gln).

Lemma. The kernel of pii′ is the C[~]–linear span of the following elements 0 (1) For any A(u , u , v) Y [u , u , v] 1 2 ∈ 1 2 2 A(σ , σ , σ ′ ) (σ σ ′ )e e ′ (σ σ ′ ~)e e ′ e i,(1) i,(2) i i,(2) − i i,0 i ,0 − i,(2) − i − i,0 i ,0 i,0 2 A(σ , σ , σ ′ ) (σ σ ′ )e e ′ e (σ σ ′ ~)e ′ e i,(1) i,(2) i i,(1) − i i,0 i ,0 i,0 − i,(1) − i − i ,0 i,0 0 (2) For any B(u , u , v)= B(u , u , v) Y [u , u , v] 1 2 2 1 ∈ 1 2 2 B(σ , σ , σ ′ )(σ σ ~)e e ′ i,(1) i,(2) i i,(1) − i,(2) − i,0 i ,0 2 B(σ , σ , σ ′ )(σ σ ~)e ′ e i,(1) i,(2) i i,(1) − i,(2) − i ,0 i,0 0 (3) For any B(u , u , v)= B(u , u , v) Y [u , u , v] 1 2 2 1 ∈ 1 2 2 2 B(σ , σ , σ ′ ) e e ′ 2e e ′ e + e ′ e i,(1) i,(2) i i,0 i ,0 − i,0 i ,0 i,0 i ,0 i,0 YANGIANSANDQUANTUMLOOPALGEBRAS 39

Corollary. The kernel of pii′ is stable under the action of A(σi,(1), σi,(2), σi′ ), for 0 any A(u , u , v)= A(u , u , v) Y [u , u , v]. 1 2 2 1 ∈ 1 2 Remark. In the next sections we use the following convention for notational con- venience: for each X Y and X = p ′ (X), we set ∈ 2αi+αi′ ii A(σi,1,σi,2,σi′ )(X)= pi,i′ A(σi,(1), σi,(2), σi′ )(X) 7.9. We shall only prove the q–Serre relations for the case of the E’s and conse- quently drop the superscript +. We need to show that the following holds for any k , k , l Z 1 2 ∈ 1 Φ(E )Φ(E )Φ(E ) (q + q− )Φ(E )Φ(E )Φ(E ) i,k1 i,k2 j,l − i,k1 j,l i,k2 + Φ(E )Φ(E )Φ(E ) + (k k ) = 0 j,l i,k1 i,k2 1 ↔ 2 As in A.5, an application of Corollary 7.8 shows that this reduces to showing that § 2 1 2 Φ(E ) Φ(E ) (q + q− )Φ(E )Φ(E )Φ(E )+Φ(E )Φ(E ) = 0 i,0 j,0 − i,0 j,0 i,0 j,0 i,0 7.10. With Corollary 7.8 in mind, we seek to factor a common symmetric function out of each of the above summands. This is achieved by the following result. 0 Lemma. There exists H(u , u , v) Y [[u , u , v]] symmetric in u u , such that 1 2 ∈ 1 2 1 ↔ 2 2 Φ(E ) Φ(E ′ )= H(σ ,σ ,σ ′ ) (σ ,σ ,σ ′ )e e e ′ i,0 i ,0 i,1 i,2 i P0 i,1 i,2 i i,0 i,0 i ,0 Φ(E )Φ(E ′ )Φ(E )= H(σ ,σ ,σ ′ ) (σ ,σ ,σ ′ )e e ′ e i,0 i ,0 i,0 i,1 i,2 i P1 i,1 i,2 i i,0 i ,0 i,0 2 Φ(E ′ )Φ(E ) = H(σ ,σ ,σ ′ ) (σ ,σ ,σ ′ )e ′ e e i ,0 i,0 i,1 i,2 i P2 i,1 i,2 i i ,0 i,0 i,0 where , , C[[u , u , v]] are given in terms of the function P0 P1 P2 ∈ 1 2 ex ey P (x,y)= − C[[x,y]]S2 x y ∈ − by = P (u + ~, u )P (u ~/2, v + ~/2)P (u ~/2, v + ~/2) P0 1 2 1 − 2 − = P (u + ~, u )P (u ~/2, v + ~/2)P (u , v) P1 1 2 1 − 2 = P (u + ~, u )P (u , v)P (u , v) P2 1 2 1 2 0 Proof. Deﬁne Gab(x,y) Y [[x,y]] by λa(x)(gb(y)) = gb(y)Gab(x,y). Then, in obvious notation, ∈

Φ(Ea,0)Φ(Eb,0)Φ(Ec,0)= ga(σa)ea,0gb(σb)eb,0gc(σc)ec,0 = g (σ )λ (σ )(g (σ ))λ (σ ) λ (σ )(g (σ ))e e e a a a a b b a a ◦ b b c c a,0 b,0 c,0 = ga(σa)gb(σb)gc(σc)Gab(σa,σb)Gac(σa,σc)λa(σa)(Gbc(σb,σc))ea,0eb,0ec,0 We record for later use the symmetry in the interchange a b of the term ↔ Gac(σa,σc)λa(σa)(Gbc(σb,σc)) = λa(σa) λb(σb)(gc(σc))/gc(σc) ◦ (7.2) = Gbc(σb,σc)λb(σb)(Gac(σa,σc)) where the second equality follows from the commutativity of λa(σa) and λb(σb). 40 S.GAUTAMANDV.TOLEDANOLAREDO

0 S Set now F = g (σ )g (σ )g ′ (σ ′ ) Y [[σ ,σ ,σ ′ ]] 2 . Then, the above yields i i,1 i i,2 i i ∈ i,1 i,2 i 2 2 Φ(Ei,0) Φ(Ei′,0)= F Gii(σi,1,σi,2)Gii′ (σi,1,σi′ )λi(σi,1)(Gii′ (σi,2,σi′ ))ei,0ei′,0 Φ(Ei,0)Φ(Ei′,0)Φ(Ei,0)= F Gii′ (σi,1,σi′ )Gii(σi,1,σi,2)λi(σi,1)(Gi′i(σi′ ,σi,2))ei,0ei′,0ei,0 2 2 Φ(Ei′,0)Φ(Ei,0) = F Gi′i(σi′ ,σi,1)Gi′i(σi′ ,σi,2)λi′ (σi′ )(Gii(σi,1,σi,2))ei′,0ei,0

We claim that Gii(u1, u2) = Gii(u1, u2)P (u1 + ~, u2), where G is symmetric in u1, u2. Indeed, by condition (C1)

Gii(u1, u2)P (u1, u2 + ~)= Gii(u2, u1)P (u2, u1 + ~)

whence the result with Gii(u1, u2)= Gii(u1, u2)/P (u1 + ~, u2). It follows that 2 2 Φ(Ei,0) Φ(Ei′,0)= H(σi,1,σi,2,σi′ )P (σi,1 + ~,σi,2)ei,0ei′,0 where 0 H(u , u , v)= g (u )g (u )g ′ (v)G (u , u )G ′ (u , v)λ (u )(G ′ (u , v)) Y [[u , u , v]] 1 2 i 1 i 2 i ii 1 2 ii 1 i 1 ii 2 ∈ 1 2 is symmetric in u1, u2 by (7.2). Next, assuming that i′ = i + 1, condition (C2) yields

G ′ (u, v)P (u, v) = G ′ (v, u)P (u ~/2, v + ~/2) ii i i − so that

Φ(Ei,0)Φ(Ei′,0)Φ(Ei,0)=

P (σi,2,σi′ ) H(σi,1,σi,2,σi′ )P (σi,1 + ~,σi,2) ei,0ei′,0ei,0 P (σ ~/2,σ ′ + ~/2) i,2 − i Finally, using (7.2) again, with a = i, b = i′, c = i and the previous calculation yields

2 Φ(Ei′,0)Φ(Ei,0) =

P (σi,1,σi′ ) P (σi,2,σi′ ) 2 H(σi,1,σi,2,σi′ )P (σi,1+~,σi,2) ei′,0ei,0 P (σ ~/2,σ ′ + ~/2) P (σ ~/2,σ ′ + ~/2) i,1 − i i,2 − i as claimed.

7.11. By Lemma 7.10 and Corollary 7.8, we are reduced to proving the following

q 2 1 = (σ ,σ ,σ ′ )e e ′ (q + q− ) (σ ,σ ,σ ′ )e e ′ e S P0 i,1 i,2 i i,0 i ,0 − P1 i,1 i,2 i i,0 i ,0 i,0 2 + (σ ,σ ,σ ′ )e ′ e = 0 P2 i,1 i,2 i i ,0 i,0 Step 1. We ﬁrst observe that 1+ e~ P (u + ~, u ) P (u , u ) (u u ~)C[[~, u , u ]]S2 1 2 − 2 1 2 ∈ 1 − 2 − 1 2 This allows us to use (2) of Lemma 7.8 to obtain q 2 2 = ′ (σ ,σ ,σ ′ )e e ′ 2 ′ (σ ,σ ,σ ′ )e e ′ e + ′ (σ ,σ ,σ ′ )e ′ e S P0 i,1 i,2 i i,0 i ,0 − P1 i,1 i,2 i i,0 i ,0 i,0 P2 i,1 i,2 i i ,0 i,0 YANGIANSANDQUANTUMLOOPALGEBRAS 41

where

~/2 ′ = e P (u , u )P (u ~/2, v + ~/2)P (u ~/2, v + ~/2) P0 1 2 1 − 2 − ′ = P (u + ~, u )P (u ~/2, v + ~/2)P (u , v)= P1 1 2 1 − 2 P1 ~/2 ′ = e P (u , u )P (u , v)P (u , v) P2 1 2 1 2 Step 2. We use next (3) of Lemma 7.8 with B = to get P2′ q 2 = ( ′ ′ )e e ′ 2( ′ ′ )e e ′ e S P0 − P2 i,0 i ,0 − P1 − P2 i,0 i ,0 i,0 One can easily check that is divisible by u v ~, which together with P1′ − P2′ 2 − − (1) of Lemma 7.8, with A = 2 P1′ − P2′ , yields u v ~ 2 − − q 1′ 2′ 2 = ′ ′ 2 P − P (u v) e e ′ S P0 − P2 − u v ~ 2 − i,0 i ,0 2 − − Step 3. Finally we can directly verify that the function

1′ 2′ = ′ ′ 2 P − P (u v) F P0 − P2 − u v ~ 2 − 2 − − is divisible by u1 u2 ~. Moreover the quotient F is symmetric in u1, u2. − − u1 u2 ~ This allows us to use (2) of Lemma 7.8 to conclude− that −q = 0. S 7.12. The variety . Fix integers 1 n d N, let F ≤ ≤ ∈ = 0= V V V = Cd F 0 ⊆ 1 ⊆···⊆ n n o be the variety of n–step ﬂags in Cd, and T its cotangent bundle. We describe ∗F below the GLd(C) C×–equivariant K–theory and cohomology of T ∗ following [15, 34]. × F The connected components of are parametrised by the set of partitions of [1, d] into n intervals, i.e., F P

= d =(0= d d d = d) P { 0 ≤ 1 ≤···≤ n } where d labels the component d consisting of ﬂags such that dim V = d . ∈ P F k k The symmetric group Sd acts on the rings 1 1 1 S = C[q± , X1± ,...,Xd± ] R = C[~,x1,...,xd] by permuting the variables X1,...,Xd and x1,...,xd and ﬁxing q, ~ respectively. For any d , denote by ∈ P

S(d)= Sd1 d0 Sdn dn−1 Sd − ×···× − ⊂ 42 S.GAUTAMANDV.TOLEDANOLAREDO the subgroup preserving the corresponding partition. Then, the following holds GL (C) C× S(d) K d × (T ∗ ) = S F ∼ d M∈P S(d) HGL (C) C× (T ∗ ) = R d × F ∼ d M∈P C× 1 where K (pt)= C[q,q− ] and HC× (pt)= C[~]. 7.13. For any partition d and i I, set ∈ P ∈ di± =(0= d0 di 1 di 1 di+1 dn = d) ≤···≤ − ≤ ± ≤ ≤···≤ if the right–hand side makes sense as a partition. If d, d are two partitions, and P is one of the rings R,S, we denote by ′ ∈ P σ(d, d′) the symmetrisation operator S(d) S(d′) S(d′) σ(d, d′) : P ∩ P , σ(d, d′)(p)= τ(p) → τ S(d′)/S(d) S(d′) ∈ X ∩

7.14. U~(Lgln)–action [15, 34]. Consider the following operators acting on S( )= SS(d) P d M∈P S(d) (1) For any j J,Ψ (Θ±(z)) acts on S as multiplication by ∈ U j dj−1 1 d qz q− Xk z Xk S(d) 1 − 1 − S [[z∓ ]] z Xk q− z qXk ∈ kY=1 − k=Ydj +1 − (2) For any i I, the operators ∈ S(d) S(d+) 1 ΨU (Ei(z)) : S S i [[z, z− ]] → − S(d) S(d ) 1 Ψ (F (z)) : S S i [[z, z− ]] U i → act by 0 if di± is not deﬁned, and by

1 + qXdi+1 q− Xk ΨU (Ei(z))p = σ(d, di ) δ(Xdi+1/z) − p  Xdi+1 Xk  k Ii Y∈ −   1 q− Xdi qXk ΨU (Fi(z))p = σ(d, di−) δ(Xdi /z) − p  Xdi Xk  k Ii ∈Y+1 −   otherwise, where Ii is the interval [di 1 + 1,...,di]. − The following result is due to Ginzburg and Vasserot and is proved in [34, 2.2]. § Theorem. The assignment ΨU extends to an algebra homomorphism

Ψ : U~(Lgl ) End −1 (S( )) U n → C[q,q ] P YANGIANSANDQUANTUMLOOPALGEBRAS 43

7.15. Y~(gln)–action. Consider the following operators acting on

R( )= RS(d) P d M∈P (1) For any j J,Ψ (θ (u)) acts on RS(d) as multiplication by ∈ Y j dj−1 d u xk + ~ u xk S(d) 1 − − R [[u− ]] u xk u xk ~ ∈ kY=1 − k=Ydj +1 − − (2) For any i I, ∈ S(d) S(d+) 1 Ψ (e (u)) : R R i [[u− ]] Y i → S(d) S(d−) 1 Ψ (f (u)) : R R i [[u− ]] Y i →

act as zero if di± is not deﬁned, and by

+ 1 xdi+1 xk + ~ ΨY (ei(u))p = ~σ(d, di ) − p u xdi+1 xdi+1 xk  k Ii − Y∈ −   1 xdi xk ~ ΨY (fi(u))p = ~σ(d, di−) − − p u xdi xdi xk  k Ii − ∈Y+1 −   otherwise. The following result is proved in a similar way to Theorem 7.14

Proposition. The assignment ΨY extends to an algebra homomorphism

Ψ : Y~(gl ) End (R( )) Y n → C[~] P Remark. The above formulae are degenerations of those of the previous section obtained by setting z = etu,q = et~/2, X = etxk and letting t 0. k → 7.16.

Lemma. The homomorphism ΨY maps the centre of Y~(gln) surjectively to Sd Z C[~,x1,...,xd] . In particular, there exists an element

r 1 1 ∆(u)=1+ ~ ∆ u− − [[u− ]] r ∈Z r 0 X≥ such that d u xk ~ ΨY (∆(u)) = − − u xk kY=1 − 44 S.GAUTAMANDV.TOLEDANOLAREDO

Proof. By [25, Cor. 1.11.8], is generated by the coeﬃcients of the element Z 1 qdet(u)= θ (u)θ (u ~) θ (u (n 1)~) Y~(gl )[[u− ]] 1 2 − · · · n − − ∈ n It readily follows from 7.15 that

d u xk ΨY (qdet(u)) = − u xk (n 1)~ kY=1 − − − By (2.7), L(v)= B(log(qdet(u))) [[v]] therefore satisﬁes ∈Z d e(xk+(n 1)~)v exkv vr 1 Ψ (L(v)) = − − = p ( x + (n 1)~ ) p ( x ) − Y v r { k − } − r { k} r! k=1 r 1 X X≥ r which yields the surjectivity since the power sums pr(x1,...,xd)= k xk generate C[x ,...,x ]Sd . 1 d P

7.17. We will need the following

r [0 Lemma. For any i I, there exists Tdi±(v)= r 0 Tdi,r± v Y [v] such that ∈ ≥ ∈ P v+xk ~ + v xk 1 e− − ΨY Tdi (v) = − v+x − 1 e− k v xk + ~ k Ii − − Y∈ v+xk+~ v xk 1 e− ΨY Tdi−(v) = − v+x − 1 e− k v xk ~ k Ii+1 − − − ∈Y The proof of this lemma is given in Section 7.20.

\ 7.18. A compatible assignment. Let Φ : Ei,0, Fi,0, Dj,r i I,j J,r Z Y~(gln) be the assignment deﬁned by { } ∈ ∈ ∈ →

Φ(Dj,0)= θj,0 B (v) Φ(D )= j j,r q q 1 − v=r − Φ(E )= e Td + q ∆0 θi,0 i,0 i,s i,s − − s 0 X≥ ∆0+θi+1,0 Φ(Fi,0)= fi,s Tdi,s− q s 0 X≥ where ∆0 is given by Lemma 7.16. Extend Φ to the generators Ei,r, Fi,r, r Z by deﬁning, as in Section 3.3, ∈

,(r) ,(r) m rv 0 Td± (v)= Td± v = e Td±(v) Y [[v]] i i,m i ∈ m 0 X≥ c YANGIANSANDQUANTUMLOOPALGEBRAS 45

and setting

+,(r) ∆0 θi,0 Φ(Ei,r)= ei,s Tdi,s q− − s 0 X≥ ,(r) ∆0+θi+1,0 Φ(Fi,r)= fi,s Td−i,s q s 0 X≥

7.19. Let R[( ) be the completion with respect to the N–grading given by deg(x )= P k deg(~) = 1. Deﬁne a homomorphism ch : S( ) R[( ) mapping each SS(d) to P → P R\S(d) by

q e~/2 and X exk 7−→ k 7−→

Theorem. The assignment Φ above intertwines the geometric realisations of U~(Lgln) and Y~(gl ) on S( ) and R( ) respectively. In other words, the following holds for n P P any X Ei,r, Fi,r, Dj,r i I,j J,r N and π S( ). ∈ { } ∈ ∈ ∈ ∈ P ch(X π)=Φ(X) ch(π) · ·

Proof. Consider ﬁrst the case X = Dj,r, j J, r Z. By deﬁnition of ΨU and ΨY , S(d) ∈ S∈(d) Dj,0 = 2/~ log(Θj,0) and θj,0 act on S and R respectively as multiplication by d (dj dj 1). Further, (2.7) yields − − −

dj−1 d 1 ~v x v ~v x v Ψ (B (v)) = (1 e− ) e k + (e 1) e k Y j v  − −  Xk=1 k=Xdj +1   Similarly, taking log in

dj−1 2 d 1 s z q− Xk z Xk ΨU exp (q q− ) Dj,sz− = − − 2   −  z Xk z q Xk s 1 k=1 k=d +1 X≥ Y − Yj −    yields

dj−1 d 1 1 2r r 2r r Ψ ((q q− )D )= (1 q− ) X + (q 1) X U − j,r r  − j − k  Xk=1 k=Xdj +1   Thus, ch(Ψ (D )π)=Ψ B (r)π/(q q 1) for any π SS(d). U j,r Y j − − ∈ We turn next to X = E . Let π SS(d) and set p = ch(π) RS(d). Since ∆ i,0 ∈ ∈ 0 acts on RS(d) as multiplication by d by Lemma 7.16, and θ acts as multiplication − j,0 46 S.GAUTAMANDV.TOLEDANOLAREDO by d (dj dj 1), we get − − − + dj dj−1 Φ(Ei,0)(p)= ei,s Tdi,s p q − s 0 X≥ ~ + + s xdi+1 xk + dj dj−1 = σ(d, di ) Tdi,s xdi+1p − q −  xdi+1 xk  s 0 k Ii X≥ Y∈ −   + + xdi+1 xk + ~ dj dj−1 = σ(d, di ) Tdi (xdi+1) p − q −  xdi+1 xk  k Ii Y∈ −   xdi+1 xk ~ + e e − dj dj−1 = σ(d, di ) p x − q −  e di+1 exk  k Ii Y∈ −   1 + qXdi+1 q− Xk = σ(d, di ) p ch −  Xdi+1 Xk  k Ii Y∈ − = ch (Ei,0 π)  The proof for the rest of the generators is identical.

7.20. Proof of Lemma 7.17. Let ∆(u) be the formal series given in Lemma 7.16, and set r 1 z(u)= ~ zru− − = log(∆(u)) r 0 X≥ For any j J, deﬁne y (u) Y 0[[u 1]] by ∈ j ∈ − j 1 − yj(u)= z(u + (j 1)~)+ dj(u)+ (dj s(u + s~) dj s(u + (s 1)~)) (7.3) − − − − − Xs=1 A computation similar to the one given in 7.19 shows that, for any j J, ∈ 1 e~v Ψ (B(y (u))) = − exkv (7.4) Y j v k Ij X∈ Set now4 v J(v) = log Q[[v]] 1 e v ∈ − − and, for any i I, deﬁne ∈ + td (v)= B(y ( ∂))J ′(v + ~) (7.5) i i − td−(v)= B(y ( ∂))J ′(v) (7.6) i − i+1 −

4Note the diﬀerence between J(v) and the function G(v) used in Section 4.1 for constructing v the solutions for simple Lie algebras: J(v)= G(v)+ 2 . YANGIANSANDQUANTUMLOOPALGEBRAS 47

where ∂ = d/dv. We claim that Tdi±(u) = exp(tdi±(u)) satisfy the conditions of the Lemma. By (7.4) we have

~∂ + 1 e− x ∂ Ψ td (v) = − e− k ∂J(v + ~) Y i  ∂  − k Ii X∈   Using e p∂f(v)= f(v p), we get − − Ψ (td+(v)) = J(v x ) J(v x + ~) Y i − k − − k k Ii X∈ v+x ~ v xk 1 e− k− = log − v+x − 1 e− k v xk + ~ k Ii X∈ − − v+xk+~ v xk 1 e− = log − v+x −  1 e− k v xk + ~  k Ii Y∈ − − The proof for the case is same.   − 7.21. Standard form of Φ. We rewrite below the assignment Φ in a form in which Theorem 7.6 can be applied, and use this to prove that Φ extends to an algebra \ homomorphism U~(Lgl ) Y~(gl ). n → n Lemma. Let yj(v) be given by (7.3), and λi±(u) the operators of Proposition 7.5. Then, ~v e 1 uv (λ±(u) 1)B(y (v)) = (δ δ ) − e i − j ± i,j − j,i+1 v The proof of this lemma essentially follows from Proposition 7.5. Corollary. For any i, i I, we have ′ ∈ + + + v+u δi,i′ δi,i′−1 λi (u)(Tdi′ (v)) Tdi′ (v) v u + ~ 1 e− − + = + = − v+u ~ − Td ′ (u) λ (u)(Td ′ (v)) 1 e v u i i− i − − − − + v+u+~ δi,i′ δi,i′+1 λi (u)(Tdi−′ (v)) Tdi−′ (v) v u 1 e− − = = − v+u − Td ′ (u) λ (u)(Td ′ (v)) 1 e v u ~ i− i− i− − − − − It follows that + kσi + + Φ(Ei,k)= e gi (σi )ei,0 − kσi Φ(Fi,k)= e gi−(σi−)fi,0 where

+ ∆0 θi, ~ + g (u)= q− − 0 Td (u) i q q 1 i − − (7.7) ∆ +θi , ~ g−(u)= q 0 +1 0 Td−(u) i q q 1 i − − 48 S.GAUTAMANDV.TOLEDANOLAREDO

7.22. We record the action of the operators λi±′ (u) on gi±(v) using Corollary 7.21 v u + ~ ev eu λ+(u)(g+(v)) = λ (u)(g+(v)) = g+(v) (7.8) i i i− 1 i i v+~/−2 u ~/2 − − e e − v u v −u ~ ev − eu λ+ (u)(g (v)) = λ (u)(g (v)) = g (v) (7.9) i+1 i− i− i− i− v ~/−2 −u+~/2 − e − e v u + − − Using the fact that λi (u)λi−(u) = id, we get four more equations from these. More- + + over, λ±′ (u)(g (v)) = g (v) for i = i, i 1 and λ±′ (u)(g−(v)) = g−(v) for i = i, i+1. i i i ′ 6 − i i i ′ 6 7.23.

Theorem. The series gi±(u) satisfy the conditions (A),(B),(C0)–(C2) of Theorem \ 7.6 and therefore give rise to an algebra homomorphism Φ : U~(Lgl ) Y~(gl ). n → n 7.24. Proof of (A). We need to prove that for every i, i I, we have ′ ∈ + + + gi (u)λi (u)(gi−′ (v)) = gi−′ (v)λi−′ (v)(gi (u)) + If i = i′, i′ + 1, both sides are equal to gi (u)gi−′ (v). For i = i′, by (7.8)–(7.9), the left6 and right–hand sides are respectively equal to v u ev ~/2 eu+~/2 u v eu+~/2 ev ~/2 − − − and − − − ev eu v u ~ eu ev u v + ~ − − − − − The case i = i′ + 1 follows in the same way. 7.25. Proof of (B). Let i I. By (7.9), ∈ 1 + + + q q− g (u)λ (u)(g−(u)) = g (u)g−(u) − i i i i i ~

θi , θi, ~ + = q +1 0− 0 Td (u)Td−(u) q q 1 i i − − ~ ~(θi+1,0 θi,0) + = exp − + td (u)+ td−(u) q q 1 2 i i − − By deﬁnition of tdi±, + td (u)+ td−(u)= By ( ∂)J ′(u)+ By ( ∂)J ′(u + ~) i i − i+1 − i − = B(yi+1(v) yi(v + ~)) v= ∂ J ′(u) − − | − ~∂ where the second equality follows from J ′(u + ~) = e J ′(u) and the fact that pv e B(f(u)) = B(f(u + p)). Next, the deﬁnition of yi yields y (v) y (v + ~)= d (v) d (v) i+1 − i i+1 − i hence + r+1 di+1,r di,r (r+1) td (u)+ td−(u)= ~ ( 1) − J (u) i i − r! r 0 X≥ which implies that

~(θi+1,0 θi,0) + r+1 di+1,r di,r (r+1) − + td (u)+ td−(u)= ~ ( 1) − G (u) 2 i i − r! r 0 X≥ YANGIANSANDQUANTUMLOOPALGEBRAS 49

and the proof of (B) follows from Proposition 4.2

7.26. Proof of (C0)–(C2). The condition (C0) follows from the fact that (λ±(u) i − 1)(gi±′ (v)) = 0 if i i′ > 1. Since the proof of (C1) is the same as the one given in the verification of| − (A),| we are left with checking (C2). We need to show that, for any i I n 1 , ∈ \ { − } u v u ~/2 v+~/2 + + + e e + + e − e g (u)λ (u)g (v) − = g (v)λ± (v)g (u) − i i i+1 u v i+1 i+1 i u v ~ − − − Using (7.8)–(7.9), the left and right–hand sides are respectively equal to u v eu ~/2 ev+~/2 eu ev eu ~/2 ev+~/2 − − − − and − − eu ev u v ~ u v u v ~ − − − − − − 7.27. Isomorphism between completions. We give below an analogue of The- orem 6.2 for g = gln. We begin by defining the appropriate completion of U~(Lgln), which differs from the one used in 6.3 due to the fact that gln is not semisimple. For each r 0, t Z and X = Ei, Fi, Θj, where i I and j J, consider the element ≥ ∈ ∈ ∈ r r X = ( 1)s X r;t − s s+t s=0 + 1X t r where Θ = (Θ Θ− )/(q q ). Note that X = x z (1 z) mod ~ where j,l j,l − j,l − − r;t ⊗ − x gl is such that X = x mod ~. Let be the two–sided ideal of U~(Lgl ) ∈ n Kr n generated by the elements Xr′;t r′ r,t Z, and ~ if r = 1. Finally, let n U~(Lgln) be the ideal { } ≥ ∈ J ⊂ = Jm Km1 ···Kmk m1,...,mk 1 m1+ X+m ≥=m ··· k Then, is a descending filtration, ′ ′ , and the completion Jm JmJm ⊂Jm+m \ U~(Lgln) = lim U~(Lgln)/ m ←− J 1 is a flat deformation of the completion of U(gln[z, z− ]) at z = 1.

Remark. The reason we use m instead of the powers of the evaluation ideal 1 as J J 1 in 6.3 can be seen at the classical level. Indeed, the correct ﬁltration of U(gln[z, z− ]) m 1 m is given by the Jm = U((z 1) gln[z, z− ]) which contains, but does not equal J1 . − 2 m For example, if I is the identity matrix, then I (z 1) J2 m 1 J1 . ⊗ − ∈ \ ≥ For each m N, set ∈ S \ ′ \ Y~(gln) m = (Y~(gln))m Y~(gln) ≥ ⊂ m′ m Y≥ where (Y~(gln))m is the subspace of degree m. The proof of the following result is similar to that of Theorem 6.2 and therefore omitted. \ Theorem. The homomorphism Φ maps m to Y~(gln)m for any m N, and induces an isomorphism of completed algebras J ∈ \ \ Φ : U~(Lgl ) ∼ Y~(gl ) n → n b 50 S.GAUTAMANDV.TOLEDANOLAREDO

Appendix A. Proof of the Serre relations A.1. Let g be a complex, semisimple Lie algebra. The aim of this appendix is to prove the following [ Proposition. Let Φ be the assignment Ei,k, Fi,k,Hi,k Y~(g) given in Sections 3.1–3.3, and assume that the relations (A){ and (B) of} Theorem → 3.4 hold. Then, Φ is compatible with the q–Serre relations (QL6). For i = j I, set m = 1 a . Define, for any k = (k ,...,k ) Zm and l Z 6 ∈ − ij 1 m ∈ ∈ m q s m ij(k, l)= ( 1) Φ(Ei,kπ(1) ) Φ(Ei,kπ(m−s) )Φ(Ej,l) S − s q · · · π Sm s=0 i X∈ X [ Φ(E ) Φ(E ) Y~(g) (A.1) i,kπ(m−s+1) · · · i,kπ(m) ∈ and let q = q (0, 0), explicitly given as follows Sij Sij m q s m m s s = ( 1) (Φ(E )) − Φ(E )(Φ(E )) (A.2) ij s i,0 j,0 i,0 S − qi Xs=0 q Our aim is to show that ij(k, l) = 0. Let us outline the main steps of the proof. S q q (1) We first reduce the proof of ij(k, l) = 0 to ij = 0. This is achieved in Lemma A.5. S S (2) By a standard argument using the representation theory of U~sl2, we deduce in Lemma A.6 that q acts by zero on any finite–dimensional representation Sij of Y~(g). (3) Finally, we show that these representations separate points in Y~(g), and hence that q = 0. A.8 and A.9 are devoted to the proof of this fact Sij § (Corollary A.9) which was communicated to us by V. G. Drinfeld.

A.2. The algebra Y . Deﬁne an auxiliary algebra Y to be the unital, associative C[~]–algebra generated by ξi,r, xi,r i I,r N subject to the following relations { } ∈ ∈ (1) For every i, j I and r, s N ∈ ∈ [ξi,r, ξj,s] = 0 (2) For every i, j I and s N ∈ ∈ [ξi,0, xj,s]= diaijxj,s (3) For every i, j I and r, s I ∈ ∈ d a ~ [ξ , x ] [ξ , x ]= i ij (ξ x + x ξ ) i,r+1 j,s − i,r j,s+1 2 i,r j,s j,s i,r 0 We denote by Y Y the commutative subalgebra generated by ξi,r i I,r N >0 ⊂ { } ∈ ∈ and by Y the subalgebra of Y generated by xi,r i I,r N. The latter is a free { } ∈ ∈ 0 >0 C[~]–algebra over this set of generators. Moreover, by Corollary 2.5, Y = Y Y . ∼ ⊗ YANGIANSANDQUANTUMLOOPALGEBRAS 51

A.3. The operators σi,(k) and σj. The algebra Y has a grading by the root lattice Q given by deg(ξi,r)=0 and deg(xi,r)= αi Fix henceforth i = j I, set m = 1 a and let Y be the homogeneous 6 ∈ − ij mαi+αj component of Y of degree mαi + αj. 0 >0 Deﬁne operators σ , σ on Y as follows. Since Y = Y Y j i,(k) mαi+αj mαi+αj ∼ mαi+αj >0 ⊗ and Y is free, we have m >0 0 m s s Y = Y Y (i)⊗ − Y (j) Y (i)⊗ mαi+αj ∼ ⊗ ⊗ ⊗ s=0 M >0 where, for a = i, j, Y (a) = Y α is spanned by xa,r r N. Let σa denote the C[~]– a { } ∈ linear map on Y (a) given by σa(xa,r) = xa,r+1. For any k = 1,...,m, deﬁne the 0 m s Y –linear operator σi,(k) on Y mαi+αj by letting it act on the summand Y (i)⊗ − s ⊗ Y (j) Y ⊗ (i) as ⊗ 1 k 1 σ 1 m+1 k if k m s σ = ⊗ − ⊗ i ⊗ ⊗ − ≤ − i,(k) 1 k σ 1 m k otherwise ⊗ ⊗ i ⊗ ⊗ − m s s m s Similarly, let σ End 0 (Y ) be given by 1 σ 1 on Y (i) j ∈ Y mαi+αj ⊗ − ⊗ j ⊗ ⊗ ⊗ − ⊗ Y (j) Y (i) s. ⊗ ⊗ A.4. The projection pij. Let p : Y Y~(g) be the algebra homomorphism ob- → tained by sending ξ ξ and x x+ for every a I and r N, and let a,r 7−→ a,r a,r 7−→ a,r ∈ ∈ pij be the restriction of p to Y mαi+αj . The following holds by Proposition 2.8.

Lemma. The kernel of pij is the C[~]–linear span of the following elements 0 (1) For any 0 s m 1 and A(u , . . . , u , w) Y [u , . . . , u , w] ≤ ≤ − 1 m ∈ 1 m m s s A(σi,(1),..., σi,(m), σj) (σi,(m s) σj a~)xi,0− xj,0xi,0 − − − m s 1 s+1 (σi,(m s) σj + a~)xi,0− − xj,0xi,0 − − − where a = diaij/2. (2) For any 0 s m, k 1,...,m 1 m s and A(u1, . . . , um, w) 0 ≤ ≤(kk+1) ∈ { − } \ { − } ∈ Y [u1, . . . , um, w] m s s A(σ ,..., σ , σ )(σ σ d ~)x − x x i,(1) i,(m) j i,(k) − i,(k+1) − i i,0 j,0 i,0 0 (3) For every A(u , . . . , u , w) Y [u , . . . , u , w]Sm 1 m ∈ 1 m m s m m s s A(σi,(1),..., σi,(m), σj) ( 1) xi,0− xj,0xi,0 − s ! Xs=0 0 Corollary. Let X Ker(p ) and A(u , . . . , u , w) Y [u , . . . , u , w]Sm . Then, ∈ ij 1 m ∈ 1 m A(σ ,..., σ , σ )X Ker(p ) i,(1) i,(m) j ∈ ij 52 S.GAUTAMANDV.TOLEDANOLAREDO

A.5. Reduction step. Let q (k, l), q denote the elements of Y deﬁned by Sij Sij mαi+αj the same expressions as (A.1)–(A.2). Then,

q q (k, l)= ekπ(1)σi,(1) ekπ(m)σi,(m) elσj Sij · · · Sij π Sm ! X∈ Using Corollary A.4, we obtain the following

Lemma. q = 0 implies q (k, l) = 0 for every k , , k , l Z. Sij Sij 1 · · · m ∈ [ A.6. By a finite–dimensional representation of Y~(g) we shall mean a finitely– generated topologically free C[[~]]–module endowed with a C[[~]]–linear action of [ Y~(g). [ q Lemma. Let be a finite–dimensional representation of Y~(g). Then, acts by V Sij zero on . V [ Proof. Let be the subalgebra of Y~(g) generated by Ui = Φ(E ) = Φ(F ) = Φ(H ) Ei i,0 Fi i,0 Hi i,0 By Lemma 3.5, , , satisfy the defining relations of the quantum group {Ei Fi Hi} U~i sl2, where ~i = di~/2. We use the following notation of q–adjoint operator (see [19, 4.18]) which gives a representation of on any algebra containing it § Ui 1 ad ( )(X)= X X − q Ei Ei − Ki Ki Ei ad ( )(X)= X X q Fi Fi Ki − FiKi ad ( )(X) = [ , X] q Hi Hi

i ~i i [ where = qH = e . Let ρ : Y~(g) End( ) be the representation. Then, Ki i H → V ad (ρ( ))ρ( ) = 0 q Fi Ej ad (ρ( ))ρ( )= a ρ( ) q Hi Ej ij Ej where the ﬁrst identity follows from Lemma 3.5. Thus, as a i–module, End( ) contains the lowest weight vector ρ( ) of weight a . By the representationU theoryV Ej ij of U~i sl2, we get ad (ρ( ))mρ( )= ρ (ad ( )m ) = 0 q Ei Ej q Ei Ej and the assertion follows from the well–known identity (see [19, Lemma 4.18])

m m s m m s s ad ( ) = ( 1) − q Ei Ej − s Ei EjEi s=0 qi X YANGIANSANDQUANTUMLOOPALGEBRAS 53

A.7. Let I~ Y~(g) be the ideal defined by ⊂ I~ = Ker(ρ) ( ,ρ) V\ where runs over all finite–dimensional graded modules over Y~(g), that is finitely– V generated torsion–free C[~]–modules admitting a C[~]–linear action ρ : Y~(g) → End( ) and a Z–grading compatible with that on Y~(g). V q Lemma. I~ = 0 implies = 0. Sij Proof. The action of Y~(g) on any finite–dimensional graded module extends to [ V one of Y~(g) on the completion of with respect to its grading. By Lemma A.6, q V V q ij acts by 0 on and therefore so do its homogeneous components ij;n Y~(g), S V q b q S ∈ n 0 on . Thus, ij;n I~ for any n and ij = 0. ≥ V b S ∈ S A.8. The following result, and its proof are due to V. Drinfeld [12]

Proposition. The ideal I~ Y~(g) is trivial. ⊂ Proof. It suﬃces to show that I = I~/~I~ is trivial. Indeed, if I~ = ~I~, then k k I~ = ~ I~ ~ Y~(g) = 0. By deﬁnition of I~, I~ ~Y~(g) = ~I~ so that I k ⊂ k ∩ embeds into Y~(g)/~Y~(g)= U(g[s]). Since graded representations are stable under T T tensor product, I~ is a Hopf ideal of Y~(g), that is

∆(I~) Y~(g) I~ + I~ Y~(g) ⊂ ⊗ ⊗ It follows that I is a co–Poisson Hopf ideal of U(g[s]). By Corollary A.9 below, any such ideal is either trivial or equal to U(g[s]). Since Y~(g) possesses non–trivial ﬁnite–dimensional graded representations, for example the action on C[~] given by the counit, I~ is a proper ideal of Y~(g) and is therefore equal to zero.

A.9. Recall that a co–Poisson Hopf algebra A is a Hopf algebra together with a Poisson cobracket δ : A A A satisfying the following compatibility condition (see [5, 6.2] for details): → ∧ § δ(xy)= δ(x)∆(y)+∆(x)δ(y) For a Lie algebra a, there is a one to one correspondence between co–Poisson structures on Ua and Lie bialgebra structures on a [5, Proposition 6.2.3]. Moreover, there is a one to one correspondence between co–Poisson Hopf ideals of Ua and Lie bialgebra ideals of a. The Lie bialgebra structure on g[s] is given by δ : g[s] g[s] g[s] = (g g)[s,t] → ⊗ ∼ ⊗ Ω δ(f)(s,t) = (ad(f(s)) 1 + 1 ad(f(t))) (A.3) ⊗ ⊗ s t − where Ω g g is the Casimir tensor. Note that δ lowers the degree by 1. ∈ ⊗ 54 S.GAUTAMANDV.TOLEDANOLAREDO

Let a g[s] be the Lie bialgebra ideal corresponding to co–Poisson Hopf ideal I U(g[s⊂]). By the discussion given in previous paragraph, ⊂ δ(a) a g[s]+ g[s] a (A.4) ⊂ ⊗ ⊗ Lemma. Let a g[s] be an ideal. Then a is of the form a = g gC[s] for some polynomial g C⊂[s]. ⊗ ∈ Proof. Let S C[s] be the set of all polynomials f such that there exists some non zero x g ⊂for which x f a. We claim that S is an ideal of C[s]. Let f S and g C[s∈]. Let 0 = x ⊗g be∈ such that x f a, and choose y g such that∈ [x,y] =∈ 0. Then 6 ∈ ⊗ ∈ ∈ 6 [x,y] fg = [x f,y g] a ⊗ ⊗ ⊗ ∈ and hence fg S. Now for any∈f S, the set x g : x f a is an ideal in g, which is non–zero and hence equal to∈ g. This proves{ ∈ that⊗a =∈g } S. Since C[s] is a principal ideal domain, the lemma is proved. ⊗

Corollary. Let a g[s] be a Lie bialgebra ideal. Then either a = 0 or a = g[s]. ⊂ Proof. Let g C[s] be such that a = g (g) g[s]. By (A.3), we know that the Lie cobracket δ∈lowers the degree by 1. Using⊗ (A.4⊂ ), we conclude that g is a constant polynomial.

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Mathematics Department, Columbia University, 2990 Broadway, New York, NY 10027

Department of Mathematics, Northeastern University, 360 Huntington Avenue, Boston, MA 02115. E-mail address: [email protected]

Department of Mathematics, Northeastern University, 360 Huntington Avenue, Boston MA 02115. E-mail address: [email protected]