Cohomological Hall algebras and affine quantum groups

COHOMOLOGICAL HALL ALGEBRAS AND AFFINE QUANTUM GROUPS

arXiv:1604.01865v1 [math.RT] 7 Apr 2016

YAPING YANG AND GUFANG ZHAO Abstract. We study the preprojective cohomological Hall algebra (CoHA) introduced by the authors in [YZ1] for any quiver Q and any one-parameter formal group G. In this paper, we construct a comultiplication on the CoHA, making it a bialgebra. We also construct the Drinfeld double of the CoHA. The Drinfeld double is a quantum affine algebra of the Lie algebra gQ associated to Q, whose quantization comes from the formal group G. We prove, when the group G is the additive group, the Drinfeld double of the CoHA is isomorphic to the Yangian.

0. Introduction In this paper, we construct comultiplications on certain cohomological Hall algebras, making them bialgebras. We also construct the Drinfeld double of these bialgebras. This gives a uniform way to construct both old and new affine-type quantum groups. The cohomological Hall algebra involved, called the preprojective cohomological Hall algebra (CoHA for short) and denoted by P(Q, A) or simply P, is associated to a quiver Q and an algebraic oriented cohomology theory A. The construction of the preprojective CoHA is given in [YZ1], as a generalization of the K-theoretic Hall algebra of commuting varieties studied by SchiffmannVasserot in [SV12]. The preprojective CoHA is defined to be the A-homology of the moduli of representations of the preprojective algebra of Q. It has the same flavor as the cohomological Hall algebra associated to Jacobian algebra defined by Kontsevich-Soibelman [KoSo11]. We also construct an action of P(Q, A) on the A-homology of Nakajima quiver varieties associated to Q. Let gQ be the corresponding symmetric Kac-Moody Lie algebra of Q and bQ ⊂ gQ be the Borel subalgebra. As is shown in [YZ1], a certain spherical subalgebra in an extension of P(Q, A), denoted by P s,e (Q, A) or P s,e , is a quantization of U (bQ [[u]]), where the quantization depends on the underlying formal group law of A. As in the case of quantized enveloping algebra of gQ , Drinfeld double of the Borel subalgebra should be the entire quantum group. This is the subject of the present paper. We construct a coproduct on P(Q, A), and define the Drinfeld double, which is a quantization of U (gQ [[u]]). Assume AGm (pt) ∼ = O(G) for some 1-dimensional algebraic group or formal group. When G is an affine algebraic group, the Drinfeld double of P(Q, A) recovers the Drinfeld realization of the quantization of Manin triple associated to P1 as described in [D86, § 4]. When G is a formal group which does not come from an algebraic group, this gives new affine quantum groups which have not been studied in literature. In the case when G is the elliptic curve, the method here gives a Drinfeld realization of the elliptic quantum group of [Fed94] which is previously unknown. In this paper, we focus on the purely algebraic description of P(Q, A) in terms of shuffle algebra without explicit reference to Nakajima quiver varieties. The shuffle algebra is reviewed in details in § 1. The algebra P s,e has two deformation parameters t1 , t2 , coming from a 2-dimensional torus Date: April 8, 2016. 2010 Mathematics Subject Classification. Primary 17B37; Secondary 14F43, 55N22. Key words and phrases. Quantum group, shuffle algebra, Hall algebra, Yangian, Drinfeld double. 1

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Y. YANG AND G. ZHAO

action. Let P s,e be the quotient of P s,e by the torsion part over the (Gm )2 -equivariant parameters t1 , t2 . In § 2 of the present paper, we prove the following. b s,e , making P s,e a bialgebra. Theorem A. There is a comultiplication ∆ : P s,e → P s,e ⊗P

b s,e is explained in § 2. Moreover, this comultiplication on the comHere the completion P s,e ⊗P pletion comes from expansion of certain rational morphism, the formula of which is in § 2. In § 3, we construct the Drinfeld double of the bialgebra P s,e , denoted by D(P s,e ), which is a quantization of U (gQ [[u]]) (in a sense explained in Remark 1.8), when G is an algebraic group over C endowed with a one form ω. Since P s,e is infinite-dimensional, we need to define a non-degenerate bialgebra pairing in order to define the Drinfeld double. The natural pairing is a residue pairing on a certain adèle version of P s,e , which is also a bialgebra with multiplication and comultiplication given by the same formulas as those on P s,e , the precise definition of which can be found in § 3.1. In § 4, we study in details this Drinfeld double D(P s,e ) in the example when the group G is additive. A quotient of D(P s,e ) is called the reduced Drinfeld double (defined in § 4.1), denoted by D(P s,e ). Theorem B. Assume Q has no edge-loops. There is an isomorphism as bialgebras of D(P s,e )|t1 =t2 =~/2 with the Yangian Y~ (gQ ) endowed with the Drinfeld comultiplication. As far as we know, this is the first construction of the Yangian as a Drinfeld double of the Borel subalgebra Y~≥0 (gQ ). The double Yangian, which is the Drinfeld double of Yangian, can also be realized in a similar way. (See Remark 4.4 for details.) Historically, there are a number of attempts to construct quantum groups using shuffle algebras. Rosso [R98] constructed the quantized enveloping algebra Uq (gQ ) as a subalgebra in a suitable a shuffle algebra, as an intrinsic description of Uq (gQ ). This description turns out to be useful in the study of q-characters of quiver Hecke algebras [Lec04]. In a seminal note [Gr94b], Grojnowski outlined the idea of constructing the positive part of the quantum loop algebra in terms of certain Ktheoretic Hall algebra, as well as the relation with Lusztig’s construction of the quantized enveloping algebra using perverse sheaves [L91]. As mentioned in [Gr94b], Feigin in an unpublished work also has a construction of quantum groups in terms of symmetric polynomial ring. In [Ne14] a K-theoretic shuffle algebra associated to the Jordan quiver has been studied. In particular, a comultiplication on this shuffle algebra has been constructed, making it a bialgebra. In the present paper, we work in the generality that G is any 1-dimensional affine algebraic group or 1-parameter formal group. Similar results hold when G is not necessarily affine. In [YZ2], we study in details the case when G is the elliptic curve. We construct the sheafified shuffle algebra. The corresponding cohomology is the equivariant elliptic cohomology of [Gr94a, GKV95]. When G is the universal elliptic curve over the (open) moduli space M1,2 of genus 1 curves with 2 marked points, suitable rational sections of this algebra object coincides with Felder’s elliptic quantum group [Fed94, GTL15]. The precise statements and proofs will be published in [YZ2]. 1. The shuffle algebra In this section, we recall the algebraic description of the cohomological Hall algebra P(Q, A) in terms of the shuffle algebra. The definition and the multiplication formula of shuffle algebra is recalled in § 1.2. Relation with the cohomological Hall algebra P(Q, A) is recalled in § 1.3. 1.1. Formal group algebras. We first fix the notations and general setup. Let R be a commutative ring of characteristic zero. Let G be either a 1-dimensional affine algebraic group or a 1-dimensional formal group over Spec R. The coordinate ring of G is denoted by S, in particular,


3

S is an R–algebra. Let l ∈ S be a local uniformizer of G at the identity section. Note that if G is an algebraic group, the expansion of the group multiplication of G with respect to the local uniformizer l gives a formal group law. Let Sn be the symmetric group of n letters, and let Λn be a lattice of rank-n, with a natural Sn -action. The coordinate ring of G ⊗Z Λn is denoted by Sn . When G is an algebraic group, Sn b is isomorphic to S⊗n ; and Sn ∼ = S⊗n if G is a formal group. The coordinate ring of G ⊗Z Λn /Sn (n) (n) is denoted by S . Similarly, S is isomorphic to the ring of invariants (S⊗n )Sn when G is an b Sn ) if G is a formal group. algebraic group, and (S⊗n For any group homomorphism λ : Λn → Z, let χλ : G ⊗Z Λn → G be the induced morphism. For the local uniformizer l ∈ S, we denote l(λ) ∈ Sn the pullback of l along χλ . Choose a natural coordinate (x1 , · · · , xn ) of Λn , such that the action of Sn is permuting the coordinate (x1 , · · · , xn ). We have a group homomorphism λij : Λn → Z, defined as (x1 , · · · , xn ) 7→ xi − xj . Let l(xi − xj ) ∈ Sn be the function on Gn associated to this group homomorphism. Similarly the function l(xi − xj − t1 − t2 ) ∈ Sn+2 is the function associated to the group homomorphism λij,t1 ,t2 : Λn ⊕ Z2 → Z given by (x1 , · · · , xn , t1 , t2 ) 7→ xi − xj − t1 − t2 . 1.2. Shuffle algebra associated to a quiver. Let Q be a quiver with vertex set I and arrow set H. We assume in this paper that I and H are finite sets. A dimension vector v = (v i )i∈I ∈ NI of Q is a collection of natural numbers, one for each vertex. For each quiver Q, and each 1-dimensional affine algebraic group G, we recall the definition of the shuffle algebra associated to Q and G defined in [YZ1], denoted by SHG Q , or simply SH if both Q and G are clear from the context. Let Rt1 ,t2 be the coordinate ring of G ⊗Z Λ2 , where (t1 , t2 ) are the coordinates of Λ2 . The shuffle algebra SH is an NI -graded Rt1 ,t2 -algebra. As an Rt1 ,t2 -module, we have M SH = SHv , v∈NI

Q i where the degree v piece SHv = O(G(v) ) is the coordinate ring of G(v) := G2 × i∈I G(v ) . For two dimension vectors v1 , v2 ∈ NI , we consider SHv1 +v2 as a subalgebra of SHv1 ⊗ SHv2 , where the algebra embedding is given by pulling back functions via the natural projection G(v1 ) × G(v2 ) → (i) G(v1 +v2 ) . The coordinates for G(v) are denoted by (t1 , t2 , (xs )i∈I,s∈[1,vi ] ), where (t1 , t2 ) are the i

(i)

(i)

coordinates for G2 and for each i ∈ I, the coordinates for G(v ) are (x1 , . . . , xvi ). In order to describe the multiplication of SH, we introduce some notations. For any dimension vector v ∈ NI , a partition of v is a pair of collections A = (Ai )i∈I and B = (B i )i∈I , where Ai , B i ⊂ [1, v i ] for any i ∈ I, satisfying the following conditions: Ai ∩ B i = ∅, and Ai ∪ B i = [1, v i ]. We use the notation (A, B) ⊢ v to mean (A, B) is a partition of v. We also write |A| = v if |Ai | = v i for each i ∈ I. For any two dimension vectors v1 , v2 ∈ NI such that v1 + v2 = v, we introduce the notation P(v1 , v2 ) := {(A, B) ⊢ v | |A| = v1 , |B| = v2 }. There is a standard element (Ao , Bo ) in P(v1 , v2 ) with Aio := [1, v1i ], and Boi := [v1i + 1, v1i + v2i ] for any i ∈ I. This standard element will be denoted by ([1, v1 ], [v1 + 1, v]) for short. For each arrow h ∈ H, we denote by in(h) (resp. out(h)) the incoming (resp. outgoing) vertex of h. Let h∗ be the corresponding reversed arrows in the opposite quiver Qop . We associate to (h, h∗ ) two integers, mh and mh∗ ∈ N.

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Assumption 1.1. We consider specializations of t1 and t2 which are compatible with the integers m(h) m(h∗ ) m(h), m(h∗ ) for any h ∈ H, in the sense that t1 t2 is a constant, i.e., does not depend on h ∈ H. Remark 1.2. Two examples of the integers m(h), m(h∗ ) satisfying Assumption 1.1 are the following. (1) Let t1 and t2 are independent variables, but m(h) = m(h∗ ) = 1 for any h ∈ H. (2) Specialize t1 = t2 = ~/2. For any pair of vertices i and j with arrows h1 , . . . , ha from i to j, the pairs of integers are mhp = a + 2 − 2p and mh∗p = −a + 2p. Consider the following two elements in the localization of O(G(v1 ) ×G2 G(v2 ) ). Recall that the ring O(G(v1 ) ×G2 G(v2 ) ) is isomorphic to S(v1 ) ⊗Rt1 ,t2 S(v2 ) if G is an algebraic group, and to b Rt ,t S(v2 ) if G is a formal group. Define S(v1 ) ⊗ 1 2 fac1 (xA |xB ) :=

Y Y Y l(xs(i) − x(i) + t1 + t2 ) t

, (i) (i) l(xt − xs ) Y in(h) − xout(h) + mh t1 ) l(xt s

i∈I s∈Ai t∈B i

fac2 (xA |xB ) :=

Y

h∈H

Y

s∈Aout(h) t∈B in(h)

Y

Y

out(h)

l(xt

s∈Ain(h) t∈B out(h)

∗ t ) − xin(h) + m h 2 . s

When mh = mh∗ = 1, the fac2 (xA |xB ) simplifies as Y Y Y (j) (j) (i) aji aij fac2 (xA |xB ) = l(xt − x(i) s + t1 ) l(xt − xs + t2 ) , i,j∈I s∈Ai t∈B j

where aij is the number of arrows from i to j of quiver Q. Let A := #{h ∈ H op | in(h) = k, out(h) = l} be the adjacency matrix of the quiver Q, and let C := I − A. The multiplication of f1 (xAo ) ∈ SHv1 and f2 (xBo ) ∈ SHv2 is defined to be 1 X f1 (xAo ) ⋆ f2 (xBo ) = (−1)(v2 ,Cv1 ) σ f1 (xAo ) · f2 (xBo ) · fac(xAo |xBo ) σ∈Sh(v1 ,v2 )

(1)

=

X

(−1)(v2 ,Cv1 ) f1 (xA ) · f2 (xB ) · fac(xA |xB ),

(A,B)∈P(v1 ,v2 )

where fac(xA |xB ) := fac1 (xA |xB ) · fac2 (xA |xB ) is an element in the localization of S(v1 ) ⊗ S(v2 ) = O(G(v1 ) × G(v2 ) ). Note that although fac(xA |xB ) has a simple pole, the product f1 (xAo ) ⋆ f2 (xBo ) is well-defined in S(v1 +v2 ) . For each k ∈ I, let ek be the dimension vector valued 1 at vertex k and zero otherwise. We define G the spherical subalgebra SHG,s Q to be the subalgebra of SHQ generated by SHek as k varies in I. Example 1.3. When G is a formal group whose coordinate ring is S = R[[x]]. The shuffle algebra can be identified as M Sv G SHQ = R[[t1 , t2 ]][[x(i) s ]]i∈I,s=1,...,vi , v∈NI

with the same multiplication formula (1). 1There is a discrepancy between the notation here and [YZ1]. The algebra denoted by SH in the present paper,

g in i.e., the one whose multiplication has a sign (−1)(v2 ,Cv1 ) coming from the Euler-Ringel form, is denoted by SH [YZ1]. We adapt this convention since the one without sign is not used in the present paper.


5

G defined above has a geometric inter1.3. Geometric interpretation. The shuffle algebra SHQ pretation. We consider equivariant oriented cohomology theory A in the sense of [CZZ14, § 2], [ZZ14, § 5.1], and [YZ1, § 1.2], which is an extension of the algebraic oriented cohomology [LM07] to the equivariant setting. Examples of which includes [HM13, Kr12, To99, Th93]. Assume there is an equivariant oriented cohomology theory A such that AGm (pt) ∼ = S. i} Fix V = {V an I-tuple of k-vector spaces with dimension vector dim(V i ) = v i . Let i∈I L out(h) in(h) Rep(Q, v) := h∈H Hom(V ,V ) be the representation space of Q with dimension vector v Q ∗ and let T Rep(Q, v) be the cotangent bundle of Rep(Q, v). The algebraic group Gv := i∈I GL(v i ) acts on Rep(Q, v) and T ∗ Rep(Q, v) by conjugation. Let a be the number of arrows in Q from vertex i to j. We fix a numbering h1 , . . . , ha of the arrows in Q. The corresponding reversed arrows in Qop are labelled by h∗1 , · · · , h∗a . For hp ∈ H, and B = (Bp ) ∈ Hom(V i , V j ), B ∗ = (Bp∗ ) ∈ Hom(V j , V i ), Define the T = G2m –action by: mhp

t1 · Bp := t1 As abelian groups, we have G SHQ =

M

mh∗p

Bp , t2 · Bp∗ := t2

Bp∗ .

AGv ×(Gm )2 (T ∗ Rep(Q, v)).

v∈NI

There is a geometrically defined multiplication mSv1 ,v2 on AGv ×T (T ∗ Rep(Q, v)) in [YZ1], such that, it coincides with the algebraically defined multiplication (1). Remark 1.4. If R contains Q, then any formal group G is isomorphic to the additive formal group Ga . However, this does not imply that (SHG , mG ) ∼ = (SHGa , mGa ), where mG is the multiplication G of SH . Instead, there is an R-module isomorphism ch : SHG → SHGa , which has the following Ga a property. For any v1 , v2 ∈ NI , there exists an element Tdv1 ,v2 ∈ SHG v1 ⊗ SHv2 , such that for any Ga Ga f ⊗ g ∈ SHv1 ⊗ SHv2 , we have mGa (ch(f ⊗ g) · Tdv1 ,v2 ) = ch(mG (f ⊗ g)). The element Tdv1 ,v2 can be chosen as Tdv1 ,v2 =

facG v1 ,v2

a facG v1 ,v2

.

L 1.4. The extended shuffle algebra. Let SH0 := SymR ( i∈I S(ei ) ) be the symmetric algebra on L L Q (ei ) . Here (ei ) is a vector subspace in the coordinate ring of hG := i∈I S i∈I S i∈I G. In this subsection, we define the extended shuffle algebra using the SH0 –action on SH. Q i For any (A, B) ⊢ v, let GA be G2 × i∈I G(|A |) and similarly we have GB . There is a natural projection GA × GB → G(v) . A function P on G(v) will be denoted by P (xA∪B ). Its pullback via the projection GA ×G2 GB → G(v) is denoted by P (xA ⊗ xB ). For any (A, B) ⊢ v, we introduce the following rational function on GA ×G2 GB (2)

b B |zA ) := fac(zB |zA ) . Φ(z fac(zA |zB )

b B |zA ) has a simple formula, which appears in [YZ1]. In the following special case, the function Φ(z For k ∈ I, we consider a special (A, B) ⊢ v with B k = {1}, Ak = [2, v k ], and B i = ∅, Ai = [1, v i ] b k (z, v). Recall that aik is the b B |zA ) = fac(zek |zA ) will be denoted by Φ for i 6= k. In this case Φ(z fac(zA |zek ) number of arrows in H from i to k. Let cik be the (i, k)–entry of the Cartan matrix of the quiver

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Y. YANG AND G. ZHAO

b k (z, v) can be Q. That is cik = −aik − aki if k 6= i, and 2 if k = i. If l is an odd function, then Φ simplified as follows.  Q Qvi l(z−xj(i) −t1 )aik l(z−xj(i) −t2 )aki Qvk l(z−x(k)  j +t1 +t2 )   · j=1 , (i) (k) (i) j=1 i∈I\{k}  a a  l(z−xj +t2 ) ik l(z−xj +t1 ) ki l(z−xj −t1 −t2 ) b k (z, v) = Φ when m(h) = m(h∗ ) = 1 as in Remark 1.2(1);  (i) ~  Qvi l(xt −z−(cki ) 2 ) k Q   , when m(h) and m(h∗ ) are as in Remark 1.2(2).  (−1)v (i) t=1 i∈I ~ l(z−xt −(cki ) 2 )

We define a SH0 action on SH as follows. For k ∈ I, consider the rational function Hk (w) = 1 on G(ek ) × G ⊆ hG × G, where h(k) is the coordinate of G(ek ) and w is the coordinate l(h(k) )−l(w) of the second G-factor. Expanding Hk (w) with respect of the local uniformizer l(w) of the second G-factor, we can consider Hk (w) as an element in the completion S(ek ) [[w]], where S(ek ) = O(G(ek ) ). For any g ∈ SHv , and any k ∈ I, we define ck (z, v). Hk (z)gHk (z)−1 := gΦ

(1) The action of SH0 on SH is well-defined. Lemma 1.5 ([YZ1]). (2) Furthermore, the action of SH0 on SH is by algebra homomorphisms. Definition 1.6. Define the extended shuffle algebra associated to G and Q to be the algebra SHe = s,e SH0 ⋉ SHG = SH0 ⋉ SHG,s Q . The spherical subalgebra in the extended shuffle algebra is SH Q . The following is the relation between the shuffle algebra and the CoHA, which is a consequence of [YZ1, Theorem C]. Proposition 1.7. There is an algebra homomorphism from the extended CoHA P e (Q, A) to the extended shuffle algebra SHe , which induces an isomorphism P s,e (Q, A) ∼ = SHs,e where P s,e (Q, A) is the quotient of P s,e (Q, A) by the torsion part over Rt1 ,t2 . Remark 1.8. Assume Q is a quiver without edge-loops. For simplicity, we take the formal completion of G in the case when G is an algebraic group. Assume G comes from an oriented cohomology theory. Then there is a natural cohomological grading on the base ring R so that the natural classifying map from the Lazard ring to R is homogeneous of degree zero. Let R be the image of the Lazard ring in R, which is non-negatively graded. There is an augmentation map to the zeroth degree piece R → R0 . For example, when G is multiplicative, the cohomology theory is the K-theory, in which case R = Z[β ± ] where β is the Bott periodicity; The image of the Lazard ring is Z[β −1 ], which has an augmentation map Z[β −1 ] → Z sending β −1 to 0. On the quotient, the image of the formal group law is additive. Therefore, on this special fiber of Spec R, the algebra SHs,e |t1 =t2 =0 becomes U (b[[u]]) as an algebra, by [YZ1, Theorem D], where b ⊂ g is the Borel subalgebra. In this sense, we say SHs,e is a quantization of U (b[[u]]). 2. The comultiplication In this section, we construct the map ∆ on a suitable localization of the extended shuffle algebra X b SHev2 )loc . ∆ : SHev → (SHev1 ⊗ v1 +v2 =v

Here loc means the localization away from the union of null-divisors of fac(xA , xB ) over all (A, B) ⊢ v. The ideal of this divisor in SHv is denoted by Ifac .


7

On the symmetric algebra SH0 , the map ∆ is given by ∆(Hk (w)) = Hk (w) ⊗ Hk (w), k ∈ I. This is understood in the following way. Recall that Hk (w) = G(ek )

1 l(h(k) )−l(w)

is a rational function on

×G ⊆ × G. The function Hk (w) ⊗ Hk (w) is the rational function on G(ek ) × G(ek ) × G ⊆ hG × hG × G defined as the product p∗1 (Hk (w)) · p∗2 (Hk (w)), where pi : G(ek ) × G(ek ) × G → G(ek ) × G is the natural projection contracting the i-th G(ek ) -factor, for i = 1, 2 and p∗i is the pullback along the projection pi . Here w is the coordinate of the last G-factor. Expanding with respect to l(w), b SH0 [[w]]. we have Hk (w) ⊗ Hk (w) is a well-defined element in the completion SH0 ⊗ The ∆ on SH is given by X H[1,v1 ] (x[v1 +1,v] )P (x[1,v1 ] ⊗ x[v1 +1,v] ) , (3) ∆(P (x)) = (−1)(v2 ,Cv1 ) fac(x[v1 +1,v] |x[1,v1 ] ) hG

{v1 +v2 =v}

which is a rational function on (hG × G(v1 ) ) × (G(v2 ) ). Here HA (xB ) := (k)

Q

k∈I

Q

(k)

j∈B k

Hk (xj ). The

function Hk (xj ) is defined by pulling back the function Hk (w) of G(ek ) × G along the identification G(ek ) × Gj ∼ = G(ek ) × G. Note that in the inclusion G(ek ) × Gj ⊂ (hG × G(v1 ) ) × (G(v2 ) ), the G(ek ) (k) factor lies in the hG -factor, and the Gj -factor lies in the G(v2 ) -factor. Expending each Hk (xj ) (k)

with respect to l(xj ), we get H[1,v1 ] (x[v1 +1,v] ) as a well-defined element in the completion X b SHev . SHev1 ⊗ 2 v1 +v2 =v

In particular, when v = ek for some k ∈ I, either A or B is empty, and P (xA∪B ) = f (xk ) a function with only one variable x(k) . We have

(4)

∆(f (x(k) )) = Hk (x(k) ) ⊗ f (x(k) ) + f (x(k) ) ⊗ 1.

Theorem 2.1. The map ∆ defined above satisfies the following: (1) (idSH ⊗ ∆) ◦ ∆ = (∆ ⊗ idSH ) ◦ ∆; (2) ∆(P ⋆ Q) = ∆(P ) ⋆ ∆(Q); (3) Let ǫ : SH → R be the map by ǫ(SH0 ) = R, id : SH0 = R, and SHv 7→ 0 for v 6= 0. Then, (idSH ⊗ ǫ) ◦ ∆ = idSH = (ǫ ⊗ idSH ) ◦ ∆. Proof. Proof of (1): For any P ∈ SH, for any A, B, C, the projection of (id ⊗∆)(∆(P )) into the component SHA ⊗ SHB ⊗ SHC is denoted by (id ⊗∆)(∆(P ))A,B,C . Similarly, we have (∆ ⊗ id)(∆(P ))A,B,C . With this notation, (1) becomes (id ⊗∆)(∆(P ))A,B,C = (∆ ⊗ id)(∆(P ))A,B,C . Similarly, the projection of ∆(P ) into the component SHA ⊗ SHB∪C is denoted by ∆(P )A,B∪C . It is clear that we only need to show (1) after dropping the sign (−1)(v2 ,Cv1 ) in ∆. Applying id ⊗∆ to ∆(P )A,B∪C , in SHA ⊗ SHB ⊗ SHC , we have HA (zB∪C )P (zA ⊗ zB∪C ) (id ⊗∆)(∆(P )A,B∪C ) =(id ⊗∆) fac(zB∪C |zA ) A,B,C =

HA (zB ) ⊗ HA (zC ) ⊗ HB (zC )P (zA ⊗ zB ⊗ zC ) . fac(zB∪C |zA ) fac(zC |zB )

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Y. YANG AND G. ZHAO

Applying ∆ ⊗ id to ∆(P )A∪B,C , in SHA ⊗ SHB ⊗ SHC , we have HA∪B (zC )P (zA∪B ⊗ zC ) (∆ ⊗ id)(∆(P )A∪BC ) =(∆ ⊗ id) fac(zC |zA∪B ) A,B,C =

HA (zC )HB (zC )HA (zB )P (zA ⊗ zB ⊗ zC ) fac(zC |zA∪B ) fac(zB |zA )

Therefore, we have (idSH ⊗ ∆) ◦ ∆ = (∆ ⊗ idSH ) ◦ ∆, since fac(zB∪C |zA ) fac(zC |zB ) = fac(zC |zA∪B ) fac(zB |zA ). Proof of (2): Note that we have the same sign (−1)(v2 ,Cv1 ) in both multiplication ⋆ (1) and comutiplication ∆ in (3) of SH. To show (2), it is clear that we could drop the sign (−1)(v2 ,Cv1 ) in both ⋆ and ∆. We need the following notations. For a pair of dimension vector (v1′ , v2′ ), with v1′ + v2′ = v, for (A, B) ∈ P(v1 , v2 ), we write A1 := A ∩ [1, v1′ ], A2 := A ∩ [v1′ + 1, v1′ + v2′ ], B1 := B ∩ [1, v1′ ], B2 := B ∩ [v1′ + 1, v1′ + v2′ ]. Then we have =

∆(P ⋆ Q) X

∆(P (zA ) · Q(zB ) · fac(zA , zB ))

(A,B)∈P(v1 ,v2 )

=

X

X

(A,B)∈P(v1 ,v2 ) v1′ +v2′ =v

=

X

X

v1′ +v2′ =v (A,B)∈P(v1 ,v2 )

HA1 (zA2 )HB1 (zB2 )P (zA1 ⊗ zA2 )Q(zB1 ⊗ zB2 ) fac(zA |zB ) fac(z[v1′ +1,v1′ +v2′ ] |z[1,v1′ ] ) HA1 (zA2 )P (zA1 ⊗ zA2 )HB1 (zB2 )Q(zB1 ⊗ zB2 ) b Φ(zB2 |zA1 ) fac(zA1 ⊔A2 |zB1 ⊔B2 ). fac(zA2 ⊔B2 |zA1 ⊔B1 )

b B |zA ). The last equality is obtained from the identity HB1 (zB2 )P (zA1 ) = P (zA1 )HB1 (zB2 )Φ(z 2 1 fac(z |z ) B A 1 2 b Recall that by definition Φ(zB2 |zA1 ) = fac(zA |zB ) . Plugging the equality 1

2

b B |zA ) fac(zA ⊔A |zB ⊔B ) Φ(z fac(zA1 |zB1 ) fac(zA2 |zB2 ) 2 1 1 2 1 2 = fac(zA2 ⊔B2 |zA1 ⊔B1 ) fac(zA2 |zA1 ) fac(zB2 |zB1 )

into the formula of ∆(P ⋆ Q), we have X X HA1 (zA2 )P (zA1 ⊗ zA2 ) HB1 (zB2 )Q(zB1 ⊗ zB2 ) fac(zA1 |zB1 ) fac(zA2 |zB2 ) ∆(P ⋆ Q) = fac(zA2 |zA1 ) fac(zB2 |zB1 ) ′ ′ v1 +v2 =v (A,B)∈P(v1 ,v2 )

= ∆(P ) ⋆ ∆(Q). Proof of (3): We have (ǫ ⊗ idSH ) ◦ ∆(P ) = (ǫ ⊗ idSH ) ◦ ∆(P )0,v = (ǫ ⊗ id)(H0 (x[1,v] ) ⊗ P ) = P. Similarly, (idSH ⊗ ǫ) ◦ ∆(P ) = P. This completes the proof.

Note that formula (4) of ∆(f ) has no denominators, for f ∈ SHek . In particular, ∆(SHek ) ⊂ b SHe . Consequently, by Theorem 2.1, we have SHe ⊗ X b SHev . ∆(SHsv ) ⊂ SHev1 ⊗ 2 v1 +v2 =v


9

In other words, the map ∆ is well-defined on the spherical subalgebra SHs,e without passing to localization. Therefore, we get the following direct corollary to Theorem 2.1. It implies Theorem A, taking Proposition 1.7 into account. b SHs,e is a well-defined algebra homomorphism Corollary 2.2. The map ∆ : SHs,e → SHs,e ⊗ s,e which is coassociative, making SH a bialgebra with counit SHs,e → R being the restriction of ǫ : SH → R. 3. A bialgebra pairing The aim of this section is to define the Drinfeld double of the extended shuffle algebra SHe . For this purpose, we introduce a bialgebra pairing on SHe . Recall that for a bialgebra (A, ⋆, ∆) with multiplication ⋆, and coproduct ∆, the Drinfeld double of the bialgebra A is DA = A ⊗ Acoop as a vector space endowed with suitable multiplication. Here Acoop is A as an algebra but with the opposite comultiplication. If dim(A) is infinite, in order to define DA as a bialgebra, we need a non-degenerate bialgebra pairing (·, ·) : A ⊗ A → R, such that (a ⋆ b, c) = (a ⊗ b, ∆(c)) and (c, a ⋆ b) = (∆(c), a ⊗ b) for all a, b, c ∈ A. The bialgebra structure of DA is uniquely determined by the following properties (see, e.g., [DJX, § 2.4]): A− = Acoop ⊗ 1 and A+ = 1 ⊗ A are both sub-bialgebras of DA; write ∆(a) = a1 ⊗ a2 , then, (5)

+ + − a− 1 ⋆ b2 · (a2 , b1 ) = b1 ⋆ a2 · (b2 , a1 ), for all a, b ∈ A,

where a− = a ⊗ 1 and b+ = 1 ⊗ b. However, a natural pairing in terms of residue is non-degenerate only on an adèle version of SHe,s, which we introduce here. 3.1. Scalar product on ad` ele ring. In this section we work in the following setup. Take R = C, and G an algebraic group. We fix a smooth compactification G of G and fix a meromorphic section ω of T ∗ G, such that ω is translation-invariant in G and non-vanishing anywhere on G. Denote by F the field of rational functions of G, Ox the Q local ring of G at a closed point x ∈ G, and Kx the local field. Let A be the restricted product ′x∈G Kx , i.e., an element in A is a collection of rational functions {rx }x∈G , so that rx ∈ Ox except for finitely many x. We will call A the ring of adèles.2 We consider a C-valued scalar product on A X Resx (ω · ux · v−x ), (u, v) := x∈G

which is a modification of the one used in [D86, Example 3.4]. Here −x is the group inverse of x extended to G. (n) We consider the following extension of the definition of adèle ring on G for any n ∈ N. For n any x ∈ G, we have (x, . . . , x) ∈ G , the complete local ring at which is denoted by Oxn . Let n−1 ). Note that ω induces an n-form Kxn be Vthe localization of Oxn at the divisor Sn · ({x} × G n n ω n := ni=1 ωi on G , where ωi is the pullback of ω to G via the i-th projection. Note that for any f ∈ Kxn , the residue of f · ω n at (x, . . . , x) is well-defined, denoted by Resx f ω. For any n-form η ∈ H 0 (U − D, Ωn ) in a neighborhood of certain divisor D ⊂ U , see [GH78, p.650] for the definition 2In literature this ring is also called the ring of repartitions or pre-ad` eles, where the completion of this ring is called

the ring of adèles. As we do not need to consider any topology on this ring, we will adapt the present terminology.

10

Y. YANG AND G. ZHAO

of the residue Res(η). Note that in the definition, η could have a higher order pole along the divisor D ⊂ U. Lemma 3.1. For any σ ∈ Sn and f ∈ Kxn , we have Resx f ω = Resx σ(f )ω. R Proof. Recall that by [GH78, p.650], there is a real n-cycle Γ so that Resx f ω = Γ f · ω. For an element σ ∈ Sn , let σ : U → U be the map induced by the Sn action on U . We have Z Z σ(f · ω). f ·ω = Γ

σ(Γ)

The lemma follows from the facts that σ(ω) = sign(σ)ω, and σ(Γ) = Γ with orientation differs by sign(σ). Lemma 3.2. The pairing (·, ·) on Kxn , sending f, g ∈ Kxn to Resx (f · g)ω is non-degenerate. Q n Proof. Let X = ni=1 xi be the function on G , where xi = p∗i (l) is the pullback of the local n uniformizer l of G along the i-th projection pi : G → G. Assume f 6= 0, then there is an integer l l so that X f ω has pole of order 1 along the divisor of X. For any g ∈ Kxn , there is some N large enough, such that X N g is regular. We may only work with regular g. By [GH78, p.659], if Resx X l f gω = 0 for any g, then X l f ω is regular along X, which contradicts with the assumption that X l f ω has pole of order 1. Let An be the restricted product of Kxn over all x ∈ G, and let A(n) = (An )Sn be the Sn -invariant Q i (v) part. Similarly, for any v ∈ NI , we can define the adèle ring A(v) = i∈I A(v ) of G . For any fx . f ∈ A(v) and any x ∈ G, the x-component of f is denoted byL We define the adèle version of the shuffle algebra SHA = v∈NI SHA,v . For any v ∈ NI , SHA,v (v)

is the localization of the adèle ring A(v) on G , localized at Ifac , where Ifac is the ideal of the union of the null-divisors of fac(xA |xB ) over all (A, B) ⊢ v. The action of SH0 on SH induces an action of SH0 on SHA . Therefore, we also have an adèle version of extended shuffle algebra SHeA = SH0 ⋊ SHA . The bialgebra structure on SHe induces a bialgebra structure on SHeA . Let e SHs,e A ⊂ SHA be the spherical subalgebra. Lemma 3.3. The subalgebra SHs,e A is a sub-bialgebra. Proof. Since ∆ is an algebra homomorphism, it suffices to show that ∆(f ) ∈ SHs,e A for any generator f ∈ SHs,e with i ∈ I. The latter follows from the formula of ∆ (4). A,ei Let |A|! = fac(xA ) :=

Q

Y i∈I

·

i∈I

|Ai |! and we introduce the notation.

Y

{s,t∈Ai |s6=t}

Y

h∈H

(i)

(i)

l(xs − xt + t1 + t2 )

Y

· (i) (i) l(xt − xs ) Y in(h) − xout(h) + mh t1 ) l(xt s

s∈Aout(h) t∈Ain(h)

Y

Y

s∈Ain(h) t∈Aout(h)

out(h)

l(xt

− xin(h) + mh ∗ t2 ) s

Lemma 3.4. For any v ∈ NI and any f, g ∈ SHs,e A,v , the vanishing locus of Ifac is not included in the polar divisor of

f ·g fac(xA ) .


11

Proof. This is clear if t1 6= 0 and t2 6= 0. Without loss of generality, we can assume t1 = 0. Then s,e f, g ∈ SHs,e A,v implies that f, g are product of elements in SHA,ei for i ∈ I. By the multiplication formula (1) and the formula of fac(xA ), the vanishing order of f · g is at least the vanishing order of fac(xA ) along Ifac . s,e We define a non-degenerate bialgebra pairing on the adèle shuffle algebra SHA s,e (·, ·) : SHs,e A ⊗ SHA → C

as follows. s,e • For f ∈ SHs,e A,v , and g ∈ SHA,w , we define (f, g) = 0 if v 6= w. • For h ∈ SH0 , and f ∈ SHsA , we define (h, f ) = 0. • For Hk (u) ∈ SH0 [[u]], we define (Hk (u), Hk (w)) = fac(u|w) fac(w|u) for any k ∈ I. Note that in particular for Hk (u) ∈ SH0 [[u]] we have (1, Hk (u)) = 1. For any i ∈ I and f, g ∈ SHA,ei , we follow Drinfeld [D86] and define X (f, g) := Resx (fx · g−x · ω). x∈G

In general, for f, g ∈

SHs,e A,v ,

we define the pairing (f, g) to be f (x ) · g(−x ) X A A Resx (f, g) := ω . |A|! fac(xA ) x∈G

Here we consider the x-component of x-component of

f (xA )·g(−xA ) |A|! fac(xA )

f (xA )·g(−xA ) |A|! fac(xA )

v

as a function on G . Lemma 3.4 implies that the

is in Kxv , and hence the residue is well-defined.

Theorem 3.5. Assume the numbers mh , mh∗ associated to h ∈ H are as in Remark 1.2(2). On s,e SHA , the pairing (·, ·) is a super-symmetric non-degenerate bialgebra pairing. This theorem is proved through the following lemmas. Lemma 3.6. The pairing (·, ·) on SHeA is super-symmetric, i.e., (f, g) equals (g, f ) upto a sign. Proof. We have (f, g) =

X

Resx

f (x ) · g(−x ) A A ω(xA ) |A|! fac(xA )

X

Resx

f (−x ) · g(x ) A A ω(−xA ) , |A|! fac(xA )

x∈G

=

x∈G

where the last equality used the fact that fac(xA ) = fac(−xA ) under the assumption of Remark 1.2(2). Note that ω and its pullback under group inverse only differ by a sign, the above is equal to (g, f ) upto sign. Lemma 3.7. The above super-symmetric pairing (·, ·) on SHeA is non-degenerate. Proof. By symmetry, we need to show that if (f, g) = 0, for any g ∈ SHeA , then f = 0. This follows from Lemma 3.2. Lemma 3.8. The above super-symmetric non-degenerate pairing (·, ·) on SHeA has the property (a ⋆ b, c) = (a ⊗ b, ∆(c)).

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Y. YANG AND G. ZHAO

Proof. Let v1 , v2 ∈ NI be two dimension vectors with v = v1 + v2 . We need to show (f1 ⋆ f2 , P ) = (f1 ⊗ f2 , ∆(P )), for f1 ∈ SHA,v1 , f2 ∈ SHA,v2 , P ∈ SHA,v , since (SHA,v , SHA,w ) = 0, if v 6= w. Let (Ao , Bo ) = ([1, v1 ], [v1 + 1, v]) be the standard element in P(v1 , v2 ). By definition, it suffices to show that X HAo (xBo )P (xAo ⊗ xBo ) (6) f1 (xA )·f2 (xB )·fac(xA |xB ), P = f1 (xAo )⊗f2 (xBo ), . fac(xBo |xAo ) (A,B)∈P(v1 ,v2 )

Using Lemma 3.1, the left hand side is the same as X X f1 (xA ) · f2 (xB ) · fac(xA |xB ) · P (−xA , −xB ) Resx wA∪B |A ∪ B|! fac(xA∪B ) {(A,B)}

x∈G

=

X

Resx

x∈G

f (x ) · f (x ) · fac(x |x ) · P (−x , −x ) 1 Ao 2 Bo Ao Bo Ao Bo wAo ∪Bo |Ao |!|Bo |! fac(xAo ∪Bo )

b Bo |xAo ), the right hand Using the equality HAo (xBo )P (xAo ⊗ xBo ) = P (xAo ⊗ xBo )HAo (xBo )Φ(x side of (6) is the same as P (xAo ⊗ xBo )HAo (xBo ) f1 (xAo ) ⊗ f2 (xBo ), fac(xAo |xBo ) X f1 (xAo ) f2 (xBo ) P (−xAo , −xBo ) Resx = · · )wAo wBo |Ao |! fac(xAo ) |Bo |! fac(xBo ) fac(−xAo | − xBo ) x∈G

=

X

x∈G

Resx

f (x ) · f (x ) · fac(x |x ) · P (−x , −x ) 1 Ao 2 Bo Ao Bo Ao Bo wAo ∪Bo , |Ao |!|Bo |! fac(xAo ∪Bo )

Here we used the fact that (1, h(w)) = 1 and fac(−xAo | − xBo ) = fac(xBo |xAo ) under assumption of Remark 1.2(2). Therefore, the equality (6) holds. This completes the proof. 3.2. Quantization of Manin triples. Using the non-degenerate bilinear pairing (·, ·), we form s,e the Drinfeld double of the bialgebra SHs,e A , denoted by D(SHA ). Again following the idea of s,e Drinfeld, we consider a subalgebra in the Drinfeld double D(SHA ). Assume Λ ⊂ A is an isotropic C-subring such that A ∼ = F ⊕ Λ. Let S ⊂ G be a non-empty finite subset. Let AS be the adèle ring without x-component for x ∈ S. Let FS be the subring {a ∈ F | im(a) ∈ AS lies in im(Λ) ⊂ AS .} For each i ∈ I, let SHei ,S ⊆ SHei ,A be the image of FS under the isomorphism SHei ,A = A. coop 0 0 coop . We consider the subalgebra of D(SHs,e A ), generated by SHei ,S , SHei ,S for i ∈ I, and SH ⊗(SH ) s,e This subalgebra is denoted by D(SHS ). This subalgebra is closed under comultiplication, hence s,e 0 a bialgebra. Let SHs,e S be the subalgebra of SHA generated by SHei ,S and SH . Clearly this is s,e s,e coop a sub-bialgebra of D(SHS ). Similarly we have a sub-bialgebra (SHS ) . As a vector space, we have s,e s,e coop ∼ D(SHs,e . S ) = SHS ⊗(SHS ) One example of D(SHs,e S ), when G = Ga is studied in § 4, which is related to the Yangian. From [Gr94b], it is expected that when G = Gm , ω = x1 dx with x being the natural coordinate of A1 ⊃ Gm , and S ⊆ P1 consists of 0, ∞, the algebra D(SHs,e S ) should be related to the quantum loop algebra. However, the proof of this statement involves calculation of non-standard generators of the quantum loop algebra, we postpone it to later investigation.


13

Example 3.9. It is equally interesting to consider example when G is beyond the additive, multiplicative, and elliptic ones. For example, let R = Q[β], and F (u, v) = u + v − βuv be a formal group law over R. An OCT with this formal group law is the connective K-theory, denoted by CK. Let P s,e (Q, CK) be the spherical extended preprojective CoHA with A = CK. Then, D(P s,e (Q, CK)) can be considered as a family of algebras over Spec R = A1 , whose generic fiber is Uq (LgQ ), and the special fiber is Y~ (gQ ). This recovers a classical theorem of Drinfeld that the quantum loop algebra degenerates to the Yangian. ± ± 1 uv There is a hyperbolic formal group law (F (v, u) = v+u−µ 1+µ2 uv , Z[µ1 , µ2 ]) studied in [LZ14]. This formal group law comes from generic singular locus of the Weierstrass elliptic curve. Generically, this formal group law is multiplicative, and at the special fiber the formal group law is additive. Therefore, let A be the OCT with this formal group law, then P s,e (Q, A) should provide another degeneration of Uq≥0 (LgQ ) to Yq≥0 (gQ ). However, the hyperbolic formal group law does not specialize to the one associated to connective K-theory. Hence, this degeneration is different than the one constructed by Drinfeld. When A = Ω is the algebraic cobordism of [LM07], the formal group law is the universal formal group law over the Lazard ring L. It is known that the Lazard ring consists of Chern numbers of smooth projective varieties. We have Ω P s,e acts on Ω M(w). The eigen-values of Ω P 0 on Ω M(w) are given by Chern numbers of smooth projective varieties. We expect this observation to be of geometric applications. Another potentially interesting OCT is the Morava K-theory, whose formal group law is the Lubin-Tate formal group law. However, as the coefficient ring of the Lubin-Tate formal group law has positive characteristic, we will not study this example in the present paper. 4. Yangian as a Drinfeld double In this section we assume the group G is the additive group Ga . We assume Q has no edge-loops, and for each h ∈ H the numbers mh and mh∗ are as in Remark 1.2. In this case, we recover the Yangian using the construction of Drinfeld double of the cohomological Hall algebra. 4.1. The Yangian. The smooth compactification of Ga is P1 . Let l = x be the natural coordinate function of Ga . The meromorphic section ω can Q be taken as ω = dx, which has an order 2 pole at ∞ ∈ Ga . Let S = {∞} ⊂ P1 , and Λ = (m∞ × x∈X\S Ox ) ⊂ A, where m∞ is the maximal ideal in s,e O∞ . Then, FS ∼ = C[x], the ring of regular functions on Ga . Hence, SHS = SHs,e , and the Drinfeld s,e s,e s,e coop endowed with a suitable bialgebra structure. double D(SHS ) is SH ⊗(SH ) e,s Define the reduced Drinfeld double D(SHe,s S ) (see e.g. [DJX]) to be D(SHS ) with the following additional relation imposed Hk+ (u) = Hk− (−u), for any k ∈ I. The main theorem of this section is Theorem B, i.e., in the setup of this section, we have an isomorphism of bialgebras ∼ D(SHe,s S )|t1 =t2 =~/2 = Y~ (g) where Y~ (g) is endowed with the Drinfeld coproduct. Recall that Yangian for a finite dimensional Lie algebra is the quantization of a Manin triple [D86]. Fix a Lie algebra a(dim a < ∞) and an invariant scalar product on it. Set p = a(u−1 ), p1 = a[u], p2 = u−1 a[u−1 ] and define the scalar product p by (f, g) := Resu=∞ (f (u), g(−u))du.

14

Y. YANG AND G. ZHAO

The Manin triple (p, p1 , p2 ) defines a Lie bialgebra structure on a[u]. The cocommutator is given by t ] a(u) 7→ [a(u) ⊗ 1 + 1 ⊗ a(u), u−v where t is the Casimir element. Identify Spec C[u] with G = Ga , then p1 can be alternatively described as the following sub-bialgebra of a ⊗C F GS = {a ∈ a ⊗C F | the image of a in a ⊗C AS belongs to the image of a ⊗C Λ in a ⊗C AS }, where AS is the ring of adèle without ∞–components. The double Yangian, defined as the Drinfeld double of the Yangian, is a quantization of p. There is an explicit Drinfeld-type presentation of the Yangian, which applies to any Kac-Moody Lie algebra. Let gQ be the symmetric Kac-Moody Lie algebra associated to the quiver Q. The Cartan matrix of gQ is (ckl )k,l∈I , which is a symmetric matrix. Recall that the Yangian of gQ , denoted by Y~ (gQ ), is an associative algebra over C[~], generated by the variables x± k,r , ξk,r , (k ∈ I, r ∈ N), −1 subject to relations described below. Take the generating series ξk (u), x± k (u) ∈ Y~ (gQ )[[u ]] by X X −r−1 ξk (u) = 1 + ~ ξk,r u−r−1 and x± x± . k (u) = ~ k,r u r≥0

r≥0

The following is a complete set of relations defining Y~ (gQ ) (see, e.g., [GTL14, § 3.4]): (Y1): For any i, j ∈ I, and h, h′ ∈ h [ξi (u), ξi (v)] = 0 [ξi (u), h] = 0 [h, h′ ] = 0 (Y2): For any i ∈ I, and h, ∈ h, ± [h, x± i (u)] = ±αi (h)xi (u)

(Y3): For any i, j ∈ I, and a =

~cij 2

± ± (u − v ∓ a)ξi (u)x± j (v) = (u − v ± a)xj (v)ξi (v) ∓ 2axj (u ∓ a)ξi (u)

(Y4): For any i, j ∈ I, and a =

~cij 2

± ± ± ± ± ± ± (u − v ∓ a)x± (u)x (v) = (u − v ± a)x (v)x (u) + ~ [x , x (v)] − [x (u), x ] i j j i i,0 j i j,0 (Y5): For any i, j ∈ I, − (u − v)[x+ i (u), xj (v)] = −δij ~(ξi (u) − ξi (v))

(Y6): For any i 6= j ∈ I, X ± ± ± [x± i (uσ(1) ), [xi (uσ(2) ), [· · · , [xi (uσ(1−cij ) ), xj (v)] · · · ]]] = 0. σ∈S1−cij


15

4.2. Relation with the shuffle algebra. In [YZ1], there is an algebra isomorphism ≥0 SHs,e |t1 =t2 = ~ ∼ = Y~ (g). 2

Under this isomorphism, SHek ∋ (x(k) )r 7→ xk,r for any k ∈ I and r ∈ N, and SH0 ⊇ Sek ∋ (h(k) )r 7→ ξk,r . This isomorphism is compatible with the action of the left hand side on cohomology of quiver varieties [YZ1] and the action of the right hand side [Va00]. Lemma 4.1. Under this isomorphism, the comultiplication ∆ defined in § 2 agrees with the Drinfeld coproduct of the Yangian. Proof. This follows from the formula (3) and the formula of Drinfeld coproduct of the Yangian [GTL14, § 4.4] (see also [Her07] for the case of quantum loop algebra). In [Nak12], the tensor structure on Yangian representations, in terms of the cohomology of quiver varieties, associated to Drinfeld comultiplication has been studied. In other words, the map from D(SHs,e ) to the convolution algebra of the Steinberg variety [YZ1] is compatible with the Drinfeld comultiplications on both sides. P Let f (x(k) ) and g(x(k) ) be any elements in SHek . We denote i∈N (h(k) )i ((x(k) )−i−1 · f, g) by Hk (x(k) )(f, g) then we have Hk (x(k) )(f (x(k) ), g(x(k) )) = Resx(k) =∞ (Hk (x(k) ) · f (x(k) ) · g(−x(k) ) · dx(k) ). P Let Ek (u) := ~ r≥0 (x(k) )r u−r−1 ∈ SHek [[u−1 ]] be the standard generating series. Note that it is the expansion of the rational function u−x~ (k) on Gα × G around u = ∞. We denote the s,e −1 −1 element −Ek (−u)− ∈ SHcoop ek [[u ]] ⊆ D(SH )[[u ]] by Fk (u). Note that as a consequence of the ≥0 − isomorphism SHs,e ∼ = Y~ (g), the map Fk (u) 7→ x− k (u), Hk (−u) 7→ ξk (u) induces an isomorphism ≤0 (SHs,e )coop ∼ = Y~ (g).

Recall the reduced Drinfeld double D(SHe,s ) (see e.g. [DJX]) is D(SHe,s) with the following additional relation imposed Hk+ (u) = Hk− (−u), for any k ∈ I. Set Hk (u) := −~Hk+ (u) − ~Hk− (−u). Theorem 4.2. The following relations hold in the reduced Drinfeld double D(SHe,s ): (7) (8)

Ek (u)Fl (v) = Fl (v)Ek (u), for k 6= l. Hk (u) − Hk (v) Ek (u)Fk (v) − Fk (v)Ek (u) = −~ . u−v

Proof. First we consider the case when k = l. In SHe , we have ∆(Ek (u)) = Hk (x(k) ) ⊗ Ek (u) + Ek (u) ⊗ 1 by (4). We use the relation (5) with a = −Ek (−u), b = Ek (v) and the fact (Ek (u), Hl (x)) = 0, (1, Ek (u)) = 0. It gives the following relation in SHe,coop ⊗ SHe : Hk− (x(k) ) ⋆ 1(−Ek (−u), Ek (v)) + Fk (u) ⋆ Ek (v)(1, Hk (x(k) )) =Ek (v) ⋆ Fk (u)(1, Hk (x(k) )) + Hk+ (x(k) ) ⋆ 1(Ek (v), −Ek (−u)).

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Y. YANG AND G. ZHAO

Therefore, we have [Fk (u), Ek (v)] = Hk+ (x(k) )(Ek (v), −Ek (−u)) − Hk− (x(k) )(−Ek (−u), Ek (v)) ~ ~ ~ ~ − (k) + (k) , − Hk (x ) , = Hk (x ) v − x(k) u + x(k) u + x(k) v − x(k) Hk+ (x(k) )dx(k) Hk− (x(k) )dx(k) 2 − ~ Res x(k) =∞ (v − x(k) )(u − x(k) ) (u + x(k) )(v + x(k) ) H + (u) − H + (v) H − (−v) − H − (−u) k k = ~2 − k + k u−v −v + u Hk (u) − Hk (v) =~ u−v The case when k 6= l is clear, which follows from the relation (5) with a = Ek (u), b = El (v), and (Ek (u), El (v)) = 0. This completes the proof. = ~2 Resx(k) =∞

The following is a direct corollary to Theorem 4.2. It implies Theorem B, taking Proposition 1.7 into account. Corollary 4.3. We have an algebra isomorphism D(SHe,s ) ∼ = Y~ (g). Proof. Construct a map from the Yangian Y~ (g) to D(SHe,s ), by − x+ k (u) 7→ Ek (u) , xk (u) 7→ Fk (u) and ξk (u) 7→ Hk (u).

By Theorem 4.2, the map respects the relations of the Yangian Y~ (g), hence is an algebra homomorphism. Be definition of D(SHe,s), this map is surjective. It also preserves the triangular decomposition Y~ (g) = Y~+ (g) ⊗ Y 0 ⊗ Y~− (g) and D(SHe,s ) = SHs,e ⊗ SH0 ⊗(SHs,e )coop , which restricts to an isomorphism on each tensor-factor. Therefore, it is an isomorphism. 4.3. The double Yangian. The double Yangian DY~ (g), which is the Drinfeld double of the Yangian Y~ (g), can also be realized as a subalgebra in the reduced version of D(SHs,e A ). The s,e construction is similar to that of D(SHS ). Remark 4.4. For each i ∈ I, let SHei ,K ⊆ SHei ,A be the image of K∞ under the isomorphism coop SHei ,A = A. We consider the subalgebra of D(SHs,e A ), generated by SHei ,K , SHei ,K for i ∈ I, and SH0 ⊗(SH0 )coop . This subalgebra, which is closed under comultiplication, is denoted by D(SHs,e K ). e,s Again, define the reduced Drinfeld double D(SHe,s ) to be D(SH ) with the following additional K K relation imposed Hk+ (u) = Hk− (−u), for any k ∈ I. s,e Clearly D(SHe,s S ) ⊆ D(SHK ) is a sub-bialgebra. Taking classical limit, we have an isomorphism −1 ∼ D(SHe,s K )|t1 =t2 =0 = U (g(u )). ∼ Therefore, we have an algebra isomorphism D(SHe,s K ) = DY~ (g), where DY~ (g) is the double Yangian.

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