Quantum Toroidal Comodule Algebra of Type 𝐴ₙ₋₁ and Integrals of Motion

We introduce an algebra 𝒦ₙ which has the structure of a left comodule over the quantum toroidal algebra of type 𝐴ₙ₋₁. Algebra 𝒦ₙ is a higher rank generalization of 𝒦₁, which provides a uniform description of deformed 𝑊 algebras associated with Lie (super)algebras of types BCD. We show that 𝒦ₙ posses...

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Published in:	Symmetry, Integrability and Geometry: Methods and Applications
Date:	2022
Main Authors:	Feigin, Boris, Jimbo, Michio, Mukhin, Evgeny
Format:	Article
Language:	English
Published:	Інститут математики НАН України 2022
Online Access:	https://nasplib.isofts.kiev.ua/handle/123456789/211736
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Cite this:	Quantum Toroidal Comodule Algebra of Type 𝐴ₙ₋₁ and Integrals of Motion. Boris Feigin, Michio Jimbo and Evgeny Mukhin. SIGMA 18 (2022), 051, 31 pages

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author	Feigin, Boris Jimbo, Michio Mukhin, Evgeny
author_facet	Feigin, Boris Jimbo, Michio Mukhin, Evgeny
citation_txt	Quantum Toroidal Comodule Algebra of Type 𝐴ₙ₋₁ and Integrals of Motion. Boris Feigin, Michio Jimbo and Evgeny Mukhin. SIGMA 18 (2022), 051, 31 pages
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container_title	Symmetry, Integrability and Geometry: Methods and Applications
description	We introduce an algebra 𝒦ₙ which has the structure of a left comodule over the quantum toroidal algebra of type 𝐴ₙ₋₁. Algebra 𝒦ₙ is a higher rank generalization of 𝒦₁, which provides a uniform description of deformed 𝑊 algebras associated with Lie (super)algebras of types BCD. We show that 𝒦ₙ possesses a family of commutative subalgebras.
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 051, 31 pages Quantum Toroidal Comodule Algebra of Type An−1 and Integrals of Motion Boris FEIGIN ab, Michio JIMBO c and Evgeny MUKHIN d a) National Research University Higher School of Economics, 20 Myasnitskaya Str., Moscow, 101000, Russia b) Landau Institute for Theoretical Physics, 1a Akademika Semenova Ave., Chernogolovka, 142432, Russia E-mail: bfeigin@gmail.com c) Department of Mathematics, Rikkyo University, Toshima-ku, Tokyo 171-8501, Japan E-mail: jimbomm@rikkyo.ac.jp d) Department of Mathematics, Indiana University Purdue University Indianapolis, 402 N. Blackford St., LD 270, Indianapolis, IN 46202, USA E-mail: emukhin@iupui.edu Received March 02, 2022, in final form June 27, 2022; Published online July 07, 2022 https://doi.org/10.3842/SIGMA.2022.051 Abstract. We introduce an algebra Kn which has a structure of a left comodule over the quantum toroidal algebra of type An−1. Algebra Kn is a higher rank generalization of K1, which provides a uniform description of deformed W algebras associated with Lie (super)al- gebras of types BCD. We show that Kn possesses a family of commutative subalgebras. Key words: quantum toroidal algebras; comodule; integrals of motion 2020 Mathematics Subject Classification: 81R10; 81R12; 17B69; 17B80 1 Introduction Integrable systems originated in quantum toroidal algebras have been drawing a lot of attention in recent years, see [1, 4, 5, 8, 9, 10, 11, 14, 15, 16], to name a few. In this paper we continue the study launched in [7] concerning deformed W algebras and a quantum toroidal version of integrable models with reflections in the spirit of [21]. The work [7] is based on a new algebra K1 generated by current E(z), Heisenberg half-cur- rents K±(z), and a central element C. In various representations the current E(z) recovers the fundamental current of known deformed W algebras of types BCD as well as their su- persymmetric analogs. It was shown that K1 has a family of commuting elements {IN}∞N=1 called integrals of motion. The element IN is written as an N -fold integral of the product E(zN ) · · ·E(z1) with the explicit elliptic kernel of [19]. Here E(z) is the dressed current of the form E(z) = E(z)K+(z)−1, where K+(z) is given in terms of K+(z). In the representations corresponding to deformed W algebras, the integrals of motion IN are deformations of local integrals of motion in the W algebras, see [2]. In the present paper we generalize this construction by introducing algebrasKn for n > 1. The algebra Kn depends on parameters q1, q2 = q2, and is generated by currents E1(z), . . . , En(z), Heisenberg half-currents K± 1 (z), . . . ,K± n (z), and a central element C. We show that it has the following key properties: mailto:bfeigin@gmail.com mailto:jimbomm@rikkyo.ac.jp mailto:emukhin@iupui.edu https://doi.org/10.3842/SIGMA.2022.051 2 B. Feigin, M. Jimbo and E. Mukhin � Algebra Kn has essentially the same size as the Borel subalgebra of quantum toroidal gln algebra En; see Remark 3.1. � For m < n, algebra Km can be obtained as a subalgebra of Kn using the fusion procedure in the same way as Em is found as a subalgebra of En in [3]; see Proposition 6.2. � Algebra Kn is a left comodule over quantum toroidal algebra En; see Theorem 3.4. � Algebra Kn has two “boundary” modules FD of level C = q−1 and FB of level C = q1/2; see Lemmas 3.5 and 3.6. � AlgebraKn has a family of commuting integrals of motion {IN}∞N=1, such that IN is an nN - fold integral of a current given by an ordered product of the form ∏↶ 1≤i≤n ∏↶ 1≤a≤N Ei(za) together with the kernel of [12]; see (4.9) and Theorem 4.2. Here the dressed currents have the form Ei(z) = K− i (z) −1Ei(z), where K− i (z) is given in terms of K− i (z).1 Here we should make a disclaimer: the relations of Kn involve infinite sums, so that an appropriate completion is necessary. The same is true with the fusion procedure. Also the comodule map is defined at the level of a class of representations called admissible ones. These properties ensure that Kn has a rich representation theory. At least, we have rep- resentations of the form V ⊗W , where W is one of the two boundary modules and V is any admissible En module. Here by an admissible module we mean a homogeneously graded module with the degree bounded from above; for example any tensor product of Fock modules (or, more generally, modules with highest weight with respect to the “rotated” set of generators) is admis- sible. Then, under a mild restriction on parameters q1, q2 and the level C, in any Kn module we have an integrable system given by action of {IN}. This is the principal result of this paper. For n = 1, this system is related to the system with reflections, see [7, Section 5.1], and the Bethe ansatz for it is studied in [16] in the setting of affine Yangians. Although the expressions of {IN} are non-local, we shall sometimes refer to them as “local” integrals of motion, since in the case n = 1 they are deformations of local ones as already mentioned. We do not completely understand the nature of algebras Kn. The Kn algebras we discuss here are of A type. Similar algebras can be defined in the same way for arbitrary simply laced types, though the construction of integrals of motion is more challenging. At first glance, algebra Kn bears resemblance to ıquantum groups; formula (3.16) for comodule structure is rem- iniscent of the Chevalley generators [13, formula (5.1)], and the defining relations (3.1)–(3.6) of Kn look similar to the Drinfeld realization given in [17, Theorem 5.1]. However, the details seem to be very different. For example, the relations in Kn are invariant under scaling z → cz but such homogeneity does not exist in those of [17]. The current appearing in the residues in the right-hand side of (5.5) in [17] is commutative, whereas our current K− i (z)K+ i ( C2z ) in (3.4) is not. The ıquantum groups are coideals in the corresponding quantum groups, while we are not aware of inclusion of Kn to En. We do not know if ıquantum groups have in- teresting families of integrals of motion in general. Natural candidates which could be com- pared to Kn are the reflection algebras and, in particular, quantum twisted Yangians, see [18]. The quantum twisted Yangians are coideal subalgebras behind integrable systems with reflec- tions. However, the exact relation to Kn algebras is unclear. The quantum twisted Yangians are given in terms of the R-matrix realization (as opposed to the Drinfeld-type realization of Kn) and the formulas for the generators of the coideal are quadratic in generators of quantum group. In quantum toroidal algebras En, the integrals of motion originate from the standard con- struction of transfer matrices and depend on n arbitrary parameters. They commute for a simple reason following from the Yang–Baxter equation. Then a computation provides the expression 1We change the convention slightly from [7]. Quantum Toroidal Comodule Algebra of Type An−1 and Integrals of Motion 3 for the integrals of motion in the form of integrals. The integrals of motion in the Kn algebra are given virtually by the same formula as in En, except that one of the parameters (the ellip- tic nome) is fixed by the central element C. But we do not have any general reason for their commutativity; we simply check it directly. It is known that in type A there is a natural duality of such models of (glm, gln) type, see [6]. Under that duality, the algebras Em and En act on the same space in such a way that the screening operators for one algebra are given in terms of generating currents of the other. Furthermore the integrals of motion of both algebras commute with each other. For K1 acting in a representation, we have a system of screening operators, see [7]. We expect that the algebras generated by the corresponding screening currents also possess a commutative family which should be called “non-local” integrals of motion, and that “local” and “non-local” integrals of motion commute. Similarly, we expect that algebras Kn acting in representations commute with a family of screening operators. Moreover the algebra generated by the screening currents does not depend on n. For the case of the tensor product of ℓ Fock En modules with a boundary Kn module, we anticipate ℓ + 1 screening currents each given as a sums of n vertex operators. The correspon- ding “non-local” integrals of motion should commute with the “local” integrals of motion {IN} constructed in this paper. We plan to address this issue in future publications. The text is organized as follows. In Section 2 we introduce our convention about the quantum toroidal algebra En of type gln. In Section 3 we define algebra Kn. Working with admissible representations, we rephrase the Serre relations as zero conditions (called wheel conditions) on matrix coefficients of products of generating currents. We show that on admissible representations, Kn may be viewed as a left comodule over En. We use this fact to construct representations starting with elementary ones. In Section 4 we introduce integrals of motion. They are integrals of products of generating currents of Kn obtained by a dressing procedure. The kernel functions entering integrals of motion are precisely the same as those used for those of En. We then prove that integrals of motion mutually commute by direct computation. For simplicity of presentation, we treat the case n ≥ 3 in the body of the text. We shall mention the modifications necessary for n = 1, 2 in Section 5. In Section 6 we show that, for any k = 1, . . . , n−1, algebra Kn = Kn(q1, q2, q3) with parame- ters q1, q2, q3 contains mutually commuting subalgebras Kk(q̄1, q̄2, q̄3) and Kn−k(¯̄q1, ¯̄q2, ¯̄q3) with appropriate parameters q̄i, ¯̄qi. In Appendix A, we give a proof of the theta function identities used in the commutativity of integrals of motion. Notation 1.1. Throughout the text we fix parameters q1/2, d1/2 ∈ C× and define q1 = q−1d, q2 = q2, q3 = q−1d−1, so that q1q2q3 = 1. We assume that qi1q j 2 = 1 for i, j ∈ Z implies i = j = 0. For a positive integer N , we write δ (N) i,k = 1 if i ≡ k mod N and δ (N) i,k = 0 otherwise. We use the standard symbols for infinite products (z1, . . . , zr; p)∞ = r∏ i=1 ∞∏ k=0 ( 1− zip k ) , Θp(z) = ( z, pz−1, p; p ) ∞. We use also the formal series δ(z) = ∑ k∈Z z k. 4 B. Feigin, M. Jimbo and E. Mukhin 2 Preliminaries In this section we fix our convention regarding the quantum toroidal algebra En of type gln. 2.1 Quantum toroidal algebra En Fix a positive integer n ≥ 3. We define gi,j(z, w), Gi,j(x) for i, j ∈ Z/nZ as follows gi,i(z, w) = z − q2w, Gi,i(x) = q2 1− q−2x 1− q2x , gi,i±1(z, w) = z − q−1d±1w, Gi,i±1(x) = q−1 1− qd±1x 1− q−1d±1x , gi,j(z, w) = z − w, Gi,j(x) = 1 if j ̸≡ i, i± 1. Here Gi,j(x) should be understood as power series expansion in x. As rational function we have Gi,j(w/z) = − gj,i(w, z) di,jgi,j(z, w) = Gj,i(z/w) −1, where di,j = d±1, i ≡ j ± 1, and di,j = 1 otherwise. Let P be a free Z module with basis {εi \| i ∈ Z/nZ}, equipped with a symmetric bilinear form ( , ) : P × P → Z such that εi are orthonormal. We write ᾱi = εi−1 − εi. The quantum toroidal algebra En of type gln is a unital associative algebra generated by ei,k, fi,k, hi,r, i ∈ Z/nZ, k ∈ Z, r ∈ Z\{0}, qh, h ∈ P, C±1. In terms of the generating series ei(z) = ∑ k∈Z ei,kz −k, fi(z) = ∑ k∈Z fi,kz −k, ψ± i (z) = ψ± i,0 exp ( ± ( q − q−1 )∑ r>0 hi,±rz ∓r ) , ψ± i,0 = q±ᾱi , the defining relations read as follows qhqh ′ = qh+h′ , h, h′ ∈ P, q0 = 1, C is central, qhei(z)q −h = q(h,ᾱi)ei(z), qhfi(z)q −h = q−(h,ᾱi)fi(z), qhψ± i (z)q −h = ψ± i (z), h ∈ P, ψ± i (z)ψ ± j (w) = ψ± j (w)ψ ± i (z), ψ+ i (z)ψ − j (w) = ψ− j (w)ψ + i (z)Gi,j(Cw/z) −1Gi,j ( C−1w/z ) , ψ+ i (z)ej(w) = ej(w)ψ + i (z)Gi,j ( C−1w/z ) , ej(w)ψ − i (z) = ψ− i (z)ej(w)Gj,i(z/w), ψ+ i (z)fj(w) = fj(w)ψ + i (z)Gi,j(w/z) −1, fj(w)ψ − i (z) = ψ− i (z)fj(w)Gj,i(Cz/w) −1, [ei(z), fj(w)] = δi,j q − q−1 (δ ( Cw/z ) ψ+ i (w)− δ ( Cz/w ) ψ− i (z)), [ei(z), ej(w)] = 0, [fi(z), fj(w)] = 0, i ̸≡ j, j ± 1, di,jgi,j(z, w)ei(z)ej(w) + gj,i(w, z)ej(w)ei(z) = 0, dj,igj,i(w, z)fi(z)fj(w) + gi,j(z, w)fj(w)fi(z) = 0, Sym z1,z2 [ei(z1), [ei(z2), ei±1(w)]q]q−1 = 0, Sym z1,z2 [fi(z1), [fi(z2), fi±1(w)]q]q−1 = 0. Quantum Toroidal Comodule Algebra of Type An−1 and Integrals of Motion 5 Here [A,B]p = AB − pBA, and the symbol Sym stands for symmetrization: Sym x1,...,xN f(x1, . . . , xN ) = 1 N ! ∑ σ∈SN f(xσ(1), . . . , xσ(N)). The relations above imply in particular [hi,r, hj,s] = δr+s,0 · ai,j(r) Cr − C−r q − q−1 , r, s ̸= 0, ai,j(r) = 1 r qr − q−r q − q−1 ( (qr + q−r)δ (n) i,j − drδ (n) i,j−1 − d−rδ (n) i,j+1 ) . (2.1) Algebra En has a Z grading which we call homogeneous grading, given by deg xi,r = r, x = e, f, h, deg qh = degC = 0. 2.2 Coproduct Quite generally, let A be a Z graded algebra with a central element C. The completion of A in the positive direction is the algebra Ã, linearly spanned by products of series of the form∑∞ k=M fkgk, where M ∈ Z, fk, gk ∈ A and deg gk = k. We call an A module V admissible if C is diagonalizable, and if V is Z graded with finite dimensional components of degree bounded from above: V = ⊕N k=−∞Vk, where Vk = {v ∈ V \| deg v = k}, dimVk < ∞. The completion Ã acts on all admissible modules. Let En⊗̃En be the tensor algebra En ⊗ En completed in the positive direction. We use the topological coproduct ∆: En → En⊗̃En ∆ei(z) = ei(C2z)⊗ ψ− i (z) + 1⊗ ei(z), ∆fi(z) = fi(z)⊗ 1 + ψ+ i (z)⊗ fi(C1z), ∆ψ+ i (z) = ψ+ i (z)⊗ ψ+ i (C1z), ∆ψ− i (z) = ψ− i (C2z)⊗ ψ− i (z), ∆x = x⊗ x, x = qh, C. Here C1 = C ⊗ 1 and C2 = 1⊗ C. Later on we shall use the elements in the completed algebra Ẽn f̃i(z) = S−1 ( fi(z) ) = −fi ( C−1z ) ψ+ i ( C−1z )−1 , where S denotes the antipode. The f̃i(z)’s satisfy the same quadratic relations and the Serre relations as do ei(z)’s. In addition dj,igj,i(w, z)ei(z)f̃j(w) + gi,j(z, w)f̃j(w)ei(z) = −δi,j gi,i(w, z) q − q−1 ( δ(w/z)− δ ( C2z/w ) ψ− i (z)ψ + i (Cz) −1) . (2.2) 2.3 Fock modules It is well known [20] that En has admissible representations given in terms of vertex operators. Consider a vector space with basis \|m⟩ labeled by n tuples of integers m = (m0,m1, . . . ,mn−1) ∈ Zn. We define linear operators e±ϵi and ∂i, 0 ≤ i ≤ n− 1, by e±ϵi \|m⟩ = (−1) ∑i−1 s=0 ms \|m± 1i⟩, ∂i\|m⟩ = mi\|m⟩, 6 B. Feigin, M. Jimbo and E. Mukhin with 1i = (δi,j)0≤j≤n−1. We have eϵieϵj = −eϵjeϵi for i ̸= j. We set F = C[{hi,−r}r>0,0≤i≤n−1]⊗ ( ⊕ m∈Zn C\|m⟩ ) . We write operators hi,±r ⊗ id, id⊗ e±ϵi and id⊗ ∂i simply as hi,±r, e ±ϵi and ∂i. We extend the range of the suffix i by periodicity, e.g., ∂i+n = ∂i. Define a vertex operator which acts on F Vi(z) = e−ϵieϵi−1z∂i−1−∂id(∂i−1+∂i)/2 exp (∑ r>0 q−r [r] hi,−rz r ) exp ( − ∑ r>0 1 [r] hi,rz −r ) . For u ∈ C×, the following assignment gives F a structure of an admissible En module denoted by F(u), ei(z) 7→ u−δi,0zVi(z), f̃i(z) 7→ −q−1uδi,0z :Vi(z) −1:, hi,r 7→ hi,r, r ̸= 0, qεi 7→ q∂i , C 7→ q. Here we use the usual normal ordering symbol to bring e±ϵi , hi,−r, r > 0, to the left and ∂i, hi,r, r > 0, to the right, except that the order between e±ϵi ’s is kept unchanged. The En module F(u) is a direct sum of irreducible submodules, see [6, Lemma 3.2]. 3 Comodule algebra Kn In this section we introduce an algebra Kn, and show that it has the structure of an En comodule in the sense to be made precise below. Algebra K1 was first introduced and studied in [7] in order to give a uniform description of deformedW algebras associated with simple Lie (super)algebras of types BCD. 3.1 Algebra Kn We introduce an algebra Kn through generating series Ei(z) = ∑ k∈Z Ei,kz −k, K± i (z) = K± i,0 exp ( ± ( q − q−1 ) ∑ ±r>0 Hi,rz −r ) , i ∈ Z/nZ, and an invertible central element C. The defining relations read as follows K± i (z)K± j (w) = K± j (w)K± i (z), (3.1) K+ i (z)K− j (w) = K− j (w)K+ i (z)Gi,j(w/z)Gi,j ( C2w/z ) , (3.2) K+ i (z)Ej(w) = Ej(w)K + i (z)Gi,j(w/z), Ej(w)K − i (z) = K− i (z)Ej(w)Gj,i(z/w), (3.3) di,jgi,j(z, w)Ei(z)Ej(w) + gj,i(w, z)Ej(w)Ei(z) (3.4) = δi,j q−q−1 ( gi,i(z, w)δ ( C2z/w ) K− i (z)K+ i ( C2z ) +gi,i(w, z)δ ( C2w/z ) K− i (w)K+ i ( C2w )) , Ei(z)Ej(w) = Ej(w)Ei(z), i ̸≡ j, j ± 1. (3.5) In addition, we impose the Serre relations Sym z1,z2 [Ei(z1), [Ei(z2), Ei±1(w)]q]q−1 = −Sym z1,z2 {( 1 1− q−1d∓1z1/w + q−1d±1w/z2 1− q−1d±1w/z2 ) × δ ( C2z1/z2 ) K− i (z1)Ei±1(w)K + i (z2) } . (3.6) Quantum Toroidal Comodule Algebra of Type An−1 and Integrals of Motion 7 In the right-hand side, the rational functions of the form 1/(1− az1/w) or 1/(1− bw/z2) stand for their expansions in non-negative powers of z1/w or w/z2, respectively. In the right-hand sides of (3.4) and (3.6), the Fourier coefficients of K− i (z)K+ i ( C2z ) are infi- nite series. Therefore the relations require some justification. We proceed as follows. We define a Z grading of generators by degEi,k = k, degHi,±r = ±r, degK± i,0 = degC = 0. We call it the homogeneous grading. Consider the free algebra A generated by Ei,k, Hi,±r, K± i,0, C ±1. Let Ã be the completion of A with respect to the grading in the positive direction. Then we define Kn to be the graded quotient algebra of Ã under the relations (3.1)–(3.6). The quadratic relations (3.4) imply di,j ĝi,j(z, w)Ei(z)Ej(w) + ĝj,i(w, z)Ej(w)Ei(z) = 0, (3.7) where ĝi,j(z, w) = {( z − C2w )( z − C−2w )( z − q2w ) , i = j, gi,j(z, w), i ̸= j. The factor ( z−C2w )( z−C−2w ) for i = j is chosen to kill the delta functions in the right-hand side of (3.4). Remark 3.1. Algebra Kn has a filtration defined by pdegEi(z) = 1, pdegK± i (z) = 0 for all i, which we call the principal filtration. In the associated graded algebra, the subalgebra genera- ted by {Ei(z)} is isomorphic to the subalgebra of the quantum toroidal algebra En generated by {ei(z)}. The relations for K± i (z) are slightly different from those of ψ± i (z) in En, since the right-hand side of (3.2) contains a product of Gi,j ’s as opposed to the ratio. In particular, while the elements ψ± i,0 ∈ En are inverse to each other, the elements K± i,0 ∈ Kn are not mutually commutative, K+ i,0K − j,0 = q2(ᾱi,ᾱj)K− j,0K + i,0. We have the relations [Hi,r, Hi,s] = −δr+s,0 · ai,j(r) 1 + C2r q − q−1 , r > 0, which are to be compared with (2.1). In the limit q → 1 with C = qc, algebra En reduces to the enveloping algebra of a Lie algebra of matrix valued difference operators, see [3, Section 3.7]. In contrast, algebra Kn does not seem to have a reasonable limit due to the presence of the sum 1 + C2r. 3.2 Wheel conditions All representations of En or Kn considered in this paper are admissible representations. In this setting, we rewrite the formal series relations of generating currents in the language of matrix coefficients. Let V =⊕k0 k=−∞Vk be a graded vector space with degrees bounded from above and dimVk<∞. We denote by V ∗ = ⊕k0 k=−∞V ∗ k the restricted dual space. Suppose that we are given a set of formal series { Ei(z) = ∑ k∈ZEi,kz −k \| i ∈ Z/nZ } , such that Ei,k ∈ EndV , Ei,kVl ⊂ Vk+l, 8 B. Feigin, M. Jimbo and E. Mukhin satisfying the relations (3.5) and (3.7) with some C ∈ C×. We assume that the zeros of ĝi,i(z, w) are distinct: C2 ̸= ±1, q±2. It follows from (3.5) and (3.7) that all matrix coefficients ⟨w,Ei1(z1) · · ·EiN (zN )v⟩, v ∈ V, w ∈ V ∗, converge in the region \|z1\| ≫ · · · ≫ \|zN \| to rational functions which have at most simple poles at ĝir,is(zr, zs) = 0, where r < s and ir ≡ is, is ± 1. Quite generally, let A1(z), A2(z) be operator-valued formal series and ri,j(x) be rational functions. We shall say that an exchange relation r1,2(z2/z1)A1(z1)A2(z2) = r2,1(z1/z2)A2(z2)A1(z1) holds as rational functions, if an arbitrary matrix coefficient of each side converges (in regions \|z1\| ≫ \|z2\| and \|z2\| ≫ \|z1\| respectively) to the same rational function. Similarly we shall use the term “exchange relation as meromorphic functions”. Equations (3.5) and (3.7) imply that we have relations as rational functions λ0i,j(w/z)Ei(z)Ej(w) = λ0j,i(z/w)Ej(w)Ei(z), (3.8) where we set λ0i,i(x) = 1− C2x 1− x 1− C−2x 1− x 1− q2x 1− x , λ0i,i±1(x) = d∓1/2 1− q−1d±1x 1− x , and λ0i,j(x) = 1 for j ̸≡ i, i± 1. Set Ei1,...,iN (z1, . . . , zN ) = ∏ 1≤r<s≤N λ0ir,is(zs/zr)Ei1(z1) · · ·EiN (zN ). (3.9) It follows from (3.8) that Eiσ(1),...,iσ(N) (zσ(1), . . . , zσ(N)) = Ei1,...,iN (z1, . . . , zN ), σ ∈ SN . (3.10) In particular (3.9) is symmetric with respect to zr and zs when they have the same “color” ir = is. Moreover all matrix coefficients of (3.9) have the form P (z1, . . . , zN )∏ r<s(zr − zs)Nr,s , P (z1, . . . , zN ) ∈ C [ z±1 1 , . . . , z±1 N ] , (3.11) where Nr,s = 2 if ir = is, Nr,s = 1 if ir ≡ is ± 1, and Nr,s = 0 otherwise. Note that, when ir = is, third order poles are absent due to the symmetry in variables of the same color. The following is easy to see. Proposition 3.2. Under the conditions (3.5) and (3.7), the condition (3.4) holds if and only if Ei,i ( z, C2z ) = 1 q − q−1 1 + C2 1− C2 1− q2C2 1− C2 K− i (z)K+ i ( C2z ) . (3.12) We have also Quantum Toroidal Comodule Algebra of Type An−1 and Integrals of Motion 9 Proposition 3.3. Under the conditions (3.5) and (3.7), the Serre relations (3.6) are satisfied if and only if the following zero conditions hold: Ei,i,i±1 ( z, q2z, qd∓1z ) = 0, (3.13) Ei,i,i±1 ( z, C2z, C2q−1d∓1z ) = 0. (3.14) If it is the case then we have also Ei,i,i±1 ( z, C−2z, C−2qd∓1z ) = 0. (3.15) We call (3.13)–(3.15) “wheel conditions”. Proof. The left-hand side and the right-hand side of the Serre relations (3.6) read respecti- vely as LHS = Sym z1,z2 ( Ei(z1)Ei(z2)Ei+1(w)− ( q+q−1 ) Ei(z1)Ei+1(w)Ei(z2)+Ei+1(w)Ei(z1)Ei(z2) ) , RHS = − Sym z1,z2 ( − q−1 3 w/z1 1− q−1 3 w/z1 + q1w/z2 1− q1w/z2 + δ(q3z1/w) ) δ ( C2z1/z2 ) K− i (z1) × Ei+1(w)K + i (z2). The case involving Ei(z1), Ei(z2) and Ei−1(w) can be treated by interchanging q3 with q1. Using the definition (3.9), LHS can be rewritten as LHS = Ei,i,i+1(z1, z2, w) ( 1 λ0i,i ( z2 z1 ) λ0i,i+1 ( w z1 ) λ0i,i+1 ( w z2 ) − q + q−1 λ0i,i ( z2 z1 ) λ0i,i+1 ( w z1 ) λ0i+1,i ( z2 w ) + 1 λ0i,i ( z2 z1 ) λ0i+1,i ( z1 w ) λ0i+1,i ( z2 w ) + 1 λ0i,i ( z1 z2 ) λ0i,i+1 ( w z1 ) λ0i,i+1 ( w z2 ) − q + q−1 λ0i,i ( z1 z2 ) λ0i+1,i ( z1 w ) λ0i,i+1 ( w z2 ) + 1 λ0i,i( z1 z2 )λ0i+1,i ( z1 w ) λ0i+1,i ( z2 w )). The matrix coefficients of Ei,i,i+1(z1, z2, w) are Laurent polynomials up to poles on z1 = z2 or zi = w, which are cancelled by the zeros of 1/λ0i,j . This is a sum of six formal series, each expanded in a different region. In order to bring them to expansions in a common region, we use the identity 1/(1− z) = δ(z)− z−1/ ( 1− z−1 ) , where 1/(1 − z) stands for its expansion in non-negative powers of z, while z−1/ ( 1 − z−1 ) means the one in negative powers of z. In the underlined factors, we substitute 1 λ0i+1,i ( z w ) = (expansion in \|z\| ≫ \|w\|) + d−1/2 ( 1− q−1 3 ) δ(q3z/w), 1 λ0i,i ( z1 z2 ) = (expansion in \|z1\| ≫ \|z2\|)− q−2 ( 1− q2 )3( 1− C−2q2 )( 1− C2q2 )δ(q2z1/z2) + 1− C2 1 + C2 1− C2 1− q2C2 δ ( C2z2/z1 ) + 1− C−2 1 + C−2 1− C−2 1− q2C−2 δ ( C−2z2/z1 ) and bring all terms into the sum of their expansions in \|z1\| ≫ \|z2\| ≫ \|w\| and additional delta functions. We then compare the coefficients of each product of delta functions. Up to the symmetry z1 ↔ z2, there are the following cases to consider: (1) 1, (2) δ(q3z1/w), (3) δ ( q2z1/z2 ) , 10 B. Feigin, M. Jimbo and E. Mukhin (4) δ ( C2z2/z1 ) , (5) δ(q3z1/w)δ ( q2z1/z2 ) , (6) δ(q3z1/w)δ ( C2z2/z1 ) , (7) δ(q3z1/w)δ ( C−2z2/z1 ) , (8) δ(q3z1/w)δ(q3z2/w). In cases (1), (2) and (3), the coefficients vanish due to the identities of rational functions 0 = Sym z1,z2 { 1 λ0i,i ( z2 z1 ) λ0i,i+1 ( w z1 ) λ0i,i+1 ( w z2 ) − q + q−1 λ0i,i ( z2 z1 ) λ0i,i+1 ( w z1 ) λ0i+1,i ( z2 w ) + 1 λ0i,i ( z2 z1 ) λ0i+1,i ( z1 w ) λ0i+1,i ( z2 w )}, 0 = 1 λ0i,i(z2/z1) 1 λ0i+1,i(z2/w) − ( q+q−1 ) 1 λ0i,i(z1/z2) 1 λ0i,i+1(w/z2) + 1 λ0i,i(z1/z2) 1 λ0i+1,i(z2/w) , 0 = 1 λ0i,i+1(w/z1) 1 λ0i,i+1(w/z2) − ( q + q−1 ) 1 λ0i+1,i(z1/w) 1 λ0i,i+1(w/z2) + 1 λ0i+1,i(z1/w) 1 λ0i+1,i(z2/w) . In case (8) the coefficient vanishes on z1 = z2 due to the triple zero of λ0i,i(z1/z2) −1 there. In case (4), the relevant terms are the 3 terms containing λ0i,i(z1/z2) −1 . Using the rela- tion (3.12), we find that the coefficient of δ ( C2z2/z1 ) is 1− C2 1 + C2 1− C2 1− q2C2 Ei,i ( C2z2, z2 ) Ei+1(w)λ 0 i,i+1(w/z1)λ 0 i,i+1(w/z2) × ( 1 λ0i,i+1(w/z1) 1 λ0i,i+1(w/z2) − ( q + q−1 ) 1 λ0i+1,i(z1/w) 1 λ0i,i+1(w/z2) + 1 λ0i+1,i(z1/w) 1 λ0i+1,i(z2/w) ) = − d(w/z1 − q2w/z2)( 1− q−1 3 w/z1 )( 1− q−1 3 w/z2 )K− i (z2)K + i ( C2z2 ) Ei+1(w), which coincides with the corresponding term in RHS. Likewise, in case (7) the coefficient of δ ( C2z2/z1 ) δ ( q−1 3 w/z1 ) comes from Ei,i,i+1(z1, z2, w) 1 λ0i,i(z1/z2) 1 λ0i+1,i(z2/w) ( − ( q + q−1 ) λ0i,i+1(w/z2) + λ0i+1,i(z2/w) ) . Computing similarly we find that it matches with the corresponding term in RHS. It remains to consider (5) and (6). The coefficients of the product of delta functions are proportional (with non-zero multipliers) to the left-hand side of (3.13) and (3.14), respectively. Since there are no terms that come from RHS, we have shown that the Serre relations are equivalent to (3.13) and (3.14). Instead of rewriting terms into expansions in \|z1\| ≫ \|z2\| ≫ \|w\|, one can equally well proceed to the opposite region \|z1\| ≪ \|z2\| ≪ \|w\|. Computing in the same way we obtain (3.15) in place of (3.14). ■ 3.3 Comodule structure In this section we show that, when we restrict to admissible modules, Kn may be viewed as a left comodule of En. To be precise we prove the following.2 2Formula for ∆ is slightly changed from the one used in [7]. The E1 comodule structure discussed there should be understood at the level of admissible representations. Quantum Toroidal Comodule Algebra of Type An−1 and Integrals of Motion 11 Theorem 3.4. Let V be an admissible En module and W an admissible Kn module, such that C acts as scalar C1, C2 respectively. We set C12 = C1C2 and assume that C2 2 ̸= ±1, q±2, C2 12 ̸= ±1, q±2. Then V ⊗W is given the structure of an admissible Kn module by the action of the following currents: ∆Ei(z) = ei(C2z)⊗K− i (z) + 1⊗ Ei(z) + f̃i ( C−1 2 z ) ⊗K+ i (z), (3.16) ∆K+ i (z) = ψ+ i ( C−1 1 C−1 2 z )−1 ⊗K+ i (z), (3.17) ∆K− i (z) = ψ− i (C2z)⊗K− i (z), (3.18) ∆C = C12. (3.19) Moreover the following “coassociativity” and “counit property” hold: (V1 ⊗ V2)⊗W = V1 ⊗ (V2 ⊗W ), C⊗W =W. Here V1, V2 are admissible En modules and C denotes the trivial En module. Proof. On each vector of an admissible module, the currents of En or Kn comprise only a finite number of negative powers in z. Hence each Fourier coefficient of (3.16)–(3.18) is a well-defined operator on V ⊗W . It is easy to verify the relations (3.1)–(3.3) and (3.5) by direct calculation. With the aid of (2.2), one can also check (3.4) for i ̸= j. Let us check (3.4) for i = j. We shall write A1(z, w) ≡ A2(z, w) if each matrix coefficient of A1(z, w)−A2(z, w) is a Laurent polynomial. Then we have on V gi,j(z, w)ei(z)ej(w) ≡ 0, gi,j(z, w)f̃i(z)f̃j(w) ≡ 0, (3.20) ei(z)f̃j(w) ≡ δi,j q − q−1 ( − 1 1− w/z + 1 1− C−2w/z ψ− i (z)ψ + i ( C−1w )−1 ) , (3.21) and on W gi,j(z, w)Ei(z)Ej(w) ≡ δi,j q − q−1 ( z − q2w 1− C−2w/z K− i (z)K+ i (w) + w − q2z 1− C2w/z K− i (w)K+ i (z) ) . (3.22) Using these we obtain( q − q−1 )( z − q2w ) ∆Ei(z)∆Ei(w) ≡ ( q − q−1 )( z − q2w )( ei(C2z)f̃i ( C−1 2 w ) ⊗K− i (z)K+ i (w) + f̃i ( C−1 2 z ) ei(C2w)⊗K+ i (z)K− i (w) + 1⊗ Ei(z)Ei(w) ) ≡ − ( z − q2w 1− C−2 2 w/z − z − q2w 1− C−2 12 w/z ψ− i (C2z)ψ + i ( C−1 12 w )−1 ) ⊗K− i (z)K+ i (w) + ( w − q2z 1− C−2 2 z/w − w − q2z 1− C−2 12 z/w ψ− i (C2w)ψ + i ( C−1 12 z )−1 ) ⊗K− i (w)K+ i (z) + z − q2w 1− C−2 2 w/z 1⊗K− i (z)K+ i (w) + w − q2z 1− C2 2w/z 1⊗K− i (w)K+ i (z) 12 B. Feigin, M. Jimbo and E. Mukhin ≡ z − q2w 1− C−2 12 w/z ψ− i (C2z)ψ + i ( C−1 12 w )−1 ⊗K− i (z)K+ i (w) − w − q2z 1− C−2 12 z/w ψ− i (C2w)ψ + i ( C−1 12 z )−1 ⊗K− i (w)K+ i (z). Interchanging z with w and summing these, we arrive at (3.4) for i = j. Next let us verify the Serre relations (3.6). By Proposition 3.3, it suffices to show that the cur- rent ∆Ei,i,i±1(z1, z1, w) satisfies the wheel conditions (3.13)–(3.14). It is comprised of 27 terms which we group together according to the principal grading in the first component, defined by pdeg ei(z) = 1, pdeg fi(z) = −1, pdegψ± i (z) = pdegC = 0. For the terms of principal degrees ±3 and ±2, the wheel conditions are easily checked. Let us verify the case of principal degree 1. There are 6 terms coming from ∆Ei,i,i±1(z1, z2, w): A1 = ϕ ei(C2z1)ei(C2z2)f̃i±1 ( C−1 2 w ) ⊗K− i (z1)K − i (z2)K + i±1(w), A2 = ϕ ei(C2z1)ei±1(C2w)f̃i ( C−1 2 z2 ) ⊗K− i (z1)K − i±1(w)K + i (z2)Gi,i±1(w/z2), A3 = ϕ ei(C2z2)ei±1(C2w)f̃i ( C−1 2 z1 ) ⊗K− i (z2)K − i±1(w)K + i (z1)Gi,i(z2/z1)Gi,i±1(w/z1), A4 = ϕ ei(C2z1)⊗K− i (z1)Ei(z2)Ei±1(w), A5 = ϕ ei(C2z2)⊗K− i (z2)Ei(z1)Ei±1(w)Gi,i(z2/z1), A6 = ϕ ei±1(C2w)⊗K− i±1(w)Ei(z1)Ei(z2)Gi,i±1(w/z1)Gi,i±1(w/z2), where ϕ = z1 − C2 12z2 z1 − z2 z1 − C−2 12 z2 z1 − z2 z1 − q2z2 z1 − z2 z1 − q−1d±1w z1 − w z2 − q−1d±1w z1 − w . From (3.20)–(3.22), we see that the sum ∑6 i=1Ai has no poles other than z1 = z2, zi = w. In fact, the only possible poles at z2 = C2 2z1 (resp. z2 = C−2 2 z1) arise from A2 and A6 (resp., A3 with A6), and they cancel by virtue of (3.21) and (3.22). Let us check (3.13). Under the specialization z2 = q2z1, w = qd∓1z1 = q−1d∓1z2, each term has a zero due to the vanishing factors( z1 − q2z2 ) Gi,i(z2/z1), z1 − q−1d±1w, ( z2 − q−1d±1w ) Gi,i±1(w/z2). To check (3.14), set z2 = C2 12z1. All Ai’s vanish at z2 = C2 12z1 with the exception of A2. Since the latter has the factor ( z2 − q−1d±1w ) Gi,i±1(w/z2), it also vanishes by setting further w = q−1d∓1z2. In the case of principal degree 0, there are 7 terms: B0 = ϕ 1⊗ Ei(z1)Ei(z2)Ei±1(w), B1 = ϕ ei(C2z1)f̃i ( C−1 2 z2 ) ⊗K− i (z1)Ei±1(w)K + i (z2)Gi,i±1(w/z2), B2 = ϕ ei(C2z2)f̃i ( C−1 2 z1 ) ⊗K− i (z2)Ei±1(w)K + i (z1)Gi,i(z2/z1)Gi,i±1(w/z1), B3 = ϕ ei(C2z1)f̃i±1 ( C−1 2 w ) ⊗K− i (z1)Ei(z2)K + i±1(w), B4 = ϕ ei(C2z2)f̃i±1 ( C−1 2 w ) ⊗K− i (z2)Ei(z1)K + i±1(w)Gi,i(z2/z1), B5 = ϕ ei±1(C2w)f̃i ( C−1 2 z2 ) ⊗K− i±1(w)Ei(z1)K + i (z2)Gi,i±1(w/z1)Gi,i±1(w/z2), B6 = ϕ ei±1(C2w)f̃i ( C−1 2 z1 ) ⊗K− i±1(w)Ei(z2)K + i (z1)Gi,i(z2/z1)Gi,i±1(w/z1)Gi,i±1(w/z2). Again possible poles at z2 = C±2 2 z1 are cancelled among B0, B1, B2 and the sum ∑6 i=0Bi has no poles other than z1 = z2, zi = w. Quantum Toroidal Comodule Algebra of Type An−1 and Integrals of Motion 13 Consider z2 = q2z1, w = qd∓1z1 = q−1d∓1z2. By the assumption on C2 2 , B0 vanishes. The rest of the terms are zero for the same reason as in principal degree 1. Next let z2 = C2 12z1. The only term which may survive is B1. Due to the factor ( z2 − q−1d±1w ) Gi,i±1(w/z2) it also vanishes by setting further w = q−1d∓1z2. The case of principal degree −1 being similar to principal degree 1, we omit the details. Finally the coassociativity and counit property can be readily checked by using (3.16)–(3.19). ■ 3.4 Representations Theorem 3.4 allows us to construct a large family of admissible representations of Kn starting from simple ones. Algebra Kn has a Heisenberg subalgebra Hn generated by Hi,r, i ∈ Z/nZ, r ̸= 0, K± i,0, and C±1. Let FD be an irreducible representation of Hn on which C acts as q−1. Lemma 3.5. With the following action, FD is an admissible Kn module: Ei(z) 7→ 0, K± i (z) 7→ K± i (z), C 7→ q−1. Proof. The defining relations are obviously satisfied except (3.4). To see the latter it suffices to note that gi,i ( z, C2z ) = 0 when C = q−1. ■ In the next case we need to adjoin ( K± i,0 )1/2 to Hn and Kn. We define K̃± i (z) by K̃± i (z)K̃± i (qz) = K± i (z). Let FB be an irreducible representation of Hn on which C acts as q1/2. Lemma 3.6. With the following action, FB is an admissible Kn module: Ei(z) 7→ kK̃− i (z)K̃+ i (qz), K± i (z) 7→ K± i (z), C 7→ q1/2, where k = 1/(q1/2 − q−1/2). Proof. We can write Gi,j(x) = G̃i,j(x)G̃i,j(q −1x), where G̃i,i(x) = q 1− q−1x 1− x 1− qx 1− q2x , G̃i,i±1(x) = q−1/2 1− qd±1x 1− d±1x , and G̃i,j(x) = 1 otherwise. The assertion can be checked by noting K̃+ i (z)K̃− j (w) = K̃− j (w)K̃+ i (z)G̃i,j(w/z). We use also Ei,i(z, w) 7→ qk2K̃− i (z)K̃− i (w)K̃+ i (qz)K̃+ i (qw) 1− q−2w/z 1− w/z 1− q2w/z 1− w/z and Proposition 3.2 to check the coefficients of delta functions in (3.4), and use Proposition 3.3 along with Ei,i,i±1(z1, z2, w) 7→ k3K̃− i (z1)K̃ − i (z2)K̃ − i±1(w)K̃ + i (qz1)K̃ + i (qz2)K̃ + i±1(qw) × 1− q−2z2/z1 1− z2/z1 1− q2z2/z1 1− z2/z1 1− d±1w/z1 1− w/z1 1− d±1w/z2 1− w/z2 for the Serre relations (3.6). ■ 14 B. Feigin, M. Jimbo and E. Mukhin Together with Theorem 3.4, the above lemmas imply Corollary 3.7. Let V be an admissible En module, where C acts as scalar C1 with q−2C2 1 ̸= ±1, q±2. Then V ⊗ FD is an admissible Kn module with the action ∆Ei(z) = ei ( q−1z ) ⊗K− i (z) + f̃i(qz)⊗K+ i (z), ∆K+ i (z) = ψ+ i ( qC−1 1 z )−1 ⊗K+ i (z), ∆K− i (z) = ψ− i ( q−1z ) ⊗K− i (z), ∆C = C1q −1. Proof. Since C2 2 = q−2, Theorem 3.4 does not literally apply. Nevertheless, in the proof there, the only place where this matters is the term B0 in degree 0. Since this term is identically zero in FD, the same proof works. ■ Corollary 3.8. Let V be an admissible En module, where C acts as scalar C1 with qC2 1 ̸= ±1, q±2. Then V ⊗ FB is an admissible Kn module with the action ∆Ei(z) = ei ( q1/2z ) ⊗K− i (z) + 1⊗ kK̃− i (z)K̃+ i (qz) + f̃i ( q−1/2z ) ⊗K+ i (z), ∆K+ i (z) = ψ+ i ( q−1/2C−1 1 z )−1 ⊗K+ i (z), ∆K− i (z) = ψ− i ( q1/2z ) ⊗K− i (z), ∆C = C1q 1/2, where K̃± i (z) and k are as in Lemma 3.6. Example 3.9. Consider the Kn module F(u1)⊗ · · · ⊗ F(uℓ)⊗ FD, ℓ ≥ 3. The central element acts as C2 = q2ℓ−2. The current Ei(z) is represented as a sum of 2ℓ vertex operators: z−1Ei(z) = − ℓ∑ k=1 qk−ℓukΛi,k(z) + ℓ∑ k=1 q−k+ℓ−1u−1 k Λi,k̄(z), (3.23) where Λi,k(z) = k−1︷︸︸︷ 1⊗ · · · ⊗ 1⊗:Vi ( qk−ℓ+1z )−1 :⊗ ψ+ i ( qk−ℓ+1z )−1 ⊗ · · · ⊗ ψ+ i (z) −1 ⊗K+ i (z), Λi,k̄(z) = k−1︷︸︸︷ 1⊗ · · · ⊗ 1⊗Vi ( q−k+ℓ−1z ) ⊗ ψ− i ( q−k+ℓ−2z ) ⊗ · · · ⊗ ψ− i ( q−1z ) ⊗K− i (z). In the right-hand side, ψ± i (z) or K ± i (z) stand for their action on F(ui) or FD, respectively. In what follows we introduce an ordering 1≺ · · ·≺ ℓ≺ ℓ̄≺ · · ·≺ 1̄ to the set {1, . . . , ℓ, ℓ̄, . . . , 1̄}. We use letters a, b, . . . for elements of the latter, with the convention that ¯̄a = a. Each Λi,a(z) is a product of oscillator part Λosc i,a (z) and the zero mode part. The contractions of the former are given by the following table. Table 1. Contractions Λosc i,a (z)Λ osc j,b (w) with x = w/z a ≺ b b ̸= ā a = b a ≻ b b ̸= ā a ≺ b b = ā a ≻ b b = ā i ≡ j 1−q−1 2 x 1− q2x (1−x)(1−q−1 2 x) 1 1− q−1 2 x 1− q2x × 1 (1!−C2q1−a 2 x)(1−C2q2−a 2 x) 1 (1−C−2qb2x)(1−C−2qb−1 2 x) i+1≡j 1−q−1 3 x 1− q1x 1 1− q1x 1 1− q−1 3 x 1− q1x (1− C2q1−a 2 q1x) 1− C−2qb−1 2 q−1 3 x Quantum Toroidal Comodule Algebra of Type An−1 and Integrals of Motion 15 The contractions in the case i− 1 ≡ j can be obtained by switching q1 and q3. Using this table one can check directly the wheel conditions, namely that Ei,i,i+1(z1, z2, w) represented by 1− C2z2/z1 1− z2/z1 1− C−2z2/z1 1− z2/z1 1− q2z2/z1 1− z2/z1 1− q1w/z1 1− w/z1 1− q1w/z2 1− w/z2 Λi,a(z1)Λi,b(z2)Λi+1,c(w) vanishes for all a, b, c ∈ {1, . . . , ℓ, ℓ, . . . , 1̄} under the following specializations: (i) (z1, z2, w) = ( z, q2z, q −1 1 z ) , (ii) (z1, z2, w) = ( z, C2z, C2q3z ) , (iii) (z1, z2, w) = ( z, C−2z, C−2q−1 1 z ) . Case (i). This follows from Λi,a(z1)Λi,b(z2) ∣∣ z2=q2z1 = 0 if a ⪯ b, Λi,b(z2)Λi+1,c(w) ∣∣ w=q3z2 = 0 if b ≺ c, (1− q1w/z1)Λi,a(z1)Λi+1,c(w) ∣∣ w=q−1 1 z1 = 0 if a ≻ c. Case (ii). This follows from( 1− C−2z2/z1 ) Λi,a(z1)Λi,b(z2) ∣∣ z2=C2z1 = 0 if (a, b) ̸= (1̄, 1), Λi,1(z2)Λi+1,c(w) ∣∣ w=q3z2 = 0 if c ̸= 1, Λi,1̄(z1)Λi+1,1(w) ∣∣ w=C2q3z1 = 0. Case (iii). This follows from( 1− C2z2/z1 ) Λi,a(z1)Λi,b(z2) ∣∣ z2=C−2z1 = 0 if (a, b) ̸= (1, 1̄), (1− q1w/z2)Λi,1̄(z2)Λi+1,c(w) ∣∣ w=q−1 1 z2 = 0 if c ̸= 1̄, Λi,1(z1)Λi+1,1̄(w) ∣∣ w=C−2q−1 1 z1 = 0. Similarly for Ei,i,i−1(z1, z2, w). Remark 3.10. As it turns out, the currents (3.23) commute with a system of screening operators derived from quantum toroidal algebra of type D. We hope to address this point elsewhere. 4 Integrals of motion We continue to work with admissible representations of Kn on which C acts as a scalar. Our goal in this section is to introduce a family of commuting operators acting on each such represen- tation (with mild restrictions on parameters), which we call integrals of motion (IMs). Loosely speaking, we may think of them as generators of a commutative subalgebra of Kn. The IMs will depend on q, d, C as well as additional arbitrary parameters µ1, . . . , µn ∈ C, n∑ i=1 µi = 0. (4.1) 16 B. Feigin, M. Jimbo and E. Mukhin In addition, they depend also on an “elliptic” parameter p. However, our experience with the algebra K1 tells that the IMs exist only when p and C are related in a special way. Guided by the result in [7], we shall impose the relation C2 = pq2 and make the following assumption: \|p\| < 1, ∣∣qd±1 ∣∣ < 1, ∣∣pq−2 ∣∣ < 1, ∣∣pq−1d±1 ∣∣ < 1. (4.2) In particular \|q2\| < 1. We use parameters τ ∈ C with Im τ > 0 and β, γ ∈ C given by p = e−2πi/τ , q2 = pγ , d = pβ. (4.3) We use also θ(u) = pu 2/2−u/2Θp(p u), which satisfies θ(u+ 1) = −θ(u), θ(u+ τ) = −e−2πiu−πiτθ(u). (4.4) 4.1 Dressed currents A basic constituent of IMs is the dressed currents, obtained by modifying the currents Ei(z) of Kn. In the following we define operators Hi,0 by (K− i,0) −1K+ i,0 = q2Hi,0 . The dressed currents of Kn are defined by Ei(z) = K− i (z) −1Ei(z)z −γHi,0−γ+µi , K− i (z) = exp ( − ( q − q−1 )∑ r>0 Hi,−r 1− pr zr ) , where γ and µi are as in (4.3) and (4.1). The dressed currents have well-defined matrix coefficients on admissible representations. They satisfy the exchange relations as meromorphic functions λi,j(z2/z1)Ei(z1)Ej(z2) = λj,i(z1/z2)Ej(z2)Ei(z1), with λi,i(x) = x−γ 1− C2x 1− x 1− C−2x 1− x 1− q2x 1− x ( q−2x; p ) ∞ (q2x; p)∞ , λi,i±1(x) = xγ/2d∓1/2 1− q−1d±1x 1− x ( qd±1x; p ) ∞( q−1d±1x; p ) ∞ , and λi,j(x) = 1 in all other cases. Up to a power of x, λi,j(x) specializes to λ 0 i,j(x) at p = 0. Each matrix coefficient of products of dressed currents is a meromorphic function (up to an overall power function). We have “elliptic” exchange relations as meromorphic functions Ei(z)Ei(w) = Ei(w)Ei(z) θ(v − u+ γ) θ(v − u− γ) , Ei(z)Ei±1(w) = Ei±1(w)Ei(z)q ±2β θ(v − u± β − γ/2) θ(v − u± β + γ/2) , where z = pu, w = pv. Quantum Toroidal Comodule Algebra of Type An−1 and Integrals of Motion 17 As before we define Ei1,...,iN (z1, . . . , zN ) = ∏ 1≤r<s≤N λir,is(zs/zr)Ei1(z1) · · ·EiN (zN ). (4.5) Then it has the same symmetry as (3.10), and we have Ei1,...,iN (z1, . . . , zN ) = ∏ r<s z −γ(ᾱir ,ᾱis ) s N∏ s=1 K− is (zs) −1Ei1,...,iN (z1, . . . , zN ) N∏ s=1 z −γHis,0−γ+µis s . The product ∏N s=1K − is (zs) −1 contains only {Hi,−r} with r > 0. Its contribution to any matrix coefficient of Ei1,...,iN (z1, . . . , zN ) is a Laurent polynomial in zs’s for degree reasons. Hence the equality above shows that, up to an overall power of zi’s, each matrix coefficient of operator (4.5) has the same functional form (3.11) as that of Ei1,...,iN (z1, . . . , zN ). In particular the wheel conditions (3.13)–(3.15) are valid also for (4.5). 4.2 Integrals of motion Given parameters (4.1), let Xn = Xn(µ1, . . . , µn) denote the space of entire functions ϑ(u1, . . . , un) satisfying ϑ(u1, . . . , ui + 1, . . . , un) = ϑ(u1, . . . , ui, . . . , un), (4.6) ϑ(u1, . . . , ui + τ, . . . , un) = e−2πi(2ui−ui−1−ui+1−µi+τ) ϑ(u1, . . . , ui, . . . , un), (4.7) where ui+n = ui. We have dimXn = n. For M ≥ 1 and ϑ ∈ Xn, we introduce the kernel functions hM (u1, . . . ,un;ϑ) = ϑ(u1, . . . , un) ∏n i=1 ∏ 1≤a<b≤M θ(ui,a − ui,b)θ(ui,a − ui,b − γ)∏n−1 i=1 ∏ 1≤a,b≤M θ ( ui,a − ui+1,b − β − γ 2 )∏ 1≤a,b≤M θ ( un,a − u1,b − β + γ 2 ) , (4.8) where ui = (ui,1, . . . , ui,M ), ui = M∑ a=1 ui,a. We define the integrals of motion GM (ϑ) of Kn by GM (ϑ) = ∫ · · · ∫ ↶∏ 1≤i≤n ↶∏ 1≤a≤M Ei(xi,a) · hM (u1, . . . ,un;ϑ) n∏ i=1 M∏ a=1 dxi,a xi,a . (4.9) Here xi,a = pui,a , and the symbol ∏↶ 1≤i≤N stands for ordered product ↶∏ 1≤i≤N Ai = ANAN−1 · · ·A1. Even though individual factors contain fractional powers, the integrand of (4.9) comprises only integer powers with respect to each xi,a. The integrals are taken over a common circle \|xi,a\| = R. The result is independent of the choice of R > 0. Formally, (4.9) looks identical to that of the integrals of motion of type A [5, 8, 12]. In par- ticular the kernel functions (4.8) are the same as those used there. The difference is hidden in the pole structure of products of dressed currents. 18 B. Feigin, M. Jimbo and E. Mukhin For later use we rewrite the integrand in terms of the current in (4.5), E(M)(x1, . . . ,xn) = E1,...,1,...,n,...,n(x1, . . . ,xn), where xi = (xi,1, . . . , xi,M ). In the right-hand side, each index i occurs M times. Proposition 4.1. Set x = pu and ξ(u) = pu 2/2(1− x) ( x, p2q2x, p; p ) ∞, η(u) = pu 2/2−(β+1/2)u d−1/2 1− x−1 ( qd−1x, qdx−1, p; p ) ∞. Then the IM’s can be expressed as GM (ϑ) = cM ∫ · · · ∫ E(M)(x1, . . . ,xn) kM (u1, . . . ,un;ϑ) n∏ i=1 M∏ a=1 dxi,a xi,a , (4.10) where cM is a constant, and kM (u1, . . . ,un;ϑ) = ϑ(u1, . . . , un) ∏n i=1 ∏ 1≤a̸=b≤M ξ(ui,a − ui,b)∏n i=1 ∏ 1≤a,b≤M η(ui,a − ui+1,b) , xi,a = pui,a , ui = M∑ a=1 ui,a. Here and after, we set un+1,a = u1,a. Proof. This follows from the relations λi,i(p −u) λi,i(pu) = θ(u+ γ) θ(u− γ) , λi,i∓1(p −u) λi,i±1(pu) = q±2β θ(u± β − γ/2) θ(u± β + γ/2) , λi,i(p u)ξ(u)ξ(−u) = −p−γ2/2−3γ/2−1θ(u)θ(u− γ), λi,i±1 ( p∓u ) θ(u± γ/2− β) = p(γ/2∓β)2/2−(γ/2−β)/2 η(u). ■ 4.3 Commutativity We are now in a position to state the commutativity of IMs. Theorem 4.2. For all M,N ≥ 1 and ϑ1, ϑ2 ∈ Xn, the integrals of motion mutually commute: [GM (ϑ1),GN (ϑ2)] = 0. Theorem 4.2 is the main result of this paper. The rest of this section is devoted to its proof. Proof. We start by examining the product of operators, dropping irrelevant constants GM (ϑ1) = ∫ · · · ∫ C1 E(M)(x1, . . . ,xn) kM (u1, . . . ,un;ϑ1) n∏ i=1 M∏ a=1 dxi,a xi,a , GN (ϑ2) = ∫ · · · ∫ CR E(N)(y1, . . . ,yn) kN (v1, . . . ,vn;ϑ2) n∏ i=1 N∏ b=1 dyi,b yi,b . Here xi,a = pui,a , yi,b = pvi,b . We choose the contours to be the unit circle C1 : \|xi,a\| = 1 for GM (ϑ1), and a circle CR : \|yi,b\| = R of radius R for GN (ϑ2). Quantum Toroidal Comodule Algebra of Type An−1 and Integrals of Motion 19 Consider first the product GN (ϑ2)GM (ϑ1) = ∫ · · · ∫ CR ∫ · · · ∫ C1 E(N)(y1, . . . ,yn)E (M)(x1, . . . ,xn) × kN (v1, . . . ,vn;ϑ2) kM (u1, . . . ,un;ϑ1) n∏ i=1 N∏ b=1 dyi,b yi,b n∏ i=1 M∏ a=1 dxi,a xi,a . (4.11) We choose R≫ 1 to ensure that the product of operators in the integrand converges absolutely. Introducing the combined variables zi = (zi,1, . . . , zi,M+N ), where zi,a = xi,a, 1 ≤ a ≤ M , zi,a+M = yi,a, and ui,M+a = vi,a, 1 ≤ a ≤ N , and substituting E(N)(y1, . . . ,yn)E (M)(x1, . . . ,xn) = E(M+N)(z1, . . . , zn) n∏ i=1 ∏ 1≤a≤M 1≤b≤N λi,i(xi,a/yi,b) −1 n∏ i=1 ∏ 1≤a≤M 1≤b≤N λi+1,i(xi,a/yi+1,b) −1 × n∏ i=1 ∏ 1≤a≤M 1≤b≤N λi−1,i(xi,a/yi−1,b) −1, we find that the integrand of (4.11) can be written as a product of two factors ST , with S(z1, . . . , zn) = E(M+N)(z1, . . . , zn) n∏ i=1 ∏ 1≤a̸=b≤M+N ξ(ui,a − ui,b) n∏ i=1 ∏ 1≤a,b≤M+N η(ui,a − ui+1,b) −1, T (u1, . . . ,un\|v1, . . . ,vn) = ϑ1(u1, . . . , un)ϑ2(v1, . . . , vn) × n∏ i=1 ∏ 1≤a≤M 1≤b≤N θ(ui,a − vi+1,b − γ/2− β)θ(vi,b − ui+1,a + γ/2− β) θ(ui,a − vi,b)θ(ui,a − vi,b − γ) . (4.12) Starting from R ≫ 1, we bring the contours CR for yi,b to the unit circle. For that purpose we need to locate the position of poles between the two circles CR and C1. The poles between y’s and x’s come from ξ, η and θ: n∏ i=1 ∏ 1≤a≤M 1≤b≤N (1− xi,a/yi,b)(1− yi,b/xi,a) ( ��xi,a/yi,b, p 2q2xi,a/yi,b, 1−yi,b/xi,a ��yi,b/xi,a , p 2q2yi,b/xi,a; p ) ∞ × n∏ i=1 ∏ 1≤a≤M 1≤b≤N (1− yi+1,b/xi,a)(1− xi+1,a/yi,b)( qd−1xi,a/yi+1,b,((((((( qdyi+1,b/xi,a,(((((((( qd−1yi,b/xi+1,a, qdxi+1,a/yi,b; p ) ∞ × n∏ i=1 ∏ 1≤a≤M 1≤b≤N ( pq−1d−1xi,a/yi+1,b,((((((( qdyi+1,b/xi,a,(((((((( qd−1yi,b/xi+1,a, pq −1dxi+1,a/yi,b; p ) ∞( ��xi,a/yi,b,��pyi,b/xi,a, q−2xi,a/yi,b, pq2yi,b/xi,a; p ) ∞ . Under our assumption on parameters (4.2), we see that the only poles between CR and C1 are yi,b = q−1 2 xi,a, p −1q−1 2 xi,a, 1 ≤ i ≤ n, 1 ≤ a ≤M, 1 ≤ b ≤ N. (4.13) 20 B. Feigin, M. Jimbo and E. Mukhin Including the poles between y’s which come from η, poles of the integrand with respect to each yi,b are as follows (see the figure below): yi,b = p−kq1yi+1,c, p−kq3yi−1,c, k ≥ 0, yi,b = pkq−1 3 yi+1,c, pkq−1 1 yi−1,c, k ≥ 0, yi,b = pkq−1 2 xi,a, k ≥ −1, yi,b = pkq−1 1 xi−1,a, pkq−1 3 xi+1,a, k ≥ 0. • • pq−1 1 yi−1,c q −1 1 yi−1,c • • q3yi−1,c p−1q3yi−1,c • • pq−1 3 yi+1,c q −1 3 yi+1,c • • q1yi+1,c p−1q1yi+1,c • • • pq−1 2 xi,a q−1 2 xi,a p−1q−1 2 xi,a • • q−1 1 xi−1,a pq−1 1 xi−1,a • • pq−1 3 xi+1,a q−1 3 xi+1,a CRC1 Figure 1. The contour in the yi,b plane. We move the contour to the unit circle, picking up residues of poles (4.13) along the way. At first glance, taking residues in one variable yi,b = p−1q−1 2 xi,a or yi,b = q−1 2 xi,a seems to produce new poles in other variables, yi+1,c = q−1 2 q−1 1 xi,a, p−1q−1 2 q−1 1 xi,a, yi−1,c = q−1 2 q−1 3 xi,a, p−1q−1 2 q−1 3 xi,a. An important point is that these poles are cancelled by the zeros of E(M+N)(z1, . . . , zn) due to the wheel conditions (3.13)–(3.15). For the product in the opposite order GM (ϑ1)GN (ϑ2) = ∫ · · · ∫ C1 ∫ · · · ∫ CR E(M)(x1, . . . ,xn)E (N)(y1, . . . ,yn) × kM (u1, . . . ,un;ϑ1)kN (v1, . . . ,vn;ϑ2) n∏ i=1 M∏ a=1 dxi,a xi,a n∏ i=1 N∏ b=1 dyi,b yi,b , (4.14) Quantum Toroidal Comodule Algebra of Type An−1 and Integrals of Motion 21 we choose R ≪ 1 for convergence. The integrand can be obtained from the one for (4.11) by switching the roles of x, M and y, N . It amounts to changing (4.12) to T ′(u1, . . . ,un\|v1, . . . ,vn) = ϑ1(u1, . . . , un)ϑ2(v1, . . . , vn) × n∏ i=1 ∏ 1≤a≤M 1≤b≤N θ(vi,b−ui+1,a−γ/2−β)θ(ui,a−vi+1,b+γ/2−β) θ(vi,b − ui,a)θ(vi,b − ui,a − γ) . Similarly as above, we start from R ≪ 1 and bring CR to the unit circle. The relevant poles in between are yi,b = q2xi,a, pq2xi,a, 1 ≤ i ≤ n, 1 ≤ a ≤M, 1 ≤ b ≤ N. Moving the contour we pick up residues of these poles. Again the wheel conditions ensure that no new poles arise for the remaining variables. In order to prove the commutativity, we must show that, after moving contours to the unit circle, (4.11) and (4.14) give the same result. First, let us compare terms that arise without taking residues. Since all integrals are taken on the unit circle, we can symmetrize the integrand. Taking into account the symmetry of S(z1, . . . , zn), we see that the equality of integrals reduces to the identity of theta functions Sym T (u1, . . . ,un\|v1, . . . ,vn) = Sym T ′(u1, . . . ,un\|v1, . . . ,vn), (4.15) where Sym stands for symmetrization in each group of variables {ui,a}Ma=1∪{vi,b}Nb=1, 1 ≤ i ≤ n. This identity was stated in [12]. Since their proof contains some gaps, we prove it in Appendix; see Theorem A.1 and Remark A.6 below. In general, we compare terms obtained by taking residues with respect to some group of variables. First consider terms coming from (4.11). In view of symmetry and the zeros of S(z1, . . . , zn) at yi,a = yi,b, it is enough to consider the case yi,a = p−1q−1 2 xi,a, 1 ≤ a ≤ ki, (4.16) yi,b = q−1 2 xi,b, ki + 1 ≤ b ≤ li, (4.17) with some 0 ≤ ki ≤ li ≤ min(M,N), 1 ≤ i ≤ n. We compare it with the corresponding residue coming from (4.14) yi,a = pq2xi,a, 1 ≤ a ≤ ki, (4.18) yi,b = q2xi,b, ki + 1 ≤ b ≤ li, (4.19) for 1 ≤ i ≤ n. We further rename variables xi,a to p−1q−1 2 xi,a for (4.18) and to q−1 2 xi,a for (4.19). The contours for them change to ∣∣p−1q−1 2 xi,a ∣∣ = 1 and ∣∣q−1 2 xi,a ∣∣ = 1, respectively. We shift them back to the unit circle \|xi,a\| = 1, noting that the wheel conditions ensure there are no poles which hinder the shift. The factor S(z1, . . . , zn) has no poles at (4.16) and (4.17) and specializes to S ( . . . , zi,a, . . . , zi,b, . . . , p −1q−1 2 zi,a, . . . , q −1 2 zi,b, . . . ) . (4.20) Similarly, at (4.18) and (4.19) it specializes to S(. . . , zi,a, . . . , zi,b, . . . , pq2zi,a, . . . , q2zi,b, . . . ). (4.21) After the renaming, (4.21) is brought to (4.20) because S(z1, . . . , zn) is symmetric in {zi,a}1≤a≤M+N for each i. 22 B. Feigin, M. Jimbo and E. Mukhin It remains to compare the residues of the functions T , T ′. Since they are periodic with period 1, their residues at vi,a = ui,a − γ − 1 are the same as those at vi,a = ui,a − γ. Hence we are to show the equality Sym∗ res vi,a=ui,a−γ, 1≤a≤li, 1≤i≤n T (u1, . . . ,un\|v1, . . . ,vn) = Sym∗ { (−1) ∑n i=1 li res vi,a=ui,a+γ, 1≤a≤li, 1≤i≤n T ′(u1, . . . ,un\|v1, . . . ,vn) }∣∣∣ ui,a→ui,a−γ, 1≤a≤li, 1≤i≤n , where Sym∗ stands for the symmetrization with respect to the remaining variables {ui,a}Ma=li+1∪ {vi,b}Nb=li+1, 1 ≤ i ≤ n. Note that moving the contours CR from R < 1 to C1 we obtain a sign factor. To check the equality, we start from the residues of the identity (4.15) res vi,a=ui,a−γ, 1≤a≤li, 1≤i≤n Sym T (u1, . . . ,un\|v1, . . . ,vn)= res vi,a=ui,a−γ, 1≤a≤li, 1≤i≤n Sym T ′(u1, . . . ,un\|v1, . . . ,vn). (4.22) Symmetrization Sym amounts to replacing variables ui, vi as ui → {ui,a}a∈Ii ∪ {vi,b}b∈J ′ i , vi → {ui,a}a∈I′i ∪ {vi,b}b∈Ji , and sum over partitions of indices Ii ⊔ I ′i = {1, . . . ,M}, Ji ⊔ J ′ i = {1, . . . , N}, \|I ′i\| = \|J ′ i \|. In order to have non-zero residues at vi,a = ui,a − γ (1 ≤ a ≤ li) in the left-hand side of (4.22), Ii and Ji must contain {1, . . . , li}. Therefore the left-hand side of (4.22) reduces to Sym∗ res vi,a=ui,a−γ, 1≤a≤li, 1≤i≤n T (u1, . . . ,un\|v1, . . . ,vn). Similarly, in the right-hand side of (4.22), non-zero residues appear only when J ′ i and I ′ i contain {1, . . . , li}. For each i we have res vi,a=ui,a−γ, 1≤a≤li T ′(. . . , vi,1, . . . , vi,li , . . . \| . . . , ui,1, . . . , ui,li , . . . ) = (−1)li res u′ i,a=v′i,a+γ, 1≤a≤li T ′(. . . , v′i,1, . . . , v ′ i,li , . . . \| . . . , u′i,1, . . . , u′i,li , . . . ) ∣∣∣v′i,a→ui,a−γ 1≤a≤li . Altogether we can rewrite the right-hand side of (4.22) as Sym∗ { (−1) ∑n i=1 li res vi,a=ui,a+γ, 1≤a≤li,1≤i≤n T ′(u1, . . . ,un\|v1, . . . ,vn) }∣∣∣ ui,a→ui,a−γ, 1≤a≤li,1≤i≤n . Proof of Theorem 4.2 is now complete. ■ 5 The cases n = 1, 2 All results discussed so far are valid also in the case n = 1, 2 with suitable modifications. In this section we indicate the necessary changes. Quantum Toroidal Comodule Algebra of Type An−1 and Integrals of Motion 23 5.1 Case n = 2 Dealing with algebras En and Kn for n = 2, there are two points to be taken care of. First, the structure functions are changed. We keep gi,i(z, w) unchanged but replace gi,j(z, w) for i ̸= j with gn≥3 i,j (z, w)gn≥3 j,i (z, w). Similar changes are due for Gi,j(x), λ 0 i,j(x) and λi,j(x). Namely we use gi,1−i(z, w) = (z − q1w)(z − q3w), Gi,1−i(x) = q−2 ( 1− q−1 1 x )( 1− q−1 3 x ) (1− q1x)(1− q3x) , λ0i,1−i(x) = 1− q1x 1− x 1− q3x 1− x , λi,1−i(x) = xγ 1− q1x 1− x 1− q3x 1− x ( q−1 1 x, q−1 3 x; p ) ∞ (q1x, q3x; p)∞ . We set also di,i = 1, di,1−i = −1. Second, the Serre relations are modified. The defining relations of E2 are the same as those for n ≥ 3 except for the Serre relations Sym z1,z2,z3 [ei(z1), [ei(z2), [ei(z3), e1−i(w)]q2 ]]q−2 = 0, (5.1) Sym z1,z2,z3 [fi(z1), [fi(z2), [fi(z3), f1−i(w)]q2 ]]q−2 = 0. (5.2) The defining relations of K2 are (3.1)–(3.4), together with the Serre relations Sym z1,z2 ( q1(z1 − q3w)(z2 − q3w)Ei(z1)Ei(z2)E1−i(w)− ( 1 + q−1 2 ) (z1 − q3w)(q1z2 − w) × Ei(z1)E1−i(w)Ei(z2) + q3(q1z1 − w)(q1z2 − w)E1−i(w)Ei(z1)Ei(z2) ) = Sym z1,z2 { q−3w (( z2 − d−2z1 )1− q−1 3 z1/w 1− q3z1/w + ( z2 − d2z1 )1− q−1 1 w/z2 1− q1w/z2 − z2 + q2z1 ) × δ ( C2z1/z2 ) K− i (z1)E1−i(w)K + i (z2) } , (5.3) and the same relation with q1 and q3 interchanged. At first glance, the quartic Serre relations (5.1) and (5.2) for E2 and the cubic Serre rela- tions (5.3) for K2 look very different. Actually the former are equivalent (under the rest of the relations) to cubic relations which are the left-hand sides of (5.3) and their “f -version”. See [3, Lemma 2.1] and remark after that. With the above changes, formula for comodule structure (Theorem 3.4), formulas for integrals of motion (4.9) and (4.10) and their commutativity (Theorem 4.2) remain valid. 5.2 Case n = 1 Algebra K1 has been discussed in [7]. For reader’s reference we mention the necessary changes. We drop suffixes i, j from structure functions and set g(z, w) = 3∏ s=1 (z − qsw), G(x) = 3∏ s=1 1− q−1 s x 1− qsx , 24 B. Feigin, M. Jimbo and E. Mukhin λ0(x) = 1− C2x 1− x 1− C−2x 1− x 3∏ s=1 1− qsx 1− x , λ(x) = 1− C2x 1− x 1− C−2x 1− x 3∏ s=1 1− qsx 1− x ( q−1 s x; p ) ∞ (qsx; p)∞ . We change also the normalization of generators of E1 slightly, so that ψ±(z) = exp ( ∑ ±r>0 κrh±rz ∓r ) , [e(z), f(w)] = 1 κ1 (δ ( Cw/z ) ψ+(w)− δ ( Cz/w ) ψ−(z)), where κr = ∏3 s=1(1− qrs). The Serre relations for E1 are Sym z1,z2,z3 z2z −1 3 [e(z1), [e(z2), e(z3)]] = 0, Sym z1,z2,z3 z2z −1 3 [f(z1), [f(z2), f(z3)]] = 0. The Serre relation for K1 reads Sym z1,z2,z3 z2 z3 [E(z1), [E(z2), E(z3)]] = Sym z1,z2,z3 { X(z1, z2, z3) 1 q − q−1 δ ( C2z1/z3 ) K−(z1)E(z2)K +(z3) } , where X(z1, z2, z3) = (z1 + z2) ( z23 − z1z2 ) z1z2z3 G(z2/z3) + (z2 + z3) ( z21 − z2z3 ) z1z2z3 G(z1/z2) + (z3 + z1) ( z22 − z3z1 ) z1z2z3 . The space of theta functions X1 is defined to be that of constant functions C. Replacing hM (u1, . . . ,un;ϑ) by hM (u) = ∏ 1≤a<b≤M θ(ua − ub)θ(ua − ub − γ) θ(ua − ub − β − γ/2)θ(ua − ub + β − γ/2) , we define integrals of motion by (4.9). Then the commutativity Theorem 4.2 holds true. 6 Fusion It is known [3] that the completed quantum toroidal algebra Ẽn with parameters q1, q2, q3 contains various subalgebras isomorphic to Ẽk, with 1 ≤ k ≤ n− 1 and suitable parameters q̄1, q̄2, q̄3. In this section we discuss an analogue of this construction for Kn. Throughout this section we take n ≥ 2, and fix an admissible Kn module V , where C acts as a scalar such that C2 ̸= ±1, q±2. Consider the following set of operators { Ēi(z), K̄ ± i (z) } 0≤i≤n−2 acting on V : Ē0(z) = c̄ En−1,0(q1z, z), K̄± 0 (z) = K± n−1(q1z)K ± 0 (z), (6.1) Ēi(z) = Ei ( q i n−1 1 z ) , K̄± i (z) = K± i ( q i n−1 1 z ) , 1 ≤ i ≤ n− 2. (6.2) Here c̄ = (qq1) 1/2 ( 1− q−1 1 ) for n ≥ 3 and c̄ = ( 1− q−1 1 )2 for n = 2. Quantum Toroidal Comodule Algebra of Type An−1 and Integrals of Motion 25 Proposition 6.1. The operators (6.1) and (6.2) together with the same C give an action of Kn−1 on V with parameters q̄1 = q1q 1 n−1 1 , q̄2 = q2, q̄3 = q3q − 1 n−1 1 . Proof. Denote by Ḡi,j(x), 0 ≤ i, j ≤ n− 2 the functions Gi,j(x) for Kn−1 with the parameters given above. Then we have Ḡi,i(x) = Gi,i(x) 2Gi+1,i(q1x)Gi,i+1 ( q−1 1 x ) , n ≥ 2, and Ḡi,i±1(x) = Gi,i±1 ( q ± 1 n−1 1 x ) , n ≥ 4, Gi,i±1 ( q ± 1 n−1 1 x ) Gi±1,i ( q ∓ 1 n−1 1 x ) , n = 3. Using these one can verify the relations (3.1)–(3.4) as rational functions. For simplicity we assume n ≥ 3 below. The case n = 2 can be treated with minor modifica- tions. Let us check the condition (3.12) for Ē0(z). We have Ē0(z1)Ē0(z2) = En−1,n−1,0,0(q1z1, q1z2, z1, z2)c̄d ( 1− q−1 1 )2 (1− z2/z1) 2 (1− C2z2/z1)2 (1− z2/z1) 2 (1− C−2z2/z1)2 × (1− z2/z1) 2( 1− q2z2/z1 )2 ( 1− q−1 1 z2/z1 ) (1− q1z2/z1)( 1− q−2z2/z1 ) (1− z2/z1) . The wheel conditions (3.13)–(3.15) ensure that the apparent double poles in the right-hand side are actually all simple. To compute the residue at z2 = C2z1, it suffices to consider 1 1− C−2z2/z1 En−1,n−1,0,0(q1z1, q1z2, z1, z2) ∣∣∣∣ z2=C2z1 =w ∂ ∂w En−1,n−1,0,0(q1z1, q1z2, w, z2) ∣∣∣∣ w=z1 z2=C2z1 +En−1,n−1,0,0 ( q1w,C 2q1z1, z1, C 2z1 )∣∣∣∣ w=z1 z2=C2z1 . The second term in the right-hand side vanishes due to the wheel condition. Substituting En−1,n−1,0,0(q1z1, q1z2, w, z2) =K − n−1(q1z1)E0,0(w, q1z1)K + n−1 ( C2q1z1 ) × 1− w/z1 1−q−1 1 w/z1 1− q2C−2w/z1 1−q−1 1 C−2w/z1 −d−2q−1 1− q−1 1 1 + C2 1− C2 1− q2C2 1−q−1 1 C2 , we arrive at λ00,0(z2/z1)Ē0(z1)Ē0(z2) ∣∣ z2=C2z1 = 1 q − q−1 1 + C2 1− C2 1− q2C2 1− C2 K̄− 0 (z1)K̄ + 0 ( C2z1 ) . The wheel conditions for Ēi(z) can be verified similarly. As an example, consider Ē0,0,1(z1, z2, w). Up to an irrelevant factor it can be written as Ē0,0,1(z1, z2, w) ∝ 1 λ00,0(z2/z1) En−1,n−1,0,0,1 ( q1z1, q1z2, z1, z2, q 1 n−1 1 w ) . (6.3) Since En−1,n−1,0(q1z, q1q2z, z) = 0 and E0,0,1(z, q2z, q2q3z) = 0, En−1,n−1,0,0,1 ( q1z1, q1z2, z1, z2, q −1 1 z1 ) is divisible by (z2−q2z1)2. Therefore (6.3) vanishes at z2 = q2z1, w = q̄−1 1 z1. Similarly, the condition En−1,n−1,0(q1z, q1C 2z, z) = 0 and E0,0,1 ( z, C2z, C2q3z ) = 0 imply that (6.3) vanishes at z2 = C2z1, w = C2q̄3z1. ■ 26 B. Feigin, M. Jimbo and E. Mukhin Interchanging the roles of q1 and q3, one can equally well consider action of Kn−1 with parameters ¯̄q1 = q1q − 1 n−1 3 , ¯̄q2 = q2, ¯̄q3 = q3q 1 n−1 3 . Note that in either case the values of q, C are unchanged. Iterating this construction we obtain in general Proposition 6.2. Let 1 ≤ k ≤ n − 1. Then the following hold with an appropriate choice of constants c̄n,k, ¯̄cn,n−k: (i) The operators Ē0(z) = c̄n,k Ek,k+1,...,n−1,0 ( qn−k 1 z, qn−k−1 1 z, . . . , q1z, z ) , K̄± 0 (z) = K± k ( qn−k 1 z ) K± k+1 ( qn−k−1 1 z ) · · ·K± n−1(q1z)K ± 0 (z), Ēi(z) = Ei ( q n−k k i 1 z ) , K̄± i (z) = K± i ( q n−k k i 1 z ) , 1 ≤ i ≤ k − 1, together with the same C give an action of Kk on V with parameters q̄1 = q1q n−k k 1 , q̄2 = q2, q̄3 = q3q −n−k k 1 . (ii) The operators ¯̄E0(z) = ¯̄cn,n−kEk,k−1,...,1,0 ( qk3z, q k−1 3 z, . . . , q3z, z ) , ¯̄K± 0 (z) = K± k ( qk3z ) K± k−1 ( qk−1 3 z ) · · ·K± 1 (q3z)K ± 0 (z), ¯̄Ei(z) = Ei+k ( q k n−k i 3 z ) , ¯̄K± i (z) = K± i+k ( q k n−k i 3 z ) , 1 ≤ i ≤ n− k − 1, together with the same C give an action of Kn−k on V with parameters ¯̄q1 = q1q − k n−k 3 , ¯̄q2 = q2, ¯̄q3 = q3q k n−k 3 . (iii) The actions (i) and (ii) mutually commute. Proof. We need only to prove the commutativity (iii). For simplicity of presentation we assume n ≥ 3. Evidently { Ēi(z), K̄ ± i (z) } 1≤i≤k−1 commute with { ¯̄Ei(z), ¯̄K± i (z) } 1≤i≤n−k−1 . To show that ¯̄K+ i (z), 1 ≤ i ≤ n− k − 1, commutes with Ē0(w), it suffices to note that K+ j (z)Ek,k+1,...,n−1,0 ( qn−k 1 w, . . . , w ) = Ek,k+1,...,n−1,0 ( qn−k 1 w, . . . , w ) K+ j (z) ×Gj,j−1 ( qn−j+1 1 w/z ) Gj,j ( qn−j 1 w/z ) ×Gj,j+1 ( qn−j−1 1 w/z ) , k + 1 ≤ j ≤ n− 1, and apply the identity Gi,i−1(q1x)Gi,i(x)Gi,i+1 ( q−1 1 x ) = 1. Commutativity of { ¯̄Ei(z), ¯̄K± i (z) } 1≤i≤n−k−1 with { Ē0(w), K̄ ± 0 (w) } (for ¯̄Ei(z) with Ē0(w) as rational functions) can be shown similarly. We have further ¯̄K0(z)Ē0(w) = Ē0(w) ¯̄K0(z)Gk−1,k ( qn1 q k 2q3w/z ) Gk,k ( qn1 q k 2w/z ) Gk,k+1 ( qn1 q k 2q −1 1 w/z ) ×G1,0 ( q−1 3 w/z ) G0,0(w/z)G0,n−1(q1w/z), Quantum Toroidal Comodule Algebra of Type An−1 and Integrals of Motion 27 so that we can apply Gi−1,i(q3x)Gi,i(x)Gi,i+1 ( q−1 1 x ) = 1. By the same token { ¯̄E0(z), ¯̄K± 0 (z) } commute with { Ē0(w), K̄ ± 0 (w) } (for ¯̄E0(z) with Ē0(w) as rational functions). To finish the proof, it remains to examine the singularities of ¯̄Ei(z)Ē0(w). For 1 ≤ i ≤ n−k−1, we use Ej(z)Ej−1,j,j+1 ( q1w,w, q −1 1 w ) = Ej−1,j,j,j+1 ( q1w, z, w, q −1 1 w ) 1− q1w/z 1− q−1 2 w/z 1− w/z 1− C2w/z × 1− w/z 1− q2w/z 1− w/z 1− C−2w/z 1− q−1 1 w/z 1− w/z , and observe that all poles are cancelled due to the wheel conditions. Finally for i = 0 we have Ek,k−1,...,1,0 ( qk3z, . . . , z ) Ek,k+1,...,n−1,0 ( qn−k 1 w, . . . , w ) = E0,0,1,...,k−1,k,k,k+1,...,n−1 ( z, w, q3z, . . . , q k−1 3 z, q3z, q n−k 1 w, . . . , q1w ) × ( λ0k−1,k ( qn1 q k 2q3w/z ) λ0k,k ( qn1 q k 2w/z ) λ0k,k+1 ( qn1 q k 2q −1 1 w/z ))−1 × ( λ00,n−1(q1w/z)λ 0 0,0(w/z)λ 0 1,0 ( q−1 3 w/z ))−1 , so the cancellation works similarly as above. ■ A Theta function identities As in Section 4.2 we use the space of theta functions Xn(µ1, . . . , µn) given by (4.6) and (4.7) for n ≥ 2. For n = 1, we define X1 = C to be the space of constant functions. In this section we prove the following identity. Theorem A.1. The following identities hold for all n,M,N ≥ 1, α, γ ∈ C and ϑ1, ϑ2 ∈ Xn:∑ I1,J1 · · · ∑ In,Jn ϑ1(u1,I1 , . . . , un,In)ϑ2(u1,J1 , . . . , un,Jn) × n∏ i=1 ∏ a∈Ii b∈Ji+1 θ(ui,a − ui+1,b − α) ∏ b∈Ji a∈Ii+1 θ(ui,b − ui+1,a − α+ γ)∏ a∈Ii b∈Ji θ(ui,a − ui,b)θ(ui,a − ui,b − γ) = ∑ I1,J1 · · · ∑ In,Jn ϑ1(u1,I1 , . . . , un,In)ϑ2(u1,J1 , . . . , un,Jn) × n∏ i=1 ∏ b∈Ji a∈Ii+1 θ(ui,b − ui+1,a − α) ∏ a∈Ii b∈Ji+1 θ(ui,a − ui+1,b − α+ γ)∏ a∈Ii b∈Ji θ(ui,b − ui,a)θ(ui,b − ui,a − γ) . (A.1) Here we set ui,I = ∑ a∈I ui,a for a subset I of {1, . . . ,M + N}, and the sum is taken over all partitions Ii ⊔ Ji = {1, . . . ,M +N} satisfying \|Ii\| =M, \|Ji\| = N, i = 1, . . . , n. For the proof of theorem, we prepare several lemmas. Lemma A.2. If ϑ ∈ Xn(µ1, . . . , µn), then ϑ(u1 + u, . . . , un + u) = ϑ(u1, . . . , un), u ∈ C. 28 B. Feigin, M. Jimbo and E. Mukhin Proof. Let ϕ(u) = ϑ(u1 + u, . . . , un + u). Then ϕ(u) is entire and satisfies ϕ(u + 1) = ϕ(u), ϕ(u+ τ) = ϕ(u). Hence it is a constant: ϕ(u) = ϕ(0). ■ Lemma A.3. Let ϕ(v1, . . . , vL) be an entire function with quasi-periodicity property ϕ(v1, . . . , vi + 1, . . . , vL) = ϕ(v1, . . . , vi, . . . , vL), ϕ(v1, . . . , vi + τ, . . . , vL) = e−2πim ∑L i=1 vi+kϕ(v1, . . . , vi, . . . , vL) with some m ∈ Z≥0 and k ∈ C. Then it is a function of ∑L i=1 vi. Proof. Fix i ̸= j, and set φ(u) = ϕ(v1, . . . , vi + u, . . . , vj − u, . . . , vL). Then φ(u) is an entire function and satisfies φ(u + 1) = φ(u), φ(u + τ) = φ(u). Hence it is a constant: φ(u) = φ(0). Since i, j, u are arbitrary, this proves the lemma. ■ Denote both sides of the identity (A.1) by LHSn,M,N and RHSn,M,N , respectively, and set Φn,M,N = LHSn,M,N − RHSn,M,N . Where necessary we exhibit also their dependence on the variables ui = (ui,1, . . . , ui,M+N ) and the choice of ϑ1, ϑ2 ∈ Xn. We set Φn,M,N = 0 if M = 0 or N = 0. Lemma A.4. For n ≥ 2, Φn,M,N has the quasi-periodicity in each variable ui,a, Φn,M,N (. . . , ui,a + 1, . . . ) = Φn,M,N (. . . , ui,a, . . . ), Φn,M,N (. . . , ui,a + τ, . . . ) = e−2πi(2ūi−ūi−1−ūi+1−µi+τ)Φn,M,N (. . . , ui,a, . . . ). For n = 1, Φ1,M,N has periods 1 and τ . Modulo Z⊕ Zτ , the only poles of Φn,M,N are ui,a = ui,b + γ with some a, b, i. Proof. The quasi-periodicity follows from (4.4), (4.6) and (4.7). Since Φn,M,N is symmetric in each group of variables {ui,a}a=1,...,M+N , it cannot have a simple pole at ui,a = ui,b. ■ Lemma A.5. We have res u1,1=u1,2+γ · · · res un,1=un,2+γ Φn,M,N (u1, . . . ,un\|ϑ1, ϑ2) = An ∏n i=1 ∏M+N a=2 θ(ui,a − ui+1,2 − α)θ(ui,2 − ui+1,a − α+ γ)∏n i=1 ∏M+N a=3 θ(ui,a − ui,2)θ(ui,a − ui,2 − γ) × Φn,M−1,N−1(u ′ 1, . . . ,u ′ n\|ϑ′1, ϑ′2), where u′ i = (ui,3, . . . , ui,M+N ), ϑ′s(v1, . . . , vn) = ϑs(v1 + u1,2, . . . , vn + un,2), s = 1, 2, and A = θ(γ)−1 resu=0 θ(u) −1. Proof. In the sum LHSn,M,N , the iterated residue is non-zero only for terms such that 1 ∈ Ii and 2 ∈ Ji for all i = 1, . . . , n. Write Ii = {1} ⊔ Īi and Ji = {2} ⊔ J̄i. Similarly, for RHSn,M,N one must have 2 ∈ Ii and 1 ∈ Ji for i = 1, . . . , n. Write Ii = {2} ⊔ Īi and Ji = {1} ⊔ J̄i in that case. In either case, the functions ϑs become ϑ1(u1,2 + ū1,Ī1 , . . . , un,2 + ūn,Īn)ϑ2(u1,2 + ū1,J̄1 , . . . , un,2 + ūn,J̄n) due to Lemma A.2. The rest is a simple computation. ■ Quantum Toroidal Comodule Algebra of Type An−1 and Integrals of Motion 29 Remark A.6. Lemma A.5 is proved in [12]. However the rest of the argument necessary for completing the proof of Theorem A.1 is missing there. Lemma A.7. Fix i ∈ {1, . . . , n}. Then the following hold: Φn,M,N (u1, . . . ,un\|ϑ1, ϑ2) ∣∣ui+1,a=ui,a−α 1≤a≤M+N = (−1)MNΦn−1,M,N (u1, . . . ,ui,ui+2, . . . ,un\|ϑ′′1, ϑ′′2), Φn,M,N (u1, . . . ,un\|ϑ1, ϑ2) ∣∣ui+1,a=ui,a−α+γ 1≤a≤M+N = (−1)MNΦn−1,M,N (u1, . . . ,ui,ui+2, . . . ,un\|ϑ′′′1 , ϑ′′′2 ), where ϑ′′1(v1, . . . , vn−1) = ϑ1(v1, . . . , vi, vi −Mα, vi+1, . . . , vn−1), ϑ′′2(v1, . . . , vn−1) = ϑ2(v1, . . . , vi, vi −Nα, vi+1, . . . , vn−1), and ϑ′′′s is given by replacing α by α− γ in ϑ′′s . Proof. Since the calculations are similar, we consider the first case. Under this specialization, terms in LHSn,M,N survive only when Ii ∩ Ji+1 = ∅, which implies Ii = Ii+1 and Ji = Ji+1. Then the factor ∏ a∈Ii b∈Ji+1 θ(ui,a − ui+1,b − α) ∏ a∈Ii+1 b∈Ji θ(ui,b − ui+1,a − α+ γ) from the numerator cancels ∏ a∈Ii+1 b∈Ji+1 θ(ui+1,a−ui+1,b)θ(ui+1,a−ui+1,b−γ) from the denominator, thereby reducing n to n− 1. For RHSn,M,N the situation is entirely similar. ■ Lemma A.8 ([19]). The identity (A.1) holds for n = 1. Proof. For n = 1 we may assume ϑ1 = ϑ2 = 1, so the identity takes the form∑ I,J ∏ a∈I b∈J θ(ua − ub − α)θ(ub − ua − α+ γ) θ(ua − ub)θ(ua − ub − γ) = ∑ I,J ∏ a∈I b∈J θ(ub − ua − α)θ(ua − ub − α+ γ) θ(ub − ua)θ(ub − ua − γ) . Lemma A.5 together with the symmetry shows that Φ1,M,N has no pole at ua = ub + γ. Hence Φ1,M,N is an elliptic function without poles, therefore it is a constant. Upon specializing ua = aα, a = 1, . . . ,M +N , we obtain LHS1,M,N = (−1)MN ∏M a=1 θ((a− 1)α+ γ)/θ(aα) ∏N a=1 θ((a− 1)α+ γ)/θ(aα)∏M+N a=1 θ((a− 1)α+ γ)/θ(aα) = RHS1,M,N , thereby proving Φ1,M,N = 0. ■ Proof of Theorem A.1. We use induction on n. The base of induction is proved in Lem- ma A.8. Assume that theorem is true for n− 1. For i = 1, . . . , n we set Zi = res ui,2=ui,1−γ res ui+1,2=ui+1,1−γ · · · res un,2=un,1−γ Φn,M,N (u1, . . . ,un, α, γ\|ϑ1, ϑ2). We show Zi = 0 by induction on i. By Lemma A.5 and the induction hypothesis Φn−1,M,N = 0, we have Z1 = 0. Suppose Zi = resui,2=ui,1−γ Zi+1 = 0. Then Zi+1 has no poles in ui. 30 B. Feigin, M. Jimbo and E. Mukhin Since ui+1,2 = ui+1,1 − γ, non-zero terms in Zi+1 are such that 1 ∈ Ii+1, 2 ∈ Ji+1 in LHS. Then in the numerator we have factors θ(ui,a − ui+1,2 − α), a ∈ Ii, and θ(ui,b − ui+1,1 − α+ γ), b ∈ Ji. Therefore each term has a factor M+N∏ a=1 θ(ui,a − ui+1,2 − α). By the same argument, RHS, and hence Zi+1, is shown to have the same factor. If Zi+1 ̸= 0, this is a contradiction because the quasi-periodicity property of Zi+1 implies that it is a function of ūi = ∑M+N a=1 ui,a, see Lemma A.3. Hence we have Zi+1 = 0. It follows that Φn,M,N has no poles at un,a = un,b − γ. Therefore, if non-zero, it is a function of the sum ūn = ∑M+N a=1 un,a. On the other hand, by Lemma A.7 and the induction hypothesis, Φn,M,N has the following zeros with respect to ūn: ūn = ū1 + (M +N)α, ūn = ū1 + (M +N)(α− γ), ūn = ūn−1 − (M +N)α, ūn = ūn−1 − (M +N)(α− γ). From the quasi-periodicity, if Φn,M,N ̸= 0, then it can have only two zeros modulo Z⊕Zτ , which contradicts to the above zeros. This completes the proof of Theorem A.1. ■ Acknowledgments The study has been funded within the framework of the HSE University Basic Research Program. MJ is partially supported by JSPS KAKENHI Grant Number JP19K03549. EM is partially supported by grants from the Simons Foundation #353831 and #709444. References [1] Aganagic M., Okounkov A., Quasimap counts and Bethe eigenfunctions, Mosc. Math. J. 17 (2017), 565–600, arXiv:1704.08746. [2] Feigin B., Frenkel E., Integrals of motion and quantum groups, in Integrable Systems and Quantum Groups (Montecatini Terme, 1993), Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 349–418, arXiv:hep- th/9310022. [3] Feigin B., Jimbo M., Miwa T., Mukhin E., Branching rules for quantum toroidal gln, Adv. Math. 300 (2016), 229–274, arXiv:1309.2147. 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Gen. 21 (1988), 2375–2389. https://doi.org/10.1007/s00220-008-0524-3 https://arxiv.org/abs/0709.2305 https://doi.org/10.1016/j.aim.2014.08.010 https://arxiv.org/abs/1207.6036 https://doi.org/10.1017/S1474748021000220 https://arxiv.org/abs/2007.11786 https://doi.org/10.1007/jhep12(2020)100 https://doi.org/10.1007/jhep12(2020)100 https://arxiv.org/abs/2007.00535 https://doi.org/10.1007/jhep08(2021)141 https://arxiv.org/abs/2105.04018 https://doi.org/10.1016/j.aim.2021.108111 https://arxiv.org/abs/2009.04542 https://doi.org/10.1142/S0129055X03001813 https://arxiv.org/abs/math.QA/0208140 https://doi.org/10.1007/BF01207363 https://doi.org/10.2977/prims/1195144759 https://arxiv.org/abs/q-alg/9611030 https://doi.org/10.1088/0305-4470/21/10/015 1 Introduction 2 Preliminaries 2.1 Quantum toroidal algebra E_n 2.2 Coproduct 2.3 Fock modules 3 Comodule algebra K_n 3.1 Algebra K_n 3.2 Wheel conditions 3.3 Comodule structure 3.4 Representations 4 Integrals of motion 4.1 Dressed currents 4.2 Integrals of motion 4.3 Commutativity 5 The cases n=1,2 5.1 Case n=2 5.2 Case n=1 6 Fusion A Theta function identities References
id	nasplib_isofts_kiev_ua-123456789-211736
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn	1815-0659
language	English
last_indexed	2026-03-14T10:08:01Z
publishDate	2022
publisher	Інститут математики НАН України
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spelling	Feigin, Boris Jimbo, Michio Mukhin, Evgeny 2026-01-09T12:56:11Z 2022 Quantum Toroidal Comodule Algebra of Type 𝐴ₙ₋₁ and Integrals of Motion. Boris Feigin, Michio Jimbo and Evgeny Mukhin. SIGMA 18 (2022), 051, 31 pages 1815-0659 2020 Mathematics Subject Classification: 81R10; 81R12; 17B69; 17B80 arXiv:2112.14631 https://nasplib.isofts.kiev.ua/handle/123456789/211736 https://doi.org/10.3842/SIGMA.2022.051 We introduce an algebra 𝒦ₙ which has the structure of a left comodule over the quantum toroidal algebra of type 𝐴ₙ₋₁. Algebra 𝒦ₙ is a higher rank generalization of 𝒦₁, which provides a uniform description of deformed 𝑊 algebras associated with Lie (super)algebras of types BCD. We show that 𝒦ₙ possesses a family of commutative subalgebras. The study has been funded within the framework of the HSE University Basic Research Program. MJ is partially supported by JSPS KAKENHI Grant Number JP19K03549. EM is partially supported by grants from the Simons Foundation #353831 and #709444. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Quantum Toroidal Comodule Algebra of Type 𝐴ₙ₋₁ and Integrals of Motion Article published earlier
spellingShingle	Quantum Toroidal Comodule Algebra of Type 𝐴ₙ₋₁ and Integrals of Motion Feigin, Boris Jimbo, Michio Mukhin, Evgeny
title	Quantum Toroidal Comodule Algebra of Type 𝐴ₙ₋₁ and Integrals of Motion
title_full	Quantum Toroidal Comodule Algebra of Type 𝐴ₙ₋₁ and Integrals of Motion
title_fullStr	Quantum Toroidal Comodule Algebra of Type 𝐴ₙ₋₁ and Integrals of Motion
title_full_unstemmed	Quantum Toroidal Comodule Algebra of Type 𝐴ₙ₋₁ and Integrals of Motion
title_short	Quantum Toroidal Comodule Algebra of Type 𝐴ₙ₋₁ and Integrals of Motion
title_sort	quantum toroidal comodule algebra of type 𝐴ₙ₋₁ and integrals of motion
url	https://nasplib.isofts.kiev.ua/handle/123456789/211736
work_keys_str_mv	AT feiginboris quantumtoroidalcomodulealgebraoftypean1andintegralsofmotion AT jimbomichio quantumtoroidalcomodulealgebraoftypean1andintegralsofmotion AT mukhinevgeny quantumtoroidalcomodulealgebraoftypean1andintegralsofmotion

Quantum Toroidal Comodule Algebra of Type 𝐴ₙ₋₁ and Integrals of Motion

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