Deformations of W algebras via quantum toroidal algebras
aa r X i v : . [ m a t h . QA ] M a r DEFORMATIONS OF W ALGEBRASVIA QUANTUM TOROIDAL ALGEBRAS
B. FEIGIN, M. JIMBO, E. MUKHIN, AND I. VILKOVISKIY
Abstract.
The deformed W algebras of type A have a uniform description in terms of the quantumtoroidal gl algebra E . We introduce a comodule algebra K over E which gives a uniform constructionof basic deformed W currents and screening operators in types B , C , D including twisted and super-symmetric cases. We show that a completion of algebra K contains three commutative subalgebras.In particular, it allows us to obtain a commutative family of integrals of motion associated with affineDynkin diagrams of all non-exceptional types except D (2) ℓ +1 . We also obtain in a uniform way deformedfinite and affine Cartan matrices in all classical types together with a number of new examples, anddiscuss the corresponding screening operators. Introduction
Commutative families of operators coming from conformal field theory (CFT), known as localintegrals of motion (IM), have attracted a lot of attention in the last quarter of a century. Theinterest was boosted by a seminal sequence of works by Bazhanov, Lukyanov, and Zamolodchikov[BLZ1]–[BLZ3] where many related structures were revealed and a number of intriguing conjectureswas put forth. One outcome of that study is the celebrated ODE/IM correspondence relating thespectrum of local IM to remarkable differential operators whose monodromy coefficients satisfy Betheansatz equations [DT], [MRV], [BLZ4], [BLZ5], see also [BL]. However, many important questionsin the area still have no answers and many key statements remain conjectures. The explicit formof the local integrals of motion is unknown even in the simplest cases, see [FF] for the discussion oftheir existence, and there are no theorems about their spectrum.From a general philosophy of quantum groups, it is quite natural to consider q -deformations oflocal IM in search for clarification of the matters. It turns out that after the q -deformation, thetype A local IM become non-local, but can be written down explicitly [FKSW]. Moreover, one candescribe their spectrum via Bethe ansatz, [FJM]. In the present article we address the problem ofconstructing the q -deformation of local IM in other types.The algebraic structure underlying the CFT in question is given by the Virasoro algebra, and moregenerally, W algebras. The q -deformations of W algebras have been provided in [AKOS] for type A ,and in [FR1] for simple Lie algebras. It has been recognized in recent years that quantum groupsprovide a natural framework for studying W algebras [MO].For the deformed W algebras of type A , the relevant quantum group is the quantum toroidal gl algebra E (also known as the Ding-Iohara-Miki algebra), see Section 2.1. Algebra E has three Fockmodules depending on a color c ∈ { , , } , see Section 2.3. Consider generating current e ( z ) of E acting on a tensor product of ℓ + 1 Fock modules associated to arbitrary colors c , . . . , c ℓ +1 . Then Date : March 10, 2020.
Key words and phrases.
Quantum toroidal algebra; W algebra; integrals of motion; qq character. e ( z ) acts as a sum of ℓ + 1 vertex operators, see Section 3.1. When all Fock modules are of the samekind, the current e ( z ) is related to the basic current T ( z ) of the deformed W algebra of type A ℓ , see[FR1], as e ( z ) = T ( z ) Z ( z ), where Z ( z ) is a vertex operator written in a single boson { h i } given bythe action of the Cartan current of E and which commutes with T ( z ). For other choices of colors,see for example (3.6), one gets currents which can be viewed as various q -deformations of W algebrasassociated to Lie superalgebras of types gl m | n with m + n = ℓ + 1. In particular, we introduce thecurrents A i ( z ), i = 1 , . . . , ℓ , given by ratios of neighboring terms in this sum of ℓ + 1 vertex operators,see (3.3). The current A i ( z ) is bosonic if c i = c i +1 , and it is fermionic otherwise. Then the table ofcontractions between the A i ( z )’s gives the deformed Cartan matrix of finite type (3.8). Moreover,the screening operators which are integrals of q -primitives of A i ( z ), see (3.16),(3.17), commute with T ( z ).Algebra E has a family of commutative algebras depending on a parameter µ given by the transfermatrices. One can “dress” current e ( z ) multiplying by an appropriate combination of Cartan currentdepending on µ , see (3.9). Then generators of the commutative algebras can be computed explicitlyand are given by multiple integrals (3.22) of the dressed current e ( z ) with Feigin-Odesskii [FO] kernel,(3.20), see [FJM]. These generators acting on tensor products of Fock spaces are deformations of thelocal IM associated to W of type A , see [FKSW]. The spectrum of transfer matrices (or of deformedIM), is computed by Bethe ansatz, see [FJMM1], [FJMM2] for two different derivations. Then onecan attempt to obtain spectrum of the original local IM by taking an appropriate limit, see [FJM].The dressed current e ( z ) has form e ( z ) ˜ Z µ ( z ) where ˜ Z µ ( z ) is a vertex operator written in terms of { h i } . Then we observe that the ratio A ( z ) of the last term in the dressed current e ( z ) and the firstterm shifted by µ produces one more screening operator which commutes with the deformed IM.Moreover, the matrix of contractions between all of A i ( z ) is given by the deformed Cartan matrix ofaffine type as in [KP]. Following the terminology of [N], [KP], we interpret the dressed current e ( z )as a qq -character of the first fundamental representation of quantum affine sl ℓ +1 , see Section 3.3.We develop a similar picture for non-exceptional types other than A .We introduce an algebra K depending on three parameters q , q , q such that q q q = 1, whichplays a role of quantum toroidal algebra, see (4.1)-(4.5). Unlike E , the algebra K is not a Hopfalgebra but a left E comodule, see (4.8). Algebra K has 6 representations in one boson denoted F B c ,see Proposition 4.2, and F CD c , c ∈ { , , } , see Proposition 4.3, which we call boundary Fock modulesof types B and CD respectively. The algebra K has a central element C which determines the levelof the module. We have C = q / c in type B and C = q − c in types CD .The comodule structure allows us to consider a tensor product of a boundary Fock module of color c ℓ +1 with ℓ Fock modules of E of colors c , . . . , c ℓ . This K module is realized in ℓ + 1 bosons. Thegenerating current E ( z ) of K acting on such a tensor product is a sum of 2 ℓ + 1 terms if the theboundary module is of type B and of 2 ℓ vertex operators if the boundary module is of type CD .Similarly to type A , if all colors of E Fock modules are the same, c = · · · = c ℓ , then up to a bosonwe recover deformed W currents of [FR1]. More precisely, if the boundary module is of type B , weobtain a deformed W current of type B when c ℓ +1 = c ℓ and of type osp (1 , ℓ ) when c ℓ +1 = c ℓ (thelatter is called A (2)2 ℓ type in [FR1]). If the boundary module is of type CD , we recover the deformed W current of type C when c ℓ +1 = c ℓ , and of type D when c ℓ +1 = c ℓ . If the colors of E Fock spaces vary,we get various q -deformations of W currents related to supersymmetric cases. We again introducecurrents A i ( z ) as ratios of neighboring terms (in terms of Dynkin diagram), see (4.14)-(4.17). Then EFORMATIONS OF W ALGEBRAS 3 we observe that the contractions give a deformed Cartan matrix of corresponding finite type, andthat the screening operators constructed from A i ( z ) commute with our E ( z ).We define a dressed current E ( z ) and consider integrals of products of E ( z ) with the Feigin-Odesskii kernel similarly to type A , see (4.40). We show that these integrals commute if C /µ = q c , c ∈ { , , } . Thus we have three families of commutative subalgebras in K and three commutingfamilies of operators acting on the representation. Naturally, we call them deformed IM. Note thatthe simplest deformed IM is always given by the constant term of the deformed W current E ( z ).We use dressed current E ( z ) to introduce current A ( z ), see (4.28)-(4.30), and observe that thecontractions of A i ( z ) give deformed affine Cartan matrices. If c = c then the affine root turns out tobe of type C and if c = c then of type D . In particular, we cover that way all non-exceptional affineDynkin diagrams except D (2) ℓ +1 . Moreover, the screening operator constructed from A ( z ) commutedwith the deformed IM.It is then natural to consider affine roots of type B . It corresponds to the case when dressingparameter µ satisfies C /µ = q − / c . In this case we introduce current A ( z ) in a similar way.Adding one more vertex operator to E ( z ) (that is “adding a one dimensional representation”) weobtain a new current ˜ E ( z ). Then ˜ E ( z ) commutes with screening operators constructed from A i ( z ), i = 1 , . . . , ℓ , and the integral of ˜ E ( z ) commutes with the screening operator corresponding to A ( z ).In particular, contractions of A i ( z ) lead to deformed affine Cartan matrices whose affine node is oftype B .We do not know know how to include the integral of ˜ E ( z ) into a family of commuting operatorsdirectly. However if the boundary module is of type CD , we can exchange the affine node with the ℓ -th node. It turns out that the integral of ˜ E ( z ) coincides with the integral of a different deformed W current for which the boundary module is of type B and the affine root is of types C or D , seeRemark 4.6. Then the integral of the latter current is a part of the family of integrals of motion asbefore. Such a recipe does not work for D (2) ℓ +1 for which the end nodes are both of type B .The full list of deformed affine Cartan matrices produced by our construction includes a numberof new examples, see SectionThere are several questions arising from our work which we plan to address in the future publica-tions. • The deformed integrals of motion related to D (2) ℓ +1 are not constructed yet. The affine rootsof type B need an additional study in general. • We introduced the commuting algebras in K as explicit integrals. We would like to constructthese algebras from some version of transfer matrices and obtain their spectrum by a Betheansatz method. • The nature of K algebra needs to be clarified. We expect that it should be recognized asa twisted version of the quantum toroidal algebra. Also, it is interesting to understand therelation of K to the universal W algebras of types B and C , D of [KL]. • There are similar comodule algebras for quantum toroidal algebras E n associated to gl n . Itis important to study the currents they produce and the corresponding integrals of motion. • The deformations of integrals of motion related to exceptional types seem to be related toother quantum algebras, see Section 5.2 for a discussion of type G .The plan of the paper is as follows. B. FEIGIN, M. JIMBO, E. MUKHIN, AND I. VILKOVISKIY
In Section 2 we introduce our convention and review generalities about the quantum toroidal gl algebra E .Section 3 is devoted to deformed W algebras and deformed integrals of motion of type A . Section3.1 reviews the non-affine case. In Section 3.2 we introduce the zeroth root current and recoverdeformed affine Cartan matrices. We discuss qq -characters in Section 3.3, screenings in Section 3.4,and the deformed IM in Section 3.5.In Section 4 we treat the case of other classical types. We introduce the algebra K (Section 4.1),along with its left E comodule structure (Section 4.2) and the boundary representations F B c , F CD c (Section 4.3). We construct the deformed W currents and obtain deformed finite type Cartan matricesin Section 4.4. We discuss the affinization in Section 4.5. Section 4.6 deals with qq -characters, Section4.7 with screenings, and Section 4.8 with the deformed commuting integrals.In Section 5 we make some additional remarks. We discuss integrals of motion of Knizhnik-Zamolodchikov type in Section 5.1 and the situation in type G in Section 5.2.The text is followed by three appendices. In Appendix A, we give a proof of the commutativity ofintegrals of motion. Appendix B discusses the existence of operator K ( u ). In Appendix C we give alibrary of deformed Cartan matrices obtained from K .2. The quantum toroidal algebra associated to gl In this section we recall the quantum toroidal algebra associated to gl and its Fock modules.2.1. Relations.
Fix q , q , q ∈ C × such that q q q = 1. We also fix values of ln q i and set q ai =exp( a ln q i ) for all a ∈ C . In this paper we assume that our choice of parameters is generic, meaningthat q a q b = 1 for a, b ∈ Z if and only if a = b = 0.We use the notation s c = q / c , t c = s c − s − c , b c = t c t t t ( c ∈ { , , } ) . We also use g ( z, w ) = Y i =1 ( z − q i w ) , ¯ g ( z, w ) = Y i =1 ( z − q − i w ) = − g ( w, z ) , κ r = Y i =1 (1 − q ri ) ( r ∈ Z ) . Note that κ = − t t t = P i =1 q − i − P i =1 q i .The quantum toroidal algebra E associated to gl is an associative algebra with parameters q , q , q generated by coefficients of the currents e ( z ) = X n ∈ Z e n z − n , f ( z ) = X n ∈ Z f n z − n , ψ ± ( z ) = exp (cid:0)X r> κ r h ± r z ∓ r (cid:1) , EFORMATIONS OF W ALGEBRAS 5 and an invertible central element C . The defining relations are as follows. ψ ± ( z ) ψ ± ( w ) = ψ ± ( w ) ψ ± ( z ) ,ψ + ( z ) ψ − ( w ) = ψ − ( w ) ψ + ( z ) g ( z, Cw )¯ g ( z, Cw ) ¯ g ( Cz, w ) g ( Cz, w ) ,g ( z, w ) ψ ± ( C ( − ∓ / z ) e ( w ) = ¯ g ( z, w ) e ( w ) ψ ± ( C ( − ∓ / z ) , ¯ g ( z, w ) ψ ± ( C ( − ± / z ) f ( w ) = g ( z, w ) f ( w ) ψ ± ( C ( − ± / z ) , [ e ( z ) , f ( w )] = 1 κ ( δ (cid:0) Cwz (cid:1) ψ + ( w ) − δ (cid:0) Czw (cid:1) ψ − ( z )) ,g ( z, w ) e ( z ) e ( w ) = ¯ g ( z, w ) e ( w ) e ( z ) , ¯ g ( z, w ) f ( z ) f ( w ) = g ( z, w ) f ( w ) f ( z ) , Sym z ,z ,z z z − [ e ( z ) , [ e ( z ) , e ( z )]] = 0 , Sym z ,z ,z z z − [ f ( z ) , [ f ( z ) , f ( z )]] = 0 , where Sym z ,...,z N F ( z , . . . , z N ) = 1 N ! X π ∈ S N F ( z π (1) , . . . , z π ( N ) ) . The relations for ψ ± ( z ) are equivalent to[ h r , h s ] = δ r + s, κ r C r − C − r r . (2.1)Let A be a Z graded algebra with a central element C . The completion of A in the positivedirection is the algebra ˜ A , linearly spanned by products of series of the form P ∞ i = M f i g i , where M ∈ Z , f i , g i ∈ A and deg g i = i .We call an A module V admissible if V is Z graded with degrees bounded from above, i.e., V = ⊕ Ni = −∞ V i , where V i = { v ∈ V | deg v = i } , and if C is diagonalizable. The completion ˜ A actson all admissible modules.Algebra E has a Z grading given bydeg e i = deg f i = deg h i = i , deg C = 0 . In other words, if we formally set deg z = 1, then e ( z ), f ( z ), ψ ± ( z ) all have degree zero.Let E ˜ ⊗ E be the tensor algebra E ⊗ E completed in the positive direction. We use the topologicalcoproduct ∆ : E → E ˜ ⊗ E as in [FJM]:∆ e ( z ) = e ( C − z ) ⊗ ψ + ( C − z ) + 1 ⊗ e ( z ) , (2.2) ∆ f ( z ) = f ( z ) ⊗ ψ − ( C − z ) ⊗ f ( C − z ) , ∆ ψ + ( z ) = ψ + ( z ) ⊗ ψ + ( C z ) , (2.3) ∆ ψ − ( z ) = ψ − ( C z ) ⊗ ψ − ( z ) , In the standard definition, the quantum toroidal algebra E associated to gl has two central elements. Here we setthe second central element to 1. B. FEIGIN, M. JIMBO, E. MUKHIN, AND I. VILKOVISKIY where C = C ⊗ C = 1 ⊗ C .Note that the coproduct can be extended to the map ∆ : ˜ E → E ˜ ⊗ E .The counit map ǫ : E → C is given by ǫ (cid:0) e ( z ) (cid:1) = ǫ (cid:0) f ( z ) (cid:1) = 0, ǫ (cid:0) ψ ± ( z ) (cid:1) = 1, ǫ (cid:0) C (cid:1) = 1.We set also ˜ f ( z ) = S (cid:0) f ( z ) (cid:1) = − ψ − ( z ) − f ( Cz )where S is the antipode. The coproduct reads∆ ˜ f ( z ) = ˜ f ( C z ) ⊗ ψ − ( z ) − + 1 ⊗ ˜ f ( z ) . Note that the topological Hopf algebra E depends on q , q , q symmetrically, in other words itdepends on the unordered set { q , q , q } .2.2. Vertex operators.
In this paper we often study vertex operators acting on bosonic Fock spaces.Here we give a brief description of these objects.A Heisenberg algebra H is an associative algebra generated by linearly independent elements h ( i ) r ,where r ∈ Z , i = 1 , . . . , N , with relations [ h ( i ) r , h ( j ) s ] = δ r + s, a ijr , where a i,jr ∈ C . Algebra H has a Z grading such that deg h ( i ) r = r .A weight α is a linear functional α : span C { h (1)0 , . . . , h ( N )0 } → C . For a weight α , we define theFock space F α as an H module generated by a vacuum vector v α such that h ( i ) r v α = 0 for r > h ( i )0 v α = α ( h ( i )0 ) v α for i = 1 , . . . , N , and such that F α = C [ h ( i ) r ] i =1 ,...,Nr ∈ Z < as a graded vector space.Given a weight α , let e Q α : F µ → F µ + α be a linear operator such that e Q α v µ = v µ + α and suchthat [ e Q α , h ( i ) r ] = 0, r = 0. We have e Q α z v = z − α ( v ) z v e Q α , v ∈ span C { h (1)0 , . . . , h ( N )0 } . In this paper, by a vertex operator we mean a formal series V ( z ) of the form V ( z ) = b exp (cid:0) X r> v − r z r (cid:1) exp (cid:0) X r ≥ v r z − r (cid:1) , v r ∈ span C { h (1) r , . . . , h ( N ) r } ( r ∈ Z ) , (2.4)where b ∈ C × . For any v ∈ F µ , V ( z ) v is a well defined Laurent series in z with values in F µ .The product of (2.4) with another vertex operator V ′ ( w ) = b ′ exp( P r> v ′− r w r ) exp( P r ≥ v ′ r w − r )has the form V ( z ) V ′ ( w ) = ϕ V,V ′ ( w/z ) : V ( z ) V ′ ( w ) : , where ϕ V,V ′ ( w/z ) ∈ C [[ w/z ]] is a formal power series called the contraction of V ( z ) and V ′ ( w ), andthe vertex operator: V ( z ) V ′ ( w ) := bb ′ exp (cid:0) X r> ( v − r z r + v ′− r w r ) (cid:1) exp (cid:0) X r ≥ ( v r z − r + v ′ r w − r ) (cid:1) is called the normal ordered product of V ( z ) and V ′ ( w ). Obviously : V ( z ) V ′ ( w ) :=: V ′ ( w ) V ( z ) :.Our vertex operators will depend on various parameters, q , q , µ , C , etc. One can think of theseparameters as variables or as generic complex numbers. We call α a formal monomial if α is a productof parameters in rational powers. For example α can be of the form q a i q b i with a i , b i ∈ Q . EFORMATIONS OF W ALGEBRAS 7
We often study the case when the contraction has the form ϕ V,V ′ ( w/z ) = Y i (1 − α i w/z ) − n i (2.5)where n i ∈ Z and α i are formal monomials. Equation (2.5) is equivalent to saying[ v r , v ′ s ] = δ r + s, P i n i α ri r ( r ∈ Z > ) . We use the notation C (cid:0) V ( z ) , V ′ ( w ) (cid:1) = X i n i α i to represent (2.5), and by abusing the language we also call the sum in the right hand side thecontraction of V ( z ) and V ′ ( w ).Note that C (cid:0) V ( pz ) , V ′ ( p ′ w ) (cid:1) = C (cid:0) V ( z ) , V ′ ( w ) (cid:1) · p ′ /p .We call a contraction rational if P i n i α i has a finite number of summands. We call a contractionelliptic if we can write P i n i α i = P j m j β j / (1 − β ) where m j ∈ Z , β i , β are formal monomials, andthe summation over j is finite.We also often study screening currents and screening operators. Screening currents S ( z ) have theform S ( z ) = e Q α z s exp (cid:0) X r> s − r z r (cid:1) exp (cid:0) X r> s r z − r (cid:1) , s r ∈ span C { h (1) r , . . . , h ( N ) r } ( r ∈ Z ) , (2.6)where α is a weight. In particular, S ( z ) = e Q α z s S osc ( z ) where S osc ( z ) is a vertex operator with¯ s = 0.Given a screening current S ( z ) = e Q α z s S osc ( z ) and a vertex operator V ( w ), the normal orderingis given by : S ( z ) V ( w ) :=: V ( w ) S ( z ) := e Q α z s : S osc ( z ) V ( w ) :.The screening operator S is the constant term of zS ( z ): S = 12 πi Z S ( z ) dz . (2.7)Note that S is a well defined operator F µ → F µ + α if and only if µ ( s ) ∈ Z .Let V ( z ), V ′ ( z ) be vertex operators of the form (2.4), and set A ( z ) =: V ( z ) V ′ ( z ) − : . Assume that there are formal monomials p , p , p such that C (cid:0) A ( z ) , V ( w ) (cid:1) = − (1 − p )(1 − p ) , C (cid:0) V ( w ) , A ( z ) (cid:1) = − (1 − p − )(1 − p − ) , (2.8) C (cid:0) A ( z ) , V ′ ( w ) (cid:1) = (1 − p − )(1 − p − ) , C (cid:0) V ′ ( w ) , A ( z ) (cid:1) = (1 − p )(1 − p ) . We also assume that if v = av ′ for some a ∈ C then a = − log p / log p .Then we can construct a screening operator commuting with V ( z ) + V ′ ( z ) as follows.We define the screening current S ( z ) of the form (2.6) by A ( z ) = 1 − p − p p − p − : S ( p − z ) S ( p z ) − : . (2.9) B. FEIGIN, M. JIMBO, E. MUKHIN, AND I. VILKOVISKIY
This amounts to the relations a r = v r − v ′ r = ( p r − p − r ) s r ( r = 0) ,e a = e v − v ′ = b ′ b − p − p p − p − p − s . In addition we choose a weight α so that α ( v ) = 2 log p , α ( v ′ ) = − p . Then we have the following well known lemma.
Lemma 2.1.
When the screening operator S is well defined, we have [ S, V ( z ) + V ′ ( z )] = 0 . Proof.
We have S ( z ) V ( w ) = 1 − p p w/z − p w/z : S ( z ) V ( w ) : , V ( w ) S ( z ) = 1 − p − p − z/w − p − z/w p : S ( z ) V ( w ) : ,S ( z ) V ′ ( w ) = 1 − p − p − w/z − p − w/z : S ( z ) V ′ ( w ) : , V ′ ( w ) S ( z ) = 1 − p p z/w − p z/w p − : S ( z ) V ′ ( w ) : . It follows that [ S ( z ) , V ( w )] = (1 − p ) δ ( p w/z ) : S ( z ) V ( w ), where δ ( z ) = P i ∈ Z z i is the formal deltafunction. Integrating over z , we obtain [ S, V ( w )] = (1 − p ) p w : S ( p w ) V ( w ) :. Similarly, we obtain[ S, V ′ ( w )] = (1 − p − ) p − w : S ( p − w ) V ′ ( w ) :.On the other hand, from the definitions we have: S ( p w ) V ( w ) := 1 − p − p p − p − : S ( p − w ) V ′ ( w ) : . Hence the lemma follows. (cid:3)
We note that in (2.8) one can replace A ( z ) with A ( pz ) for any formal monomial p . Then thescreening current S ( z ) will be also shifted to S ( pz ) and S will be replaced by a constant multiple p − S . Therefore Lemma 2.1 will still hold.When p = p = p , the right hand sides of (2.8) become symmetric in p and p . Interchangingthe roles of p and p , one can construct another screening operator commuting with V ( z ) + V ′ ( z ).It is easy to check that if one has p p p = 1 in (2.8), then the corresponding screening current isan ordinary fermion [BFM].2.3. The Fock modules.
Algebra E has three families of Fock representations F c ( u ), where c ∈{ , , } and u ∈ C × . We call c the color of the Fock module and u the evaluation parameter. Quitegenerally, if the central element C acts on an E module M by a scalar k , then we say that M haslevel k and often write C = k . The Fock module F c ( u ) has level s c .The Fock modules are irreducible with respect to the Heisenberg subalgebra of E generated by ψ ± ( z ). Thus we have the identification of vector spaces F c ( u ) = C [ h − r ] r> v c , where v c is the vacuumvector such that h r v c = 0 ( r >
0) and Cv c = s c v c . Then the generators e ( z ) and ˜ f ( z ) are given byvertex operators e ( z ) b c : V c ( z ; u ) : , ˜ f ( z ) b c : V c ( z ; u ) − : , EFORMATIONS OF W ALGEBRAS 9 where b c = − t c /κ and V c ( z ; u ) = u exp (cid:16)X r> κ r h − r − q rc z r (cid:17) exp (cid:16)X r> κ r h r − q rc q r/ c z − r (cid:17) . (2.10)Note that V c ( s − c z ; u ) ψ + ( z ) = ψ − ( s − c z ) V c ( s c z ; u ) . (2.11)For vertex operators (2.10), the contractions are rational. The non-trivial ones are C (cid:0) V c ( z ; u ) , V c ( w ; v ) (cid:1) = − κ − q c , C (cid:0) ψ + ( q − / c z ) , V c ( w ; u ) (cid:1) = C (cid:0) V c ( z ; u ) , ψ − ( w ) (cid:1) = − κ , (2.12) C (cid:0) ψ + ( q − / c z ) , ψ − ( w ) (cid:1) = − (1 − q c ) κ . W algebras and Integrals of Motion of type A
The deformed W algebras are introduced in [FR1] starting from a deformed Cartan matrix, or in amore general context, a Dynkin quiver in [KP]. Now it is well known that the deformed W currentsof type A are essentially the images of the current e ( z ) of the quantum toroidal gl algebra E actingon tensor products of Fock modules. In this section we revisit this connection, and explain how theCartan matrix and integrals of motion are recovered from the data of Fock modules. Here we use[FKSW], [FHHSY], [FJM], [BFM].3.1. The current e ( z ) and root currents A i ( z ) . Fix a tensor product of ℓ + 1 Fock modules F c ( u ) ⊗ · · · ⊗ F c ℓ +1 ( u ℓ +1 ) , c , . . . , c ℓ +1 ∈ { , , } , (3.1)which has level C = Q ℓ +1 j =1 s c j . By coproduct formulas (2.2), (2.3), the current e ( z ) acts as a sum ofvertex operators in ℓ + 1 bosons e ( z ) = b c Λ ( z ) + · · · + b c ℓ +1 Λ ℓ +1 ( z ) , Λ i ( z ) = 1 ⊗ · · · ⊗ ⊗ i ⌣ V c i ( a A i z ; u i ) ⊗ ψ + ( s − c i +1 a A i +1 z ) ⊗ · · · ⊗ ψ + ( s − c ℓ +1 a A ℓ +1 z ) , where a A i = ℓ +1 Y j = i +1 s − c j . (3.2)Note that the current e ( z ) with evaluation parameters au i where a ∈ C × , is obtained from e ( z )with evaluation parameters u i by scalar multiplication by a .To each neighboring pair of Fock spaces · · · ⊗ i ⌣ F c i ( u i ) ⊗ i +1 ⌣ F c i +1 ( u i +1 ) ⊗ · · · , i = 1 , . . . , ℓ , we associate a current A i ( z ). Namely, for i = 1 , . . . , ℓ , let A i ( z ) be given by a normally ordered ratio: A i ( z ) =: Λ i (( a A i ) − z )Λ i +1 (( a A i ) − z ) : , (3.3)where a A i is given in (3.2). We call A i ( z ) a root current.From (2.12) we have the following contractions: C (cid:0) Λ i ( z ) , Λ j ( w ) (cid:1) = − κ ( i < j ) , − κ / (1 − q c i ) ( i = j ) , i > j ) , where i, j = 1 , . . . , ℓ + 1.Denote the contractions between root currents by B i,j = C (cid:0) A i ( z ) , A j ( w ) (cid:1) , ( i, j = 1 , . . . , ℓ ) . (3.4)The only non-trivial ones are B i − ,i = B i,i − = t t t t c i , B j,j = − t t t t c j t c j +1 ( s c j s c j +1 − s − c j s − c j +1 ) , (3.5)where i = 2 , . . . , ℓ , j = 1 , . . . , ℓ .In other words, only neighboring A i ( z ) have non-trivial contractions which can be described by2 × A matrices in AppendixC.2.Note that all these contractions are rational and do not depend on evaluation parameters u i .The matrix of contractions B = (cid:0) B i,j (cid:1) i,j =1 ,...,ℓ should be compared with the Cartan matrix of type A . We say that A i ( z ) is a bosonic current of type c if it corresponds to a pair F c ⊗ F c , and a fermioniccurrent of type c if it corresponds to a pair F c ⊗ F c , where { c , c , c } = { , , } . Example 3.1.
Consider the tensor product F ( u ) ⊗ F ( u ) ⊗ F ( u ) ⊗ F ( u ) ⊗ F ( u ) ⊗ F ( u ) , C = s s s . (3.6)The matrix B is given by B = − t t ( s + s − ) t t t t − t t ( s + s − ) t t t t t t t
00 0 t t t − t t ( s + s − ) t t t t t . (3.7)We picture this matrix as a Dynkin diagram where a circle represents a bosonic node and a crossedcircle a fermionic node. We also attach a marking by 1 , , (cid:3) EFORMATIONS OF W ALGEBRAS 11
The diagram in the example looks like a diagram of sl | but it is different because of the markings.In the diagram of sl | all bosonic nodes have the same markings and all fermionic nodes have thesame markings too: And if all colors are the same, e.g. c i = 2, i = 1 , . . . , ℓ + 1, then B i,j = − t t (cid:0) ( s + s − ) δ ij − δ i,j +1 − δ i,j − (cid:1) . (3.8)Up to an overall multiple − t t , this coincides with the matrix B ( q, t ) of type A ℓ in [FR1], eq.(2.4)with the identification s = qt − . However, in contrast to [FR1], we have one extra boson. Indeed, by construction our root currentscommute with the diagonal Heisenberg algebra[ A i ( z ) , ∆ ( ℓ ) h r ] = 0 ( i = 1 , . . . , ℓ, r = 0) , where ∆ ( ℓ ) signifies the iterated coproduct with ∆ (1) = ∆. This extra boson allows us to have rationalcommutation relations between Λ i ( z )’s, and will play a role in the affinization to be discussed below.3.2. Root current A ( z ) . Fix an arbitrary µ ∈ C × with | µ | <
1. We define the dressed version ofthe current e ( z ) of the algebra E by e ( z ) = e ( z ) ψ + µ ( C − z ) − , ψ + µ ( z ) = ∞ Y s =0 ψ + ( µ − s z ) = exp (cid:0)X r> κ r − µ r h r z − r (cid:1) . (3.9)The coefficients of the dressed current e ( z ) are elements of e E .While the current e ( z ) has rational commutation relations, the dressed current e ( z ) satisfies theelliptic commutation relations: e ( z ) e ( w ) = e ( w ) e ( z ) Y i =1 Θ µ ( q i w/z )Θ µ ( q − i w/z ) . Here we use symbols for infinite products and theta functions( z , . . . , z r ; p ) ∞ = r Y i =1 ∞ Y k =0 (1 − z i p k ) , Θ p ( z ) = ( z, pz − , p ; p ) ∞ . These relations have to be understood as equality of matrix elements. Namely, if M is an admissible E module then the coefficients of z − i w − j of both sides are well defined operators of degree i + j inEnd M and they are equal.There are two separate motivations for the definition of the dressed current. One is the form of theintegrals of motion, see Section 3.5. In this section we discuss a different reason: the appearance of thecurrent A ( z ) corresponding to the affine node and ultimately of the screening operator corresponding to the affine node, see Section 3.4. In type A these two points lead to the same definition of thedressed current.The dressed current e ( z ) has the form e ( z ) = b c Λ ( z ) + · · · + b c ℓ +1 Λ ℓ +1 ( z ) , where Λ i ( z ) = Λ i ( z )∆ ( ℓ ) ψ + µ ( C − z ) − .For i = 1 , . . . , ℓ , the ratios of the terms in dressed and undressed currents are the same:: Λ i ( z ) Λ i +1 ( z ) :=: Λ i ( z )Λ i +1 ( z ) := A i ( a Ai z ) , i = 1 , . . . , ℓ . However the contractions of individual terms change to elliptic ones: C (cid:0) Λ i ( z ) , Λ j ( w ) (cid:1) = C (cid:0) Λ i ( z ) , Λ j ( w ) (cid:1) + κ − µ . (3.10)This allows us to define a new current A ( z ) which has rational contractions with all root currents A i ( z ), i = 1 , . . . , ℓ and is independent of them. Namely, define the zeroth root current by A ( z ) =: Λ ℓ +1 ( z ) Λ ( µz ) :=: Λ ℓ +1 ( z )Λ ( µz ) ∆ ( ℓ ) ψ + ( C − µz ) : . The contractions involving A ( z ) are as follows. Retaining notation (3.4) we have B i, = B ,i = 0if i = 0 , , ℓ . The non-trivial ones are: B , = − t t t t c ℓ +1 t c ( s c ℓ +1 s c − s − c ℓ +1 s − c ) ,B ,ℓ = B ℓ, = t t t t c ℓ +1 , B , = t t t t c Cµ − , B , = t t t t c C − µ , where C = Q ℓ +1 j =1 s c j is the total level and ℓ > ℓ = 1, B , and B , take the form: B , = − κ (cid:16) t c + Cµ − t c (cid:17) , B , = − κ (cid:16) t c + C − µt c (cid:17) . Thus we have a U q b sl type deformed Cartan matrix corresponding to c = c = 2 and a U q b gl | typedeformed Cartan matrix corresponding to c = 1, c = 2: − t t (cid:18) s + s − − − Cµ − − − C − µ s + s − (cid:19) , t (cid:18) t t + t Cµ − t + t C − µ t (cid:19) . We call the extended matrix of contractions ˆ B = (cid:0) B i,j (cid:1) i,j =0 , ,...,ℓ the symmetrized deformed affineCartan matrix. EFORMATIONS OF W ALGEBRAS 13
Example 3.2.
We continue to consider the tensor product (3.6). The matrix ˆ B is given byˆ B = t t t Cµ − t t t t C − µ − t t ( s + s − ) t t t t − t t ( s + s − ) t t t t t t t
00 0 0 t t − t t ( s + s − ) t t t t t t t . (3.11)With the root A ( z ) the Dynkin diagram becomes the following.
21 1 3 2 1 (3.12)The affine node is fermionic and we label it by 2, since it corresponds to the ratio of the last andfirst terms obtained from F ( u ) and F ( u ). However, note that there is an arbitrary parameter µ which does not appear in the diagram. (cid:3) One can easily describe the restrictions on the markings of the Dynkin diagram which can appear.There is a single global condition: if there are a c fermionic nodes of type c , c = 1 , ,
3, then a ≡ a ≡ a modulo 2. In addition, there are several local conditions. For example, neighboring bosonicnodes have to have the same marking, neighboring bosonic and fermionic nodes cannot have thesame marking.In particular, (3.12) is not a diagram of affine sl | , as in any affine Dynkin diagram of type sl m | n the number of simple odd roots is even (which is true in our setting if all bosonic nodes have thesame type).If all colors are the same, e.g. c i = 2, i = 1 , . . . , ℓ + 1, then the matrix ˆ B is, up to a scalar multiple,the deformed affine Cartan matrix of type A in [KP].The following lemma shows that in all cases the currents A i ( z ), i = 0 , , . . . , ℓ , are independentgenerating currents of the Heisenberg algebra provided µ = 1 , C . Lemma 3.3.
We have det ˆ B = κ ℓ +11 ℓ +1 Y j =1 t − c j × (1 − µ )( C − − Cµ − ) . Proof.
It is easy to see that det ˆ B as a function of x = C − µ is a linear function of x + x − . One cancheck that if x = C − then the vector ( y , y , . . . , y ℓ ) T with y = C and y i = s c · · · s c i (1 ≤ i ≤ ℓ ) isin the kernel of ˆ B . It follows that det ˆ B = a ( C + C − − x − x − ) with some a . The coefficient a canbe determined from the behavior as x → ∞ det ˆ B = ( − ℓ ˆ B , ˆ B ,ℓ ℓ Y j =2 ˆ B j,j − + O (1) = − x ℓ +1 Y j =1 κ t c j + O (1) . (cid:3) Currents Y i ( z ) and qq -characters. In this section we describe current e ( z ) as a qq -characterin the spirit of [N], [KP].In what follows we write ˆ B = ˆ D ˆ C , ˆ C = ( C i,j ) ℓi =0 , choosing a diagonal matrix ˆ D = diag( d , . . . , d ℓ )in such a way that d i = − t t t t c , C i,i = s c + s − c , if A i ( z ) is bosonic of type c,d i = t c , C i,i = s c − s − c , if A i ( z ) is fermionic of type c. The currents A i ( z ) correspond to roots. In order to understand the combinatorics of deformed W -currents, it is convenient to introduce currents Y i ( z ), i = 0 , . . . , ℓ , which correspond to the fun-damental weights.Define the modes a i,r by setting A i ( z ) = e a i, : e P r =0 a i,r z − r : . Write elements of non-symmmetrized deformed affine Cartan matrix as a sum of formal monomials: C i,j = P k m ( k ) i,j − P s n ( s ) i,j and define Y i ( z ) by the set of equations A j ( z ) =: Y i Y k Y i ( m ( k ) i,j z ) Y s (cid:16) Y i ( n ( s ) i,j z ) (cid:17) − : , j = 0 , . . . , ℓ, where Y i ( z ) are of the form Y i ( z ) =: e P r =0 y i,r z − r : × ( e y i, if A i ( z ) is bosonic ,e Q y,i z y i, if A i ( z ) is fermionic , and the zero mode operator e Q y,i is defined by the condition e a j, e Q y,i = q − δ i,j c e Q y,i e a j, if A i ( z ) is afermionic current of type c .Due to Lemma 3.3, such Y i ( z )’s exist and are unique up to an overall shift of zero modes y i, .Moreover, we have C ( Y i ( z ) , A j ( w )) = ± C ( A j ( w ) , Y i ( z )) = d i δ i,j , (3.13)where the sign is + for bosonic nodes and − for fermionic nodes.We explain this definition in an Example 3.4 below.Let us set up some language convenient for discussing combinatorics. We adopt the shorthandnotation l a,b = Y l ( s a s b z ), l = 0 , , . . . , ℓ .Let A be the commutative ring in formal free variables l ± a,b , l = 0 , , . . . , ℓ , a, b ∈ C .Let π : A → A be the ring homomorphism sending a,b l a,b l a,b , l = 0.For a formal monomial α = s a s b define a ring homomorphism τ α : A → A shifting indices by( a, b ). For example, we have τ s a s b c,d = a + c,b + d .The combinatorial study of the qq -characters and q -characters is based on the concept of dominantmonomial, see [FR2]. The characters are obtained by a combinatorial algorithm “expanding thedominant monomials”, see [FM], [KP].If current A l ( z ) is bosonic, then a monomial m ∈ A is called l -dominant if m does not containnegative powers of l a,b . If current A l ( z ) is fermionic, then we suggest to call a monomial m ∈ A l -dominant if m is a product of ˜ l a,b with ˜ l = l and monomials of the form l a,b τ q c ( l − a,b ), where in theCartan matrix we have either ˆ C l,l − = t c or ˆ C l,l +1 = t c . EFORMATIONS OF W ALGEBRAS 15
Example 3.4. . We continue to work with (3.7). In this case we have d = t , d = d = − t t , d = t , d = − t t , d = t , and ˆ C = t t Cµ − t − C − µ s + s − − − s + s − − t t t
00 0 0 − s + s − − t t t . Reading off the columns of ˆ C , we obtain A i ( z ) currents in terms of the Y j ( z )’s: A ( z ) = , − , − − − γ, , − , − , A ( z ) = γ − , − − γ +1 , +1 , − , − , , A ( z ) = − , , − , , − , − ,A ( z ) = − , − , − − , − , , A ( z ) = , − − , , , − − , − − , , A ( z ) = , − − , − , , − − , , where we set s γ = Cµ − .Set δ i = ln q i , δ + δ + δ = 0. Then zero modes are given by a , = δ y , − y , + δ y , , a , = δ y , + 2 y , − y , , a , = − y , + 2 y , + δ y , , a , = − y , + δ y , − y , , a , = δ y , + 2 y , + δ y , , a , = δ y , − y , + δ y , . These conditions uniquely determine y i, ’s up to a common additive shift.The current e ( z ) can be written: e ( a − z ) = Λ ( a − z ) (cid:16) b c b c A − ( s z ) + b c A − ( s z ) A − ( s z ) + b c A − ( s z ) A − ( s z ) A − ( s z )+ b c A − ( s z ) A − ( s z ) A − ( s z ) A − ( s s z ) + b c A − ( s z ) A − ( s z ) A − ( s z ) A − ( s s z ) A − ( s s z ) (cid:17) , which we symbolically depict as follows. Λ Λ Λ . . . Λ ℓ Λ ℓ +1 A − A − A − A − ℓ − A − ℓ Ignoring constants b c i we write it using Y i ( z ) in the form: χ = γ, − − γ +2 , , + − , , + − , , − − , + , − , , + − , , − , + , − , , − , . (3.14)Following [N], we call the expression χ the qq -character of the vector representation correspondingto the Dynkin diagram (3.12).According to the general rule, for l = 1 , , l -dominantif it is a product of monomials l a,b and ˜ l ± a,b with a, b ∈ C and ˜ l = l . A monomial is -dominant if itis a product of monomials of the form a,b − a +2 ,b , a,b − a − ,b − , and l ± a,b with l = 0; -dominant if itis a product of monomials of the form a,b − a +2 ,b , a,b − a,b +2 , and l ± a,b with l = 3; -dominant if it is aproduct of monomials of the form a,b − a,b +2 , a,b − a − ,b − , and l ± a,b with l = 5.In (3.14) the l -th term is l -dominant for 1 ≤ l ≤
5, and the last term is -dominant. The ( l + 1)stterm is obtained by multiplying the l -th term by the inverse of current A l ( z ) with an appropriateshift, see (3.3). Moreover, the last term and the first term are connected as follows A − ( s s s z ) , − , , − , = , − , γ +2 , = τ µ ( γ, − − γ +2 , , ) . (3.15) (cid:3) While our qq -character is closely connected to that of [KP], they are different in several ways. First,our first monomial is l -dominant for l = 1 , . . . , ℓ , but not -dominant. Second, our qq -character is offinite type, namely, applying π we obtain the qq -character corresponding to the non-affine Cartanmatrix. In particular, our qq -character contains finitely many terms. Third, the fermionic nodes arenot considered in [KP] . Fourth, the variables a,b do play an important role as they correspond tothe dressing of the current e ( z ) and the finite type qq -character “closes up” in the sense of (3.15).The qq -characters of [KP] commute with all screenings operators including the one correspondingto the zeroth node and correspond to modules of quantum toroidal algebra E ℓ +1 associated to gl ℓ +1 .If we start with our qq -character and formally require such commutativity, we would have to addinfinitely many terms which correspond to the vector representation of E ℓ +1 , while [KP] starts witha dominant monomial which produces a qq -character corresponding to a Fock module of E ℓ +1 .The usual q -character of the evaluation vector representation of U q b sl ℓ in the sense of [FR1] isrecovered after applying π in the case of ⊗ ℓ +1 i =1 F ( u i ), when all Fock spaces are of the same sort. Inthis case we have b − e ( z ) = − ,γ +1 , + , − , + , − , + · · · + ℓ − ,ℓ +1 ,ℓ where s γ = Cµ − .3.4. Screenings.
We discuss the screening operators.First, we clearly have [ A i ( z ) , Λ j ( w )] = 0 whenever j i, i + 1 mod ℓ + 1.Moreover, for i = 0 , , . . . , ℓ , the non-trivial contractions of A i ( z ) have the form (2.8) where A ( z ) = b c i b c i +1 A i ( z ) , V ( z ) = b c i Λ i (( a Ai ) − z ) , V ′ ( z ) = b c i +1 Λ i +1 (( a Ai ) − µ δ ,i z ) . Here we set Λ ℓ +1 ( z ) = Λ ( z ). If the node is bosonic, c i = c i +1 , then p = p = s c , p = s b ,where { c i , b, c } = { , , } . If the node is fermionic, then p = s c i +1 , p = s d , p = s c i , where { c i , c i +1 , d } = { , , } .Accordingly we define two screening currents S ± i ( z ) for each bosonic node and one screening current S fi ( z ) for each fermionic node, see (2.9): c i = c i +1 : A i ( z ) = q c i : S + i ( s − b z ) S + i ( s b z ) : = q c i : S − i ( s − c z ) S − i ( s c z ) : , (3.16) c i = c i +1 : A i ( z ) = s − d : S fi ( s − d z ) S fi ( s d z ) : , (3.17)where ( c i , b, c ) = cycl (1 , ,
3) and ( c i , c i +1 , d ) = { , , } . If c i = c i +1 , then S fi ( z ) is a fermioniccurrent.Introduce the corresponding screening operators S ± i , S fi by (2.7) when well-defined.Let S i stand for either S ± i when A i ( z ) is a bosonic current, or S fi when A i ( z ) is a fermionic current.It follow from Lemma 2.1 that the currents e ( z ) and e ( z ) both commute with all S i with i = 0, and Fermionic root currents A i ( z ) appeared in [BFM]. EFORMATIONS OF W ALGEBRAS 17 that e ( z ) commutes with S up to a µ -difference:[ S i , e ( z )] = [ S i , e ( z )] = 0 , i = 1 , . . . , ℓ , (3.18) [ S , e ( z )] = b c [ S , Λ ( z ) − Λ ( µz )] . (3.19)The relation (3.19) implies the commutativity of S with integrals of motion, see Theorem 3.7 below.In view of these relations, we call e ( z ) the deformed W -current of type A . Example 3.5.
We again illustrate the construction of screenings on the example of (3.6), see alsoExample 3.4. In this case Λ ( z ) Λ ( µz ) = s − S f ( s − z ) S f ( s z ) , Λ ( z ) Λ ( z ) = q S ± ( s − ± / a A z ) S ± ( s (5 ± / a A z ) , Λ ( z ) Λ ( z ) = q S ± ( s − ± / a A z ) S ± ( s (5 ± / a A z ) , Λ ( z ) Λ ( z ) = s − S f ( s − a A z ) S f ( s a A z ) , Λ ( z ) Λ ( z ) = q S ± ( s − ± a A z ) S ± ( s ± a A z ) , Λ ( z ) Λ ( z ) = s − S f ( s − a A z ) S f ( s a A z ) . In particular the zero modes of screening operators satisfy u u = s − q − s f , , u u = q q − s ± , (5 ± / , u u = q q − s ± , (5 ± / ,u u = s − q − s f , , u u = q q − s ± , ± , u u = s − q − s f , . Recall that we need s i, to act as an integer in order to have well define screening operators. It dictatessome conditions on u i . Note that there is no choice of u i such that s i, = 0 for i = 0 , , . . . , (cid:3) Integrals of motion.
One of our primary interests is in constructing commuting families of op-erators. In the case of type A the source of such families is the standard transfer matrix construction,see [FJM]. Here we recall the answer, which appeared first in [FKSW].Define the Feigin-Odesskii kernel function [FO]: ω ( z ) = Θ µ ( z )Θ µ ( q − z )Θ µ ( q z )Θ µ ( q z ) . (3.20)The function ω ( z ) satisfies a series of identities parametrized by m, n ∈ Z ≥ , see [FKSW]:Sym z , ··· ,z m + n Y ≤ i ≤ mm +1 ≤ j ≤ m + n ω ( z j /z i ) − = Sym z , ··· ,z m + n Y ≤ i ≤ mm +1 ≤ j ≤ m + n ω ( z i /z j ) − . (3.21)The following theorem describes the integrals of motion. Theorem 3.6. [FKSW] , [FJM] The following elements { I n } ∞ n =1 are mutually commutative: I n = Z · · · Z e ( z ) · · · e ( z n ) · Y j
In [FKSW] the theorem is proved directly (on tensor products of Fock spaces) with the useof the identity (3.21).In [FJM], it is shown that I n , up to a constant, are Taylor coefficients of the transfer matrixcorresponding to the Fock space F ( u ), where µ is the twist parameter. Then the commutativityfollows from the standard argument with R-matrix. (cid:3) We note that in the construction of integrals of motion one can replace ω ( z ) with functions ω ( z ) = Θ µ ( z )Θ µ ( q − z )Θ µ ( q z )Θ µ ( q z ) , ω ( z ) = Θ µ ( z )Θ µ ( q − z )Θ µ ( q z )Θ µ ( q z ) . It is known that while individual integrals I n with n ≥ I n does not.Let us verify the commutativity with screenings. Theorem 3.7.
The integrals of motion commute with all screening operators, [ S i , I n ] = 0 , i = 0 , , . . . , ℓ, n ≥ . Proof.
For i = 0, this follows readily from (3.18). We check the case i = 0 using (3.19). Byreplacing ω ( x ) by ω ( x ) or ω ( x ) if necessary, it suffices to consider the two cases, c = c ℓ +1 = 2, or c = 1 , c ℓ +1 = 3. We assume | q | , | q | > c = c ℓ +1 = 2, S = S − . Then[ S , Λ ( z )] = const. A (1) ( z ) , A (1) ( z ) = z : S ( s − µ − z ) Λ ( z ) : . Noting the symmetry of the integrand and the commutativity S ( z ) e ( w ) = e ( w ) S ( z ) as meromor-phic functions, we obtain[ S , I n ] ∝ n X i =1 Z · · · Z e ( z ) · · · ( A (1) ( z i ) − A (1) ( µz i )) · · · e ( z n ) · Y j
1, one easily checks that the z integral vanishes.Otherwise one has to pick the residues at z = q − z , µq z coming from f (1) l ( z /z ). Using again the EFORMATIONS OF W ALGEBRAS 19 relation q : S ( s − z ) Λ ( µz ) :=: S ( s z ) Λ ℓ +1 ( z ) :, we find that[ S , I n ] ∝ Z · · · Z ( A (2) ( z ) − A (2) ( µq z )) e ( z ) · · · e ( z n ) · n Y k =3 ω (2)2 ( z k /z ) · Y ≤ j 7→ − E ( z ) , K ± ( z ) K ± ( z ) , (4.6) τ a : K → K , E ( z ) E ( az ) , K ± ( z ) K ± ( az ) ( a ∈ C × ) . (4.7)4.2. Left comodule structure. The algebra K does not seem to have a natural coproduct. Insteadit is a comodule over the quantum toroidal algebra E .Consider the tensor product algebra of E and K . We denote by E ˜ ⊗ K the completion of the tensoralgebra E ⊗ K with respect to the homogeneous grading in the positive direction. EFORMATIONS OF W ALGEBRAS 21 Proposition 4.1. The following map ∆ : K → E ˜ ⊗ K endows K with a structure of a left E -comodule: ∆ E ( z ) = e ( C − z ) ⊗ K + ( z ) + 1 ⊗ E ( z ) + ˜ f ( C z ) ⊗ K − ( z ) , ∆ K + ( z ) = ψ + ( C − C − z ) ⊗ K + ( z ) , (4.8) ∆ K − ( z ) = ψ − ( C z ) − ⊗ K − ( z ) , ∆ C = C ⊗ C , where C = C ⊗ , C = 1 ⊗ C . (cid:3) Proof. Checking that ∆ preserves the relations (4.1)–(4.5), especially the last one, demands a straight-forward but long calculation. We use the identity (cid:16) z z − z z (cid:17) ( g , g , − g , g , ) + (cid:16) z z − z z (cid:17) g , ( g , + g , ) + (cid:16) z z − z z (cid:17) ( g , + g , ) g , = κ (cid:16) − z z z (cid:17) ( z + z )( z + z ) z g , , where g i,j = g ( z i , z j ).Coassociativity (∆ ⊗ id) ◦ ∆ = (id ⊗ ∆) ◦ ∆ mapping K → E ˜ ⊗ E ˜ ⊗ K is a short direct calculation.The counit property ( ǫ ⊗ id) ◦ ∆ = id mapping K → K is obvious. (cid:3) Boundary Fock modules. We describe several representations of algebra K , given in one freeboson.For a complex number k ∈ C × , let H k be the Heisenberg algebra generated by { H r } r =0 with therelations [ H r , H s ] = − δ r + s, κ r (1 + k r ) /r .Keeping in mind the Dynkin types B and C , D , for c ∈ { , , } we set H B c = H s / c , H CD c = H s − c , and denote by F B c , F CD c the corresponding Fock modules of the Heisenberg algebra.We have three K modules of type B . Set k B c = (1 + s c )( s d − s b ) κ , ( c, d, b ) = cycl (1 , , . Define a vertex operator ˜ K ± c ( z ) by˜ K ± c ( z ) = exp (cid:16) X ± r> 11 + s − rc H r z − r (cid:17) . We have ˜ K ± c ( z ) ˜ K ± c ( s c z ) = K ± ( z ). Set ˜ K c ( z ) = ˜ K − c ( z ) ˜ K + c ( s c z ). Proposition 4.2. For c ∈ { , , } , the map K → H B c sending E ( z ) k B c ˜ K c ( z ) , K ± ( z ) K ± ( z ) , C s / c , endows F B c with a structure of a K module of level s / c .Proof. The proposition is proved by a direct computation. (cid:3) We also have three K modules of type CD . Proposition 4.3. For c ∈ { , , } , the map K → H CD c sending E ( z ) , K ± ( z ) K ± ( z ) , C s − c , endows F CD c with a structure of a K module of level s − c .Proof. The proposition is proved by a direct computation. (cid:3) Using the comodule map ∆ we obtain the following corollary. Corollary 4.4. There exists an algebra homomorphism K → E ˜ ⊗ H B c such that E ( z ) e ( s − / c z ) ⊗ K + ( z ) + k B c ⊗ ˜ K c ( z ) + ˜ f ( s / c z ) ⊗ K − ( z ) ,K + ( z ) ψ + ( C − s − / c z ) ⊗ K + ( z ) , K − ( z ) ψ − ( s / c z ) − ⊗ K − ( z ) ,C s / c C . Similarly, there exists an algebra homomorphism K → E ˜ ⊗ H CD c such that E ( z ) e ( s c z ) ⊗ K + ( z ) + ˜ f ( s − c z ) ⊗ K − ( z ) ,K + ( z ) ψ + ( C − s c z ) ⊗ K + ( z ) , K − ( z ) ψ − ( s − c z ) − ⊗ K − ( z ) ,C s − c C . (cid:3) It seems likely that the homomorphisms in Corollary 4.4 are injective. If so, it would give us aninclusion of algebra K into algebra E extended by an extra Heisenberg algebra.Note that twisting the boundary modules by automorphism (4.6) we obtain a new set of boundarymodules. Twisting by automorphisms (4.7) leads to isomorphic modules.4.4. Root currents A i ( z ) of types B , C , D . Fix a sequence of colors c , . . . , c ℓ +1 ∈ { , , } , andconsider a K module defined as a tensor product of ℓ Fock modules of E with a boundary Fockmodule F B c or F CD c : F c ( u ) ⊗ · · · ⊗ F c ℓ ( u ℓ ) ⊗ F B c ℓ +1 , (4.9) F c ( u ) ⊗ · · · ⊗ F c ℓ ( u ℓ ) ⊗ F CD c ℓ +1 . (4.10)We say that the tensor product hastype B for (4.9) , type C for (4.10) with c ℓ = c ℓ +1 , type D for (4.10) with c ℓ = c ℓ +1 . Let ˜ s denote the level of the boundary Fock module: ˜ s = s / c ℓ +1 for type B and ˜ s = s − c ℓ +1 for type C , D . The total level is C = ˜ s ℓ Y i =1 s c i . (4.11) EFORMATIONS OF W ALGEBRAS 23 By the comodule formula (4.8) and the coproduct formulas (2.2), (2.3), current E ( z ) acts as a sumof vertex operators in ℓ + 1 bosons of the form E ( z ) = ℓ X i =1 b c i Λ i ( z ) + k B c ℓ +1 Λ ( z ) + ℓ X i =1 b c i Λ ¯ i ( z ) for type B ,E ( z ) = ℓ X i =1 b c i Λ i ( z ) + ℓ X i =1 b c i Λ ¯ i ( z ) for type C , D . In these formulas,Λ i ( z ) = 1 ⊗ · · · ⊗ ⊗ i ⌣ V c i ( a i z ; u i ) ⊗ ψ + ( s − c i +1 a i +1 z ) ⊗ · · · ⊗ ψ + ( s − ℓ a ℓ z ) ⊗ K + ( z ) ( i = 1 , . . . , ℓ ) , Λ ( z ) = 1 ⊗ · · · ⊗ ⊗ ˜ K c ℓ +1 ( z ) , Λ ¯ i ( z ) = 1 ⊗ · · · ⊗ ⊗ i ⌣ V − c i ( a − i z ; u i ) ⊗ ψ − ( a − i +1 z ) − ⊗ · · · ⊗ ψ − ( a − ℓ z ) − ⊗ K − ( z ) ( i = 1 , . . . , ℓ ) , where a i ’s are given by a i = ˜ s − ℓ Y j = i +1 s − c j . (4.12)We have the following contractions: C (Λ i ( z ) , Λ j ( w )) = ( − κ ( i ≺ j , i = ¯ j )0 ( i ≻ j , i = ¯ j ) , C (Λ i ( z ) , Λ i ( w )) = ( − κ − q ci ( i = 0) − κ s cℓ +1 ( i = 0) , (4.13) C (Λ i ( z ) , Λ ¯ i ( w )) = − κ + κ q c i − q c i a − i (1 ≤ i ≤ ℓ ) , C (Λ ¯ i ( z ) , Λ i ( w )) = κ − q c i a i (1 ≤ i ≤ ℓ ) . Here the indices are ordered as 1 ≺ · · · ≺ ℓ ≺ ≺ ¯ ℓ ≺ · · · ≺ ¯1, and we set c ¯ i = c i .As before, to each neighboring pair of Fock spaces we associate a current A i ( z ). Namely, for i = 1 , . . . , ℓ − 1, we define A i ( z ) similarly as in (3.3), A i ( z ) =: Λ i ( a − i z )Λ i +1 ( a − i z ) :=: Λ i +1 ( a i z )Λ ¯ i ( a i z ) : . (4.14)The second equality is due to the identity (2.11). In addition we define a current A ℓ ( z ) for each type as follows. A ℓ ( z ) =: Λ ℓ ( a − ℓ z )Λ ( a − ℓ z ) :=: Λ ( a ℓ z )Λ ¯ ℓ ( a ℓ z ) : for type B , (4.15) A ℓ ( z ) =: Λ ℓ ( z )Λ ¯ ℓ ( z ) : for type C , (4.16) A ℓ ( z ) =: Λ ℓ − ( z )Λ ¯ ℓ ( z ) :=: Λ ℓ ( z )Λ ℓ − ( z ) : for type D . (4.17)We study the contractions of the root currents A i ( z ). Denote as before B i,j = C ( A i ( z ) , A j ( w )).The contractions of B i,j with i, j = ℓ are clearly the same as in type A and are given in (3.5). Thenew feature is the contractions B i,ℓ , B ℓ,i .First, we have B i,ℓ = B ℓ,i = 0 , ( i < ℓ − . Then for type B we have B ℓ − ,ℓ = B ℓ,ℓ − = 0 , B ℓ − ,ℓ = B ℓ,ℓ − = t t t t c ℓ , (4.18) B ℓ,ℓ = − t t t t c ℓ t c ℓ +1 ( s / c ℓ +1 − s − / c ℓ +1 )( s c ℓ s / c ℓ +1 + s − c ℓ s − / c ℓ +1 ) . (4.19)For type C we have B ℓ − ,ℓ = B ℓ,ℓ − = 0 , B ℓ − ,ℓ = B ℓ,ℓ − = t t t t c ℓ ( s c ℓ +1 + s − c ℓ +1 ) , (4.20) B ℓ,ℓ = − t t t t c ℓ ( s c ℓ +1 + s − c ℓ +1 )( s c ℓ s − c ℓ +1 + s − c ℓ s c ℓ +1 ) . (4.21)Finally, for type D we have B ℓ − ,ℓ = B ℓ,ℓ − = t t t t c ℓ − , B ℓ − ,ℓ = B ℓ,ℓ − = t t t t c ℓ − t c ℓ ( s c ℓ − s − c ℓ − s − c ℓ − s c ℓ ) , (4.22) B ℓ,ℓ = − t t t t c ℓ − t c ℓ ( s c ℓ − s c ℓ − s − c ℓ − s − c ℓ ) . (4.23)Note that, in particular, B ij = B ji in all cases.We illustrate these contractions in the library of Appendix C. Here we consider the simplest casewhen all Fock spaces are of the same color to describe the connection to Cartan matrices of types B , C , D .Let c i = 2, i = 1 , . . . , ℓ . Then the only nonzero B i,j with both i = ℓ and j = ℓ are B i,i = − t t ( s + s − ) and B i,i − = B i − ,i = t t .Consider now the case of type B . We have C = q ℓ s c ℓ +1 .Let c ℓ +1 = 1. Then B ℓ,ℓ − = B ℓ − ,ℓ = t t , and B ℓ,ℓ = − t ( s / − s − / )( s s / + s − s − / ). Theother entries involving index ℓ are zero.Thus the matrix − Bt − ( s / − s − / ) − coincides with type B matrix B ( q, t ) (2.4) of [FR1] underthe identification q = s / , t = s − . In particular, in the limit s = s = s = 1 it recovers thesymmetrized Cartan matrix of type B . EFORMATIONS OF W ALGEBRAS 25 Moreover, all the terms in E ( z ) are obtained from the first one by multiplications by A i ( z ) − as inthe natural representation of B ℓ shown below (we do not show the arguments of currents or constantsin front of vertex operators here). Λ Λ . . . Λ ℓ Λ Λ ¯ ℓ . . . Λ ¯2 Λ ¯1 A − A − A − ℓ − A − ℓ A − ℓ A − ℓ − A − A − Therefore we depict this case by the following Dynkin diagram. (4.24)The case c ℓ +1 = 3 is similar.However, the deformed Cartan matrix in the case of c ℓ +1 = 2 is essentially different; it correspondsto the case studied in [BrL], see also Section 7 of [FR1]. We have B ℓ,ℓ − = B ℓ − ,ℓ = t t , and B ℓ,ℓ = − t t ( s − s − ). In this case − Bt − t − in the limit s = s = s = 1 is equal tosymmetrized Cartan matrix of type osp (1 , ℓ ). We associate to this case the following diagram. (4.25)Note that the symmetrized Cartan matrix of type so (2 ℓ + 1) is twice the symmetrized Cartan matrixof type osp (1 , ℓ ). Moreover, the weight diagrams of the so (2 ℓ + 1) modules and osp (1 , ℓ ) modulesare the same and the nilpotent subalgebras in both cases are the same. Therefore, for the purposes ofthis paper either diagram fits. However, we prefer not to use the affine node to distinguish betweenthese two finite types which was suggested by [FR1]. Instead the affine node distinguishes betweenaffine versions of the deformed Cartan matrices, see Section 4.5.Let us now switch to the types C and D . We have C = q ℓ q − c ℓ +1 .When c ℓ +1 = 1, we make the choice of type C . Then the nontrivial entries with index ℓ are B ℓ,ℓ − = B ℓ − ,ℓ = t ( s − s − ), and B ℓ,ℓ = − t ( s − s − )( s s − + s − s ).Thus the matrix − Bt − t − coincides with type C matrix B ( q, t ) of [FR1] under the identification q = s , t = s − . In particular, in the limit s = s = s = 1 it recovers the symmetrized Cartanmatrix of type C .Again, current E ( z ) looks as a vector representation of type C . Λ Λ . . . Λ ℓ Λ ¯ ℓ . . . Λ ¯2 Λ ¯1 A − A − A − ℓ − A − ℓ A − ℓ − A − A − We associate to it the following Dynkin diagram. (4.26)The case c ℓ +1 = 3 is similar.When c ℓ +1 = 2, we make a choice of type D . Then nonzero entries involving index ℓ are B ℓ,ℓ − = B ℓ − ,ℓ = t t , and B ℓ,ℓ = − t t ( s + s − ).Thus the matrix − Bt − t − coincides with type D matrix B ( q, t ) of [FR1] under the identification q = s , t = s − . In particular, in the limit s = s = s = 1 it recovers the symmetrized Cartanmatrix of type D .Again, current E ( z ) looks as a vector representation of type D . Λ ℓ Λ Λ . . . Λ ℓ − Λ ℓ − . . . Λ ¯2 Λ ¯1 Λ ¯ ℓ A − ℓ A − A − A − ℓ − A − ℓ A − ℓ − A − ℓ − A − A − A − ℓ − Thus we have the following Dynkin diagram. (4.27)Thus we recover deformed W -algebras of types B , C , D described in [FR1] in the case when allFock spaces are of the same type. We remark that as in type A , we have the diagonal Heisenberg∆ ( ℓ ) H r commuting with A i ( z ), i = 1 , . . . , ℓ . This extra boson allows us to have rational contractionsbetween all the terms.4.5. Root current A ( z ) of types B , C , D . Define the dressed current E ( z ) depending on µ ∈ C × , | µ | < E ( z ) = E ( z ) K + µ ( z ) − , K + µ ( z ) = ∞ Y s =0 K + ( µ − s z ) . The Fourier coefficients of E ( z ) are elements of the algebra ˜ K , the algebra K completed with respectto homogeneous grading in the positive direction.Similarly to type A , our motivation for the definition of the dressing current is twofold : we wouldlike to have integrals of motion and we also would like to have the current A ( z ) which produces ascreening operator corresponding to the affine node. In contrast to type A , we cannot have arbitrary µ , and moreover, the two requirements above seem to be different at the moment. In this section,we discuss the current A ( z ) obtained from the dressed current acting on the modules (4.9), (4.10).For that there are 6 possible choices of µ : C /µ = s − c or C /µ = s c , c ∈ { , , } , which match thevalues of C in 6 boundary modules we have defined in Section 4.3. Only the last 3 of them (types C and D below) correspond to the dressings which give integrals of motion (if in addition | µ | < EFORMATIONS OF W ALGEBRAS 27 We make the choice for µ and label our choices by types B , C , D as we did for the ℓ -th node.Namely, we have the following choices for the affine node:type B : µ = C s c , type C : µ = C s − c , c = c , type D : µ = C s − c , c = c ,C being the level given by (4.11). We stress that the choice of c is independent of other c i , thatis, it is independent of the choice of the K representation. The type of the zeroth node is not to beconfused with the type of tensor product modules (4.9), (4.10).To that we associate the affine Dynkin diagram, where the affine node has the prescribed type andthe color is given the same way as for the n ℓ -th node, see C.1. Namely, we depict the affine node asfollows. B , c = c : c B , c = c : b { c , c , b } = { , , } C , c = c : b { c , c , b } = { , , } D , c = c : c D , c = c : b { c , c , b } = { , , } Current E ( z ) has the form E ( z ) = ℓ X i =1 b c i Λ i ( z ) + k B c ℓ +1 Λ ( z ) + ℓ X i =1 b c i Λ ¯ i ( z ) for type B , E ( z ) = ℓ X i =1 b c i Λ i ( z ) + ℓ X i =1 b c i Λ ¯ i ( z ) for types C , D , where Λ i ( z ) = Λ i ( z )∆ ( ℓ ) K + µ ( z ) − . For Λ i ( z ) the contraction rule (3.10) applies.Define the current A ( z ) of each type as follows.: A ( s − / c z ) A ( s / c z ) :=: Λ ¯1 ( µ − / z ) Λ ( µ / z ) : for type B , (4.28) A ( z ) =: Λ ¯1 ( µ − / z ) Λ ( µ / z ) : for type C , (4.29) A ( z ) =: Λ ¯2 ( µ − / z ) Λ ( µ / z ) :=: Λ ¯1 ( µ − / z ) Λ ( µ / z ) : for type D . (4.30)These definitions are to be compared with (4.15)–(4.17). Note that, for type B , (4.15) implies: A ℓ ( s − / c ℓ +1 z ) A ℓ ( s / c ℓ +1 z ) :=: Λ ℓ ( z )Λ ¯ ℓ ( z ) : . The corresponding contractions B i,j are given by the same rule as (4.18)–(4.22) if we interchange A i ( z ) with A ℓ − i ( z ) and c j with c ℓ +1 − j .Namely we have B , = B , = t t t t c , B , = − t t t t c t c ( s / c − s − / c )( s / c s c + s − / c s − c ) for type B ;(4.31) B , = B , = t t t t c ( s c + s − c ) , B , = − t t t t c ( s c + s − c )( s c s − c + s − c s c ) for type C ;(4.32) B , = B , = t t t t c , B , = B , = t t t t c t c ( s − c s c − s c s − c ) , (4.33) B , = − t t t t c t c ( s c s c − s − c s − c ) for type D . For ℓ > 3, all other B i,j involving index 0 are zero. The full list of B i,j for small values of ℓ is givenin Appendix C.We think of the matrix ˆ B = ( B i,j ) ℓi,j =0 as the affinization of matrix B . In case all colors except0 and ℓ are equal, e.g. c = · · · = c ℓ = 2, the corresponding Dynkin diagrams are those of non-exceptional affine type, where the colors c and c ℓ determine the ends of the diagram, see Table 1below. For comparison we include type A in the table.type Fock space C C /µ det ˆ C A (1) ℓ F ⊗ ℓ ⊗ F q ℓ +12 arbitrary (1 − µ )( s − ℓ − − µ − s ℓ +12 ) B (1) ℓ F ⊗ ℓ ⊗ F B q ℓ q / q ( s − s − )( s ℓ − s / − s − ℓ +12 s − / ) C (1) ℓ F ⊗ ℓ ⊗ F CD q ℓ q − q ( s − s − )( s ℓ s − − s − ℓ s ) D (1) ℓ F ⊗ ℓ ⊗ F CD q ℓ − q ( s + s − )( s − s − )( s ℓ − − s − ℓ +22 ) A (2)2 ℓ F ⊗ ℓ ⊗ F B q ℓ q / q ( s − s − )( s ℓ s − / − s − ℓ s / ) A (2)2 ℓ − F ⊗ ℓ ⊗ F CD q ℓ q − q ( s − s − )( s ℓ − s − − s − ℓ +12 s ) D (2) ℓ +1 F ⊗ ℓ ⊗ F B q ℓ q / q − / ( s − s − )( s ℓ s − s − ℓ s − ) Table 1. We draw the corresponding Dynkin diagrams, where labels are as in the finite case and the doublecircle denotes the affine node which corresponds to the dressing rather than a pair of modules. EFORMATIONS OF W ALGEBRAS 29 A (1) ℓ 22 2 2 2 B (1) ℓ 22 2 2 2 3 C (1) ℓ D (1) ℓ 22 2 2 2 22 A (2)2 ℓ A (2)2 ℓ − 22 2 2 2 3 D (2) ℓ +1 The qq -characters. As in type A we write ˆ B = ˆ D ˆ C , choosing ˆ D = diag( d , . . . , d ℓ ) in such away that C i,i becomes one of the following, see Appendix C: s c + s − c , s b s − c + s − b s c , s b s / c + s / b s c , s / c + s − / c , s c − s − c . We then introduce the currents Y i ( z ) following the rule (3.13). Then the current E ( z ) follows the qq -character. We give the following example illustrating various phenomena. Example 4.5. Consider the following affine Dynkin diagram. 22 2 1 We label the nodes in the standard way: double circled affine node is the zeroth node, the middlepoint is node 2 and the shorter node is node 3.This diagram corresponds to the F ⊗ F ⊗ F ⊗ F B of level C = s s / with the dressing parameter µ = C s − = s s .We have the following deformed Cartan matrix (see Appendix C).ˆ C = s + s − − s + s − − − − s + s − − 10 0 − s / − s − / s / s + s − / s − . We use notation Y l ( s a/ s b/ z ) = l a,b , l = 0 , , , 3. Then we have A ( z ) = , − , − , , A ( z ) = , − , − , ,A ( z ) = − , − , , − , − , − − , , A ( z ) = − , , − , − . Then the dressed current E ( z ) takes the form (ignoring the constants and the overall shift) of the qq -character with 7 monomials: χ = − , , + − , − , , + − , − , , + , − , + , − , − , + , , − , + , − , . Note the following features of this qq -character. We follow the terminology of Section 3.3.Let C P be the group ring of the weight lattice generated by fundamental weights ω l , l = 1 , . . . , ℓ .Let ρ : A → C P be the ring homomorphism sending l a,b ω l , l = 1 , . . . , ℓ , a,b 1. Then ρ ( χ ) isthe character of vector representation of B .The initial monomial has , , the final monomial has − , . The shift s s is C .The ratios between neighboring monomials are A ( s z ), A ( s z ), A ( s z ), A ( s s z ), A ( s s z ),and A ( s s z ) respectively. The first and the 6th monomial are -dominant, the second and the5th are -dominant, all these “generate” q -characters of 2 dimensional U q b sl -modules. The thirdmonimial is -dominant and “generates” the q -character of a 3 dimensional U q b sl -module. This isthe reason E ( z ) commutes with screening operators S i , i = 1 , . . . , ℓ , see Section 4.7.The 6ht and the 7th monomial are -dominant.They “generate” monomials which are µ -shifts ofthe first and the second monomial: A − ( s s ) , , − , = − , , = τ µ ( − , , ) ,A − ( s s ) , − , = − , − , , = τ µ ( − , − , , ) . This is the reason for our choice of µ and why the screening S commutes with the integral of E ( z ). EFORMATIONS OF W ALGEBRAS 31 Now we consider the same Dynkin diagram and the same deformed Cartan matrix but treat node3 as affine: 22 2 1 Then we consider the D current corresponding to F ⊗ F ⊗ F ⊗ F CD with the dressing of type B .The new level C = s is different but µ = C s = s s is the same.Then the E ( z ) current corresponds to the following qq -character with 6 summands:¯ χ = , − − , − , + − , , , + − , , + , − , + , − , − , + − , , − , . Note also that D = A . Here we consider the deformed W current corresponding to the secondfundamental module.The first and the fifth monomials are 2-dominant, the second and the sixth are 2-antidominant.Moreover the ratio of the first to the second is A ( s z ) and the fifth to the sixth is A ( s z ). Similarly,in direction 1 the ratio of the second to the third is A ( s ) and the fourth to the fifth is also A ( s z ).Finally, in the direction 0, the ratio of the second to the fourth equals to the ratio of the third tothe fifth and equal to A ( s z ). It means that ¯ χ is “closed” in the directions 0 , 1, and 2 (and that ¯ χ commutes with screenings S , S , S ).In the direction of 3, we do not have such a property. However, one can add a monomial andconsider ˜ χ = ¯ χ + , − , . (4.34)With respect to the D algebra it correspond to adding a trivial 1 dimensional representation (inparticular, the new current still commutes with screenings S , S , S ).On the other hand, the last monomial − , , , , the new monomial , − , and the shiftedmonomial τ ¯ µ ( , − − , − , ) together correspond to the 3-dimensional evaluation U q b sl module in thethird direction (in particular, the new current also commutes with the screening S , see Theorem4.8).Finally we note that the two currents corresponding to qq -characters χ and ¯ χ are closely related.Namely each monomial of χ is a shift of a monomial in ˜ χ . Namely, if we apply τ µ to the lastthree monomials of χ we obtain τ s ( ˜ χ ). In particular, the integrals (constant terms) of ˜ χ and χ coincide. (cid:3) Remark 4.6. The phenomena described in the example is rather general. Namely, if one can choosethe affine node in several different ways then the corresponding deformed W currents are all obtainedfrom each other by shifting some of the monomials. In particular, the integrals (constant terms) ofall these currents coincide.In this paper we study commutative families of operators which include the integral of a deformed W current. Thus we expect that the families for different choices of the affine node all commute.While it seems to be difficult to check directly, this fact would follow if the integral of the deformed W current had simple spectrum for generic evaluation parameters u i . We expect this is the case andwe intend to return to this question when we study the spectrum of IM. The family of commuting operators correspond to the choices of the affine node of type other than B . Such a choice exists in all cases except when both zeroth and ℓ -th nodes are of B type (e.g. intype D (2 ℓ +1 ). (cid:3) Screenings. The screening operators are defined in the same way as for type A , see (3.16),(3.17).We call the matrix ˆ B stable if B ,ℓ = B ℓ, = 0. Most of the deformed affine Cartan matrices oftypes B , C , D are stable except for a few low rank cases, see Appendix C.In what follows we often assume ˆ B is stable to simplify the considerations. We expect that in allsuch cases this assumption can be dropped.Let ˆ B be stable. For i = 1 , · · · , ℓ − 1, the screenings are defined by (3.16), (3.17). The screeningcurrents for the ℓ -th node are given as follows.For type B , c ℓ = c ℓ +1 : A ℓ ( z ) = s c ℓ s c : S + ℓ ( s − b z ) S + ℓ ( s b z ) : (cid:16) ( c ℓ , b, c ) = cycl (1 , , (cid:17) ,c ℓ = c ℓ +1 : A ℓ ( z ) = s c ℓ s c ℓ +1 : S + ℓ ( s − b z ) S + ℓ ( s b z ) := s c ℓ s c ℓ +1 : S − ℓ ( s − / c ℓ +1 z ) S − ℓ ( s / c ℓ +1 z ) : (cid:16) { c ℓ , c ℓ +1 , b } = { , , } (cid:17) . For type C , c ℓ = c ℓ +1 : A ℓ ( z ) = q c ℓ q − c ℓ +1 : S + ℓ ( s − b z ) S + ℓ ( s b z ) := q c ℓ q − c ℓ +1 : S − ℓ ( s − c ℓ +1 z ) S − ℓ ( s c ℓ +1 z ) : (cid:16) { c ℓ , c ℓ +1 , b } = { , , } (cid:17) . For type D , c ℓ − = c ℓ = c ℓ +1 : A ℓ ( z ) = q c ℓ : S + ℓ ( s − b z ) S + ℓ ( s b z ) := q c ℓ : S − ℓ ( s − c z ) S − ℓ ( s c z ) : (cid:16) ( c ℓ , b, c ) = cycl (1 , , (cid:17) ,c ℓ − = c ℓ = c ℓ +1 : A ℓ ( z ) = s − b : S fℓ ( s − b z ) S fℓ ( s b z ) : (cid:16) { c ℓ − , c ℓ , b } = { , , } (cid:17) . The zeroes screening currents S ± ( z ) are defined in the same way (changing indices ℓ , ℓ − ℓ − , , Theorem 4.7. Assume ˆ B is stable. The screening operators S i with i = 0 commute with both E ( z ) and E ( z ) : [ S i , E ( z )] = [ S i , E ( z )] = 0 , i = 1 , . . . , ℓ . For the zeroth screening we have [ S , E ( z )] = [ S , b c ( Λ ( z ) − Λ ( µz ))] for type C , [ S , E ( z )] = [ S , b c ( Λ ( z ) − Λ ( µz )) + b c ( Λ ( z ) − Λ ( µz ))] for type D . Proof. For i = 1 , · · · , ℓ − A computation, so it reduces to Lemma 2.1.For type C , one can check that the triples ( A ℓ ( z ) , Λ ℓ ( z ) , Λ ¯ ℓ ( z )) and ( A ( z ) , Λ ¯1 ( z ) , Λ ( µz )) satisfythe conditions of Lemma 2.1.Likewise, for type D , one checks that the triples ( A ℓ ( z ) , Λ ℓ − ( z ) , Λ ¯ ℓ ( z )), ( A ℓ ( z ) , Λ ℓ ( z ) , Λ ℓ − ( z )),( A ( z ) , Λ ¯2 ( z ) , Λ ( µz )), ( A ( z ) , Λ ¯1 ( z ) , Λ ( µz )) satisfy the conditions of Lemma 2.1. EFORMATIONS OF W ALGEBRAS 33 The remaining case is type B with i = ℓ . The relevant contractions are C ( A ℓ ( z ) , Λ ℓ ( w )) = − t t t t c ℓ s − c ℓ s − / c ℓ +1 , C (Λ ℓ ( z ) , A ℓ ( w )) = − t t t t c ℓ s c ℓ s / c ℓ +1 , C ( A ℓ ( z ) , Λ ( w )) = − C (Λ ( z ) , A ℓ ( w )) = t t t s / c ℓ +1 + s − / c ℓ +1 , C ( A ℓ ( z ) , Λ ¯ ℓ ( w )) = t t t t c ℓ s c ℓ s / c ℓ +1 , C (Λ ¯ ℓ ( z ) , A ℓ ( w )) = t t t t c ℓ s − c ℓ s − / c ℓ +1 . Suppose c ℓ = c ℓ +1 . We take c ℓ = c ℓ +1 = 2 for concreteness. From the above contractions we computethe singular parts of the operator products, S + ℓ ( z )Λ ℓ ( w ) = − t s − / wz − s s − / w : S + ℓ ( s s − / w )Λ ℓ ( w ) : + O (1) ,S + ℓ ( z )Λ ℓ ( w ) = k (cid:16) s − s − / wz − s − s − / w : S + ℓ ( s − s − / w )Λ ( w ) : − s s / wz − s s / w : S + ℓ ( s s / w )Λ ( w ) : (cid:17) + O (1) ,S + ℓ ( z )Λ ¯ ℓ ( w ) = t s / wz − s − s / w : S + ℓ ( s − s / w )Λ ¯ ℓ ( w ) : + O (1) , where k = t ( s / − s / ) / ( s s / − s − s − / ) = t b /k B . In view of the relations: S + ℓ ( s z )Λ ℓ ( s / z ) : = s − s : S + ℓ ( s − z )Λ ( s / z ) : , : S + ℓ ( s − z )Λ ¯ ℓ ( s − / z ) : = s s − : S + ℓ ( s z )Λ ( s − / z ) : , we conclude that [ S + ℓ , b Λ ℓ ( w ) + k B Λ ( w ) + b Λ ¯ ℓ ( w )] = 0 . Note that we can define another screening current S − ℓ ( z ) by interchanging q ↔ q in S + ℓ ( z ). Notethat k B = (1 + s )( s − s ) /κ changes sign under the swap q ↔ q . So, the screening operator S − ℓ commutes with a different current (which is also a representation of K , see (4.6))[ S − ℓ , b Λ ℓ ( w ) − k B Λ ( w ) + b Λ ¯ ℓ ( w )] = 0 . That the present case admits only one screening has been observed in [FR1], see the end of Section7 thereof.The calculation for c ℓ = c ℓ +1 is entirely similar, cf. [FR1], Theorem 3. Alternatively one can writethe relevant three terms as a “fusion” and reduce the calculation to two-dimensional representationof sl , see the last line of the proof of Proposition B.1 below. (cid:3) For type B , the current E ( z ) does not “close up” under S . However, definition (4.28) suggeststhat we introduce an operator Λ ¯0 ( z ) =: Λ ¯1 ( z ) A − ( µ / s − / c z ) : =: Λ ( µz ) A ( µ / s / c z ) :and consider an extended current ˜ E ( z ) = E ( z ) + k B c Λ ¯0 ( z ) . (4.35)On can think that this current corresponds to a direct sum of the vector representation and thetrivial representation, see (4.34). Theorem 4.8. Assume ˆ B is stable. The screening operators S i with i = 0 commute with ˜ E ( z ) : [ S i , ˜ E ( z )] = 0 , i = 1 , . . . , ℓ . For the zeroth screening we have [ S , ˜ E ( z )] = b c [ S , Λ ( z ) − Λ ( µz )] . Proof. The proof is parallel to that of Theorem 4.7. (cid:3) Integrals of motion associated with K . In this subsection we show that algebra K possessesa family of integrals of motion when C = µq c , namely if the zeroth node is type C or D .First, we prepare a lemma about matrix elements of products of E ( z ). Set f ( x ) = (1 − C x )(1 − C − x )(1 − x ) × Y s =1 ( q − s x ) ∞ ( µq s x ) ∞ . Using the defining relations, we find f ( w/z ) E ( z ) E ( w ) = f ( z/w ) E ( w ) E ( z ) . With the definition E ( z , . . . , z n ) = Y i We have E ( z, q c z, q c q c z ) = 0 ( c = c , c , c ∈ { , , } ) , (4.37) E ( z, C ± z, C ± q ± c z ) = 0 ( c ∈ { , , } ) , (4.38) E ( C − z, z, C z ) = 0 . (4.39) Proof. We can write E ( z , . . . , z n ) = E ( z , . . . , z n ) n Y i =1 K + ∞ ( z i ) − , where E ( z , . . . , z n ) = Y i The left hand side has the form LHS = Sym z ,z ,z (cid:20)(cid:16) z z − z z − z z + z z (cid:17) E ( z ) E ( z ) E ( z ) (cid:21) , while the right hand side is RHS = Sym z ,z ,z (cid:20)(cid:0) − z z z (cid:1) z ( z + z )( z + z )¯ g ( z , z )¯ g ( z , z ) g ( z , z ) δ (cid:16) C z z (cid:17) E ( z ) K ( z ) (cid:21) . Let us consider the left hand side. Each term E ( z i ) E ( z j ) E ( z k ) is defined in the region | z i | ≫| z j | ≫ | z k | . It can be rewritten as E ( z i ) E ( z j ) E ( z k ) = E ( z , z , z ) × f ( z j /z i ) − f ( z k /z i ) − f ( z k /z j ) − , where all matrix elements of E ( z , z , z ) are symmetric Laurent polynomials multiplied by Q i 3) and δ ( C ± z /z ).We observe that the terms without delta functions cancel out, due to the identity of rationalfunctions Skew z ,z ,z (cid:16) z z − z z − z z + z z (cid:17) g g g = 0 , where g ij = g ( z i , z j ).The terms with one delta function also cancel out.For example, the coefficient of δ ( q z /z ) comes from 3 terms and cancel out: h(cid:0) z z − z z − z z + z z (cid:1) g g + (cid:0) z z − z z − z z + z z (cid:1) g g + (cid:0) z z − z z − z z + z z (cid:1) g g i(cid:12)(cid:12)(cid:12) z = q z = 0 . Similarly, the coefficient of δ ( C z /z ) comes from 3 terms. In the left hand side, we collect allcontributions to κ − δ ( C z /z ) E ( z ) K ( z ) and find (cid:16) z z − z z − z z + z z (cid:17) + (cid:16) z z − z z − z z + z z (cid:17) ¯ g g + (cid:16) z z − z z − z z + z z (cid:17) ¯ g g g ¯ g = κ (cid:0) − z z z (cid:1) z ( z + z )( z + z ) g g g , which coincides with the first term in the right hand side.Finally, we consider terms with two delta functions.Consider the term δ ( q c z /z ) δ ( q c z /z ) ( c = c ). It comes only from the last term on the lefthand side with non-zero coefficient, (cid:16) z z − z z − z z + z z (cid:17) g (cid:12)(cid:12)(cid:12) z = q c z ,z = q c z = 0 , which implies the wheel condition (4.37).Consider δ ( q i z /z ) δ ( C z /z ). Again, only the last term from LHS contributes, with non-zerocoefficient, which yields E ( z, q i z, C q i z ) = 0 . Likewise, only the last term contributes to the coefficient of δ ( C z /z ) δ ( q i z /z ), giving the rela-tion E ( z, q − i z, C − q − i z ) = 0 . Consider δ ( C z /z ) δ ( C z /z ). Again only the last term contributes and we find the equality E ( C − z, z, C z ) = 0. (cid:3) First, we consider the case C = µq . . Theorem 4.10. Assume that C = µq . Then the following elements { I n } ∞ n =1 are mutually commu-tative, I n = Z · · · Z E ( z ) · · · E ( z n ) · Y j Assume that C = µq , and define I ′ n = Z · · · Z E ( z ) · · · E ( z n ) · Y j 1, thenthe integrals (4.41) over the unit circle give a commutative family. Being an analytic continuationin the parameter q from | q | > | q | < | q | − , the integrals (4.41) remain commutative. (cid:3) Thus in a generic admissible K module, we have three commutative families of operators. We callthem deformed integrals of motion due to the following theorem. Theorem 4.12. Assume ˆ B is stable. For C = µq , the elements I n commute with all screeningoperators, [ S i , I n ] = 0 , i = 0 , , . . . , ℓ, n ≥ . Proof. The argument for type C is similar to that in type A , see Theorem 3.7.For type D the formulas become a little more cumbersome. We illustrate it in the case c = c = c = 2. We start from [ S , E ( z )] = const. ( A (1) ( z ) − A (1) ( µz )) , EFORMATIONS OF W ALGEBRAS 37 where now A (1) ( z ) is a sum of two terms A (1) ( z ) = z : S ( s − µ − / z ) (cid:0) Λ ( z ) + Λ ( z ) (cid:1) : . Taking residues repeatedly we obtain currents for m = 1 , , . . . A ( m ) ( z ) = z : S ( s − q − m +11 µ − / z ) m X k =0 a ( m ) k Λ ( q − m +13 z ) · · · Λ ( q − m + k z ) Λ ( q − m + k +13 z ) · · · Λ ( z ) : ,a ( m ) k = k Y j =1 − q m − j +13 − q j − q q j − q q m − j +13 . The rest of the argument is the same. (cid:3) It would be interesting to study the spectrum of deformed integrals of motion.Theorem 4.12 deals with affine nodes of types C and D . If the affine node is of type B , the integralof current ˜ E ( z ), see (3.7), commutes with all the screenings. In the case the ℓ -th node is not of type B , this integral coincides with the integral of another current for which the affine node is of type C or D (the same as the ℓ -th node for the original current, see (4.34)). Therefore, we can include theintegral of ˜ E ( z ) into a family of integral of motions corresponding to that current. We expect thisfamily commutes with all screening operators.5. Additional remarks Integrals of motion of KZ type. In this subsection we continue with the Cartan matrices inTable 1 except types A (1) ℓ , D (2) ℓ +1 , and construct another set of commuting operators which commutewith integrals I n . We hope that these integrals may be more convenient for Bethe ansatz study inthe future.The results in this section depend on the following technical statement which we do not discuss. Conjecture 5.1. The K module F = F ( u ) ⊗ · · · ⊗ F ( u ℓ ) ⊗ F Xc is irreducible for generic parameters u , . . . , u ℓ . (cid:3) The vector space F is an irreducible representation of the set of ℓ + 1 bosons { a i,r | i = 0 , . . . , ℓ, r =0 } . We regard { a i, | i = 0 , . . . , ℓ } as functions of the parameters u , . . . , u ℓ through (4.14), (4.15)–(4.17) and (4.28)–(4.30). Separating the zero modes we shall write A i ( z ) = e a i, A osci ( z ), Y i ( z ) = e y i, Y osci ( z ). To each i = 0 , . . . , ℓ we associate a reflection operator R i ∈ End F defined as follows. Proposition 5.2. There exist operators R i ∈ End F with the properties R i Y j ( z ) = Y j ( z ) R i ( j = i ) ,R i (cid:16) e y i, Y osci ( z ) + e y i, − a i, : Y osci ( z ) A osci (ˆ s i z ) : (cid:17) = (cid:16) e y i, − a i, Y osci ( z ) + e y i, : Y osci ( z ) A osci (ˆ s i z ) : (cid:17) R i , where ˆ s i = s for i = 0 , ℓ , ˆ s i = s s / , s s − , s if i ∈ { , ℓ } is of type B , C , D , respectively. In other words, R i is an operator depending only on one boson { a i,r } r =0 as well as the zero modes { a j, } j =1 ,...,ℓ , and implements the Weyl reflection on the latter. Proof. It is clear that for i = 1 , . . . , ℓ − R matrix R i = ˇ R i,i +1 ( u i /u i +1 ), obtained from theuniversal R matrix of E , has the required properties. For i = ℓ and type B , C , we use PropositionB.1 to get R ℓ = K ℓ ( u ℓ ). In the case of type D , from the remark after Proposition B.1 we have K ℓ A ℓ − ( z ) K ℓ = A ℓ ( z ), K ℓ = 1. Hence we can take R ℓ = K ℓ ˇ R ℓ − ,ℓ ( u ℓ − u ℓ ) K ℓ .The case i = 0 is similar. (cid:3) For i = 0, they are intertwiners of K modules. Writing the E ( z ) current as E ( z ; u , . . . , u ℓ ) wehave ˇ R i,i +1 ( u i /u i +1 ) E ( z ; u , . . . , u i , u i +1 , . . . , u ℓ )(5.1) = E ( z ; u , . . . , u i +1 , u i , . . . , u ℓ )ˇ R i,i +1 ( u i /u i +1 ) ( i = 1 , . . . , ℓ − , K ℓ ( u ℓ ) E ( z ; u , . . . , u ℓ − , u ℓ ) = E ( z ; u , . . . , u ℓ − , u − ℓ ) K ℓ ( u ℓ ) ( i = ℓ ) . (5.2)For i = 0 the corresponding relation takes the form K ( u ) E ∗ ( z ; u , u , . . . , u ℓ ) = E ∗ ( z ; u − , u , . . . , u ℓ ) K ( u ) , (5.3)where E ∗ ( z ; u , . . . , u ℓ ) means E ( z ; u , . . . , u ℓ ) − b (cid:0) Λ ( z ) − Λ ( µz ) (cid:1) or E ( z ; u , . . . , u ℓ ) − b (cid:0) Λ ( z ) − Λ ( µz ) (cid:1) − b (cid:0) Λ ( z ) − Λ ( µz ) (cid:1) , depending on whether the zeroth node is of type C or of type D .We now introduce operators T i , i = 1 . . . , ℓ , by T i = T + i T − i , T + i = ˇ R i − ,i ( u i /u i − ) · · · ˇ R , ( u i /u ) K ( u i )ˇ R , ( u u i ) · · · ˇ R i − ,i ( u i − u i ) , T − i = ˇ R i,i +1 ( u i u i +1 ) · · · ˇ R ℓ − ,ℓ ( u i u ℓ ) K ℓ ( u i )ˇ R ℓ − ,ℓ ( u i /u ℓ ) · · · ˇ R i,i +1 ( u i /u i +1 ) . We call T i integrals of motion of KZ type for the following reason. Theorem 5.3. The operators T i and I n in Theorems 4.10, 4.11 are mutually commutative: [ T i , T j ] = 0 ( i, j = 1 , . . . , ℓ ) , [ T i , I n ] = 0 ( i = 1 , . . . , ℓ, n ≥ . Proof. The commutativity of T i ’s is a simple consequence of the (ordinary and boundary) Yang-Baxter equations.To see the second statement, we use (5.1), (5.2) and (5.3). The only issue is to check that one cansafely shift Λ ( z j ) to Λ ( µz j ) without encountering poles in between. This is straightforward. (cid:3) Exceptional types. The W algebras and integrals of motion of type A are obtained from thequantum toroidal algebra E . In this paper we have introduced an algebra K which allows us to treatdeformed W algebras of non-exceptional types uniformly. A natural question is what happens inexceptional types.For an exceptional type, one can consider a similar algebra by taking the current T ( z ) in the senseof [FR1], together with a vertex operator Z ( z ) in one extra boson such that Z ( z ) T ( w ) = T ( w ) Z ( z )and such that all terms in E ( z ) = T ( z ) Z ( z ) have rational contractions. This gives the quantumalgebra in this type similar to E and K . EFORMATIONS OF W ALGEBRAS 39 For example, in the case of the seven dimensional representation of G , one obtains the relationwith four δ -functions g ( z, w ) E ( z ) E ( w ) + g ( w, z ) E ( w ) E ( z )= c (cid:16) δ (cid:0) q q wz (cid:1) w K ( w ) + δ (cid:0) q q zw (cid:1) z K ( z ) (cid:17) + c (cid:16) δ (cid:0) q q wz (cid:1) w : E ( q q w ) ˜ K ( w ) : + δ (cid:0) q q zw (cid:1) z : E ( q q z ) ˜ K ( z ) : (cid:17) , where K ( z ) =: Z ( z ) Z ( q q z ) :, ˜ K ( z ) =: Z ( z ) Z ( q q z ) Z − ( q q z ) :, and c , c are constants and C ( Z ( z ) , Z ( w )) = (1 − q )(1 − q ) 1 + q q q q ( q q − q ) , C ( K ( z ) , E ( w )) = (1 − q )(1 − q )(1 + q q )( q q − q ) , C ( ˜ K ( z ) , E ( w )) = (1 − q )(1 − q )( q q − q ) . The role of this algebra is not clear since G is not a part of any family. In particular, we do notexpect any comodule or coalgebra structure. Appendix A. Proof of Theorem 4.10 In this Section we prove Theorem A. We have C = µq .A.1. Commutativity [ I , I ] = 0 . As an illustration, let us verify the commutativity of I and I .We note that, by making use of the decomposition ω ( x ) = q C f ( x ) σ ( x ) σ ( x − ) , σ ( x ) = (1 − x ) ( µx, µ q x ) ∞ ( q − x, q − x ) ∞ , the integral (4.40) can be rewritten in terms of the currents (4.36) as const. I n = Z · · · Z E ( z , . . . , z n ) · Y j = k σ ( z k /z j ) n Y j =1 dz j z j . Consider the products I I = Z Z Z E ( z , z , z ) × f ( z /z ) − f ( z /z ) − σ ( z /z ) σ ( z /z ) Y j =1 dz j πiz j , I I = Z Z Z E ( z , z , z ) × f ( z /z ) − f ( z /z ) − σ ( z /z ) σ ( z /z ) Y j =1 dz j πiz j . The integral in I I is initially defined for | z | ≫ | z | = | z | = 1, while in I I it is defined for | z | ≪ | z | = | z | = 1. In both cases we move the contour for z to the unit circle. Along the way wepick up residues at the poles z = q − z i , µ − q − z i or z = q z i , µq z i , i = 2 , 3, respectively.When all variables are on the unit circle, the two integrals coincide thanks to the identity (3.21). Let us compare the residues at z = q ± z . We obtain respectively J = Z Z | z | = | z | =1 E ( z , q − z , z ) f ( q z /z ) − σ ( z /z ) σ ( z /z ) Y i =2 dz i πiz i ,J = Z Z | z | = | z | =1 E ( z , z , q z ) f ( q z /z ) − σ ( z /z ) σ ( z /z ) Y i =2 dz i πiz i . If we rename z in J to q z (so that q z is on the unit circle), then the two integrands become thesame thanks to the identity f ( x ) − σ ( q − x ) = σ ( x ) (1 − x ) (1 − q − x ) (1 − µx )(1 − µ − q − x )(1 − q x )(1 − q x ) . (A.1)The integrand of J has poles at (see Figure 1 below) z = µ m q − s z ( s = 1 , , m ≥ z = µ − m q s q − z ( s = 1 , , m ≥ z = q s z ( s = 1 , . Among them the points z = q s z are inside the contour for J (after renaming) and outside that for J . However these poles are actually absent due to the zero condition (4.37). Hence we have J = J . • • µq − s z q − s z × × µ − q s q − z µ − q s q − z ◦ q s z J J Figure 1. Integration contours on the z -plane ( s = 1 , 3) for J , J .Next let us consider the residues at z = ( µq ) ± z . Similarly as above we obtain J ′ = Z Z | z | = | z | =1 E ( z , µ − q − z , z ) f ( µq z /z ) − σ ( z /z ) σ ( z /z ) Y i =2 dz i πiz i ,J ′ = Z Z | z | = | z | =1 E ( z , z , µq z ) f ( µq z /z ) − σ ( z /z ) σ ( z /z ) Y i =2 dz i πiz i . We have another identity f ( x ) − σ ( µ − q − x ) = σ ( x ) (1 − x ) (1 − µ − q − x ) (1 − µ − q x )(1 − µ − q x )(1 − q x )(1 − q x ) . (A.2) EFORMATIONS OF W ALGEBRAS 41 After renaming z → µq z in J ′ , the two integrands become the same. The integrand of J ′ haspoles at (see Figure 2) z = µ m q − s z ( m ≥ , s = 1 , z = µ − m q − q s z ( m ≥ , s = 1 , z = q s z , µ − q s z ( s = 1 , . The last ones z = q s z , µ − q s z ( s = 1 , 3) are inside the contour for J ′ , while they are outside for J ′ . Using the zero conditions (4.37),(4.38), we conclude that J ′ = J ′ . (cid:3) • • µq − s z q − s z × × µ − q s q − z µ − q s q − z ◦ ◦ µ − q s z q s z J ′ J ′ Figure 2. Integration contours on the z -plane ( s = 1 , 3) for J ′ , J ′ .A.2. The general case. We consider the general case [ I m , I n ] = 0. Call the integration variables z , . . . , z m for I m and w , . . . , w n for I n . We proceed in the same way, rewriting I m I n and I n I m asintegrals over the unit circle and picking residues with respect to some groups of variables.First consider I m I n . In view of symmetry and zeros on the diagonal, it is sufficient to considerresidues from z i = q − w i (1 ≤ i ≤ k ) ,z i = µ − q − w i ( k + 1 ≤ i ≤ k + l ) . The result is (we write only the integrand) J = E ( { q − w i , w i } ki =1 , { µ − q − w i , w i } k + li = k +1 , { z j } mj = k + l +1 , { w j } nj = k + l +1 ) × F G H , with F = Y ≤ i = j ≤ k f ( q w j /w i ) − σ ( w j /w i ) Y k +1 ≤ i = j ≤ k + l f ( µq w j /w i ) − σ ( w j /w i ) × Y ≤ i ≤ kk +1 ≤ j ≤ k + l f ( q w j /w i ) − σ ( µ − w j /w i ) σ ( w i /w j ) Y k +1 ≤ i ≤ k + l ≤ j ≤ k f ( µq w j /w i ) − σ ( µw j /w i ) σ ( w i /w j ) ,G = m Y j = k + l +1 (cid:16) Y ≤ i ≤ k f ( w i /z j ) − σ ( q − w i /z j ) σ ( q z j /w i ) Y k +1 ≤ i ≤ k + l f ( w i /z j ) − σ ( µ − q − w i /z j ) σ ( µq z j /w i ) (cid:17) × n Y j = k + l +1 (cid:16) Y ≤ i ≤ k f ( q w j /w i ) − σ ( w j /w i ) σ ( w i /w j ) Y k +1 ≤ i ≤ k + l f ( µq w j /w i ) − σ ( w j /w i ) σ ( w i /w j ) (cid:17) H = Y k + l +1 ≤ i ≤ mk + l +1 ≤ j ≤ n f ( w j /z i ) − Y k + l +1 ≤ i = j ≤ m σ ( z j /z i ) Y k + l +1 ≤ i = j ≤ n σ ( w j /w i ) . For I n I m we work similarly, picking poles at z i = q w i (1 ≤ i ≤ k ) ,z i = µq w i ( k + 1 ≤ i ≤ k + l ) , ending up with J = E ( { q w i , w i } ki =1 , { µq w i , w i } k + li = k +1 , { z j } mj = k + l +1 , { w j } nj = k + l +1 ) × F G H , where F = Y ≤ i = j ≤ k f ( q w j /w i ) − σ ( w j /w i ) Y k +1 ≤ i = j ≤ k + l f ( µq w j /w i ) − σ ( w j /w i ) × Y ≤ i ≤ kk +1 ≤ j ≤ k + l f ( q w i /w j ) − σ ( w i /w j ) σ ( µw j /w i ) Y k +1 ≤ i ≤ k + l ≤ j ≤ k f ( µq w j /w i ) − σ ( w j /w i ) σ ( µ − w i /w j ) ,G = m Y i = k + l +1 (cid:16) Y ≤ j ≤ k f ( z i /w j ) − σ ( q w j /z i ) σ ( q − z i /w j ) Y k +1 ≤ i ≤ k + l f ( z i /w j ) − σ ( µq w j /z i ) σ ( µ − q − z i /w j ) (cid:17) × n Y j = k + l +1 (cid:16) Y ≤ i ≤ k f ( q w i /w j ) − σ ( w i /w j ) σ ( w j /w i ) Y k +1 ≤ i ≤ k + l f ( µq w i /w j ) − σ ( w i /w j ) σ ( w j /w i ) (cid:17) ,H = Y k + l +1 ≤ i ≤ mk + l +1 ≤ j ≤ n f ( z i /w j ) − Y k + l +1 ≤ i = j ≤ m σ ( z j /z i ) Y k + l +1 ≤ i = j ≤ n σ ( w j /w i ) . Using the identities (A.1),(A.2) one can check that J (cid:12)(cid:12) w i → q w i (1 ≤ i ≤ k ) , w j → µq w j ( k +1 ≤ j ≤ k + l ) = J . The contours can be chosen to be the unit circle by the same mechanism observed above. It remainsto show that under symmetrization with respect to { z i } mi = k + l +1 and { w j } nj = k + l +1 we haveSym H = Sym H . EFORMATIONS OF W ALGEBRAS 43 This reduces to identity (3.21). The proof of Theorem 4.10 is now complete. (cid:3) Appendix B. K matrices It is well known that the Hopf algebra E is equipped with the universal R matrix, which gives riseto an intertwiner of E modulesˇ R ( u /u ) : F ( u ) ⊗ F ( u ) → F ( u ) ⊗ F ( u ) . (B.1)Since ˇ R ( u /u ) commutes with the diagonal Heisenberg ∆ h r , it is written in terms of a single boson { a A r } entering the root current A ( z ) =: Λ ( s z )Λ ( s z ) − :. Exhibiting the dependence on parameterswe shall write ˇ R ( u /u ) = ˇ R ( u /u ; q , q , q , { a A r } ) . Proposition B.1. Consider a K module F ( u ) ⊗ F Xc ( X = B , CD ), and let A ( z ) =: e P r ∈ Z a r z − r : bethe root current associated with E ( z ) . Then there exists an intertwiner of K modules K ( u ) : F ( u ) ⊗ F Xc → F ( u − ) ⊗ F Xc in the following cases.type B ( X = B , c = 3) : K ( u ) = ˇ R ( u ; q , q q / , q / , { a r } ) , (B.2) type C ( X = CD , c = 3) : K ( u ) = ˇ R ( u ; q , q q − , q , { a r } ) , (B.3) type D ( X = CD , c = 2) : K ( u ) = ( − N . (B.4) In the last line, N denotes the number operator P r> ν − r a − r a r , ν r = [ a r , a − r ] . Note that K ( u ) isindependent of u in this case.Proof. First note that, by extracting the diagonal Heisenberg, the intertwining relation (B.1) for type A reduces to ˇ R ( u /u ) (cid:0) u Λ A + ( z ) + u Λ A − ( z ) (cid:1) = (cid:0) u Λ A + ( z ) + u Λ A − ( z ) (cid:1) ˇ R ( u /u ) , where Λ A ± ( z ) are vertex operators in { a A r } r =0 such thatΛ A − ( z ) =: Λ A + ( q z ) − : , (B.5) C (cid:0) Λ A + ( z ) , Λ A + ( w ) (cid:1) = − (1 − q )(1 − q )1 + q q . (B.6)For type C and type D we proceed the same way as in type A to obtain the reduced intertwiningrelation K ( u ) (cid:0) u Λ + ( z ) + u − Λ − ( z ) (cid:1) = (cid:0) u − Λ + ( z ) + u Λ − ( z ) (cid:1) K ( u ) , where Λ ± ( z ) are vertex operators in { a r } r =0 such thatΛ − ( z ) =: Λ + ( q q − c z ) − : . (B.7)For type C ( c = 3), we have further C (cid:0) Λ + ( z ) , Λ + ( w ) (cid:1) = − (1 − q )(1 − q )1 + q q − q q − . (B.8)Comparing (B.7), (B.8) with (B.5), (B.6), we obtain (B.3). For type D ( c = 2), (B.7) becomes Λ − ( z ) =: Λ + ( z ) − :, so that the intertwining relation reducesfurther to K ( u ) a r = − a r K ( u ) for all r = 0. This leads to the solution (B.4).For type B , the reduced intertwining relation involves three terms K ( u ) (cid:0) u Λ ++ ( z ) + k Λ ( z ) + u − Λ −− ( z ) (cid:1) = (cid:0) u − Λ ++ ( z ) + k Λ ( z ) + u Λ −− ( z ) (cid:1) K ( u ) , which corresponds to the qq character of the three dimensional representation of U q b sl . One canreduce it further to intertwining relation for the two dimensional one K ( u ) (cid:0) u / Λ + ( z ) + u − / Λ − ( z ) (cid:1) = (cid:0) u − / Λ + ( z ) + u / Λ − ( z ) (cid:1) K ( u )by introducing Λ ± ( z ) such that Λ ±± ( z ) =: Λ ± ( s / z )Λ ± ( s − / z ) :, Λ ( z ) =: Λ + ( s / z )Λ − ( s − / z ) :.The rest is similar to type C . (cid:3) On tensor products (4.9), (4.10), we write the intertwiners as ˇ R i,i +1 ( u i /u i +1 ) ( i = 1 , . . . , ℓ − K ℓ ( u ℓ ). The standard argument (with ℓ = 2) leads to the boundaryYang-Baxter equationˇ R , ( u /u ) K ( u )ˇ R , ( u u ) K ( u ) = K ( u )ˇ R , ( u u ) K ( u )ˇ R , ( u /u ) . (B.9)As noted above, the K matrix K ℓ of type D is independent of u ℓ and satisfies K ℓ = 1. Comparingthe intertwining relation with the definition of the currents A ℓ − ( z ) (4.14) and A ℓ ( z ) (4.17), we seethat K ℓ A ℓ − ( z ) = A ℓ ( z ) K ℓ . Though the zeroth node of the Dynkin diagram is not associated with boundary Fock modules,one can consider K matrices depending only on A ( z ) and satisfying the intertwining relations, forexample for type C K ( u ) (cid:0) Λ ¯1 ( z ) + Λ ( µz ) (cid:1) = (cid:0) u Λ ¯1 ( z ) + u − Λ ( µz ) (cid:1) K ( u ) . Appendix C. The library of Cartan matrices C.1. Conventions. The matrix of contractions ˆ B is the deformed version of the symmetrized Cartanmatrix. We will give a list of explicit deformed Cartan matrices ˆ C of low rank which can be used towrite all others as explained in Appendix C.4.The deformed symmetrized Cartan matrix ˆ B and the deformed Cartan matrix ˆ C are related by adiagonal matrix ˆ D , namely ˆ B = ˆ D ˆ C , where the diagonal entries d i of ˆ B and the diagonal entries ofˆ C are given as follows.The nodes of type A (corresponding to F c i ⊗ F c i +1 ). A , c i = c i +1 : d i = − t t t t c i , C i,i = s c i + s − c i . c i A , c i = c i +1 : d i = t b , C i,i = t b ( b = c i , c i +1 ) . b EFORMATIONS OF W ALGEBRAS 45 The nodes of type B (corresponding to F c ℓ ⊗ F B c ℓ +1 ). B , c ℓ = c ℓ +1 : d ℓ = − t t t t c ℓ +1 ( s / c ℓ +1 + s − / c ℓ +1 ) , C ℓ,ℓ = s / c ℓ +1 + s − / c ℓ +1 . c ℓ B , c ℓ = c ℓ +1 : d ℓ = − t b ( s / c ℓ +1 − s − / c ℓ +1 ) , C ℓ,ℓ = s / c ℓ +1 s − / b + s − / c ℓ +1 s / b ( b = c ℓ , c ℓ +1 ) . b The nodes of type C (corresponding to F c ℓ ⊗ F CD c ℓ +1 , c ℓ +1 = c ℓ ). C , c ℓ = c ℓ +1 : d ℓ = − t b ( s c ℓ +1 − s − c ℓ +1 ) , C ℓ,ℓ = s c ℓ +1 s − c ℓ + s − c ℓ +1 l s c ℓ ( b = c ℓ , c ℓ +1 ) . b Type D nodes (corresponding to F c ℓ − ⊗ F c ℓ ⊗ F CD c ℓ +1 , c ℓ +1 = c ℓ ). D , c ℓ +1 = c ℓ − : d ℓ = − t t t t c ℓ +1 , C ℓ,ℓ = s c ℓ +1 + s − c ℓ +1 c ℓ D , c ℓ +1 = c ℓ − : d ℓ = t b , C ℓ,ℓ = t b ( b = c ℓ +1 , c ℓ − ) b Affine nodes (corresponding to dressing in non A types) are the same as B , C , D nodes after thechange c i → c ℓ +1 − i , d i → d ℓ − i , C i,i → C ℓ − i,ℓ − i . The deformed affine Cartan matrices we obtain have “local” form: C i,j = 0 if | i − j | is sufficientlylarge. Moreover, the non-zero terms stabilize with ℓ increased. We give here a number of explicitdeformed Cartan matrices of low rank which can be used to write all others as explained in AppendixC.4.We use the following notation. We write X ( c ; c , . . . , c ℓ − ; c ℓ +1 ) Y for the choice of the module andthe dressing and give the ( ℓ + 1) × ( ℓ + 1) matrix ˆ C . Here Y can be B , C , or D depending on whichboundary module we consider F B c ℓ +1 or F CD c ℓ +1 . As always, in the latter case we choose D if c ℓ +1 = c ℓ and C otherwise. Similarly X can be B , C , or D depending on which dressing we choose. Namely C /µ = q − / c corresponds to X = B while C /µ = q c corresponds to X = C , if c = c and to X = D if c = c .We skip writing X and c in finite types. In addition we also skip c ℓ +1 and Y in A type.C.2. Finite types. Type A . (2 , , (cid:18) s + s − − − s + s − (cid:19) (1 , , (cid:18) t t t t (cid:19) (1 , , (cid:18) t t t t (cid:19) (2 , , (cid:18) s + s − − t t (cid:19) (1 , , (cid:18) t t − s + s − (cid:19) Type B . (2 , 2; 2) B (cid:18) s + s − − − s / − s − / s / + s − / (cid:19) (2 , 2; 3) B (cid:18) s + s − − − s / − s − / s s / + s − s − / (cid:19) (1 , 2; 2) B (cid:18) t t − s / − s − / s / + s − / (cid:19) (1 , 2; 1) B (cid:18) t t − s / − s − / s s / + s − s − / (cid:19) (1 , 2; 3) B (cid:18) t t − s / − s − / s s / + s − s − / (cid:19) Type C . (2 , 2; 3) C (cid:18) s + s − − s − s − − s s − + s − s (cid:19) (1 , 2; 1) C (cid:18) t s − s − − s s − + s − s (cid:19) (1 , 2; 3) C (cid:18) t − t ( s + s − ) − s s − + s − s (cid:19) Type D . EFORMATIONS OF W ALGEBRAS 47 (2 , , 2; 2) D s + s − − − − s + s − − s + s − (1 , , 2; 2) D t t t − s + s − − s + s − (2 , , 2; 2) D t t t t t s s − − s − s t s s − − s − s t (1 , , 2; 2) D s + s − − − t t s s − − s − s t s s − − s − s t (3 , , 2; 2) D t t t t t s s − − s − s t s s − − s − s t C.3. Affine types. × cases. B − B types. B (2; 2; 2) B s / + s − / − s / − s − / − s / − s − / s / + s − / ! B (2; 2; 1) B s / + s − / − s / − s − / − s / − s − / s / s − / + s − / s / ! B (1; 2; 1) B s / s − / + s − / s / − s / − s − / − s / − s − / s / s − / + s − / s / ! B (1; 2; 3) B s / s − / + s − / s / − s / − s − / − s / − s − / s / s − / + s − / s / ! B − C types. B (2; 2; 1) C (cid:18) s / + s − / − ( s + s − )( s / + s / ) − s s − + s − s (cid:19) B (1; 2; 1) C (cid:18) s / s − / + s − / s / − ( s + s − )( s / + s − / ) − s s − + s − s (cid:19) B (3; 2; 1) C (cid:18) s / s − / + s − / s / − ( s + s − )( s / + s − / ) − s s − + s − s (cid:19) C − C types. C (1; 2; 1) C (cid:18) s s − + s − s − s − s − − s − s − s s − + s s − (cid:19) C (1; 2; 3) C (cid:18) s s − + s − s − s − s − − s − s − s s − + s s − (cid:19) × cases. B − D types. B (2; 2 , 2; 2) D s / + s − / − s / − s − / − s / − s − / − s + s − − s + s − B (1; 2 , 2; 2) D s / s − / + s − / s / − s / − s − / − s / − s − / − s + s − − s + s − B (2; 1 , 2; 2) D s / s − / + s − / s / − s / − s − / − s / − s − / s − s − s − s − s s − − s − s s − s − s s − − s − s s − s − B (1; 1 , 2; 2) D s / + s − / − s / − s − / − s / − s − / s − s − s − s − s s − − s − s s − s − s s − − s − s s − s − B (1; 3 , 2; 2) D s / s − / + s − / s / − s / − s − / − s / − s − / s − s − s − s − s s − − s − s s − s − s s − − s − s s − s − C − D types. C (1; 2 , 2; 2) D s s − + s − s − − − s − s − s + s − − s − s − s + s − C (1; 3 , 2; 2) D s s − + s − s − − s + s − )( s − s − ) s − s − s − s − s s − ( s + s − )( s − s − ) s − s − s s − s − s − C (2; 1 , 2; 2) D s s − + s − s − − s − s − s − s − s − s − s s − s − s − s − s − s s − s − s − D − D types. EFORMATIONS OF W ALGEBRAS 49 D (2; 2 , 2; 2) D s + s − − s − s − s + s − − s − s − s + s − D (1; 1 , 2; 2) D s − s − s − s − s s − − s + s − s − s − s s − s − s − − s − s + s s − − s + s − − s − s + s s − s − s − × cases D − D types. D (2; 2 , , 2; 2) D s + s − − − s + s − − − − − s + s − − − s + s − D (2; 2 , , 2; 2) D s − s − s s − − s − s s − s − s − s − s s − − s − s s − s − s − s − s − s − s − s − s − s − s − s − s s − − s − s s − s − s − s − s s − − s − s s − s − D (1; 1 , , 2; 2) D s − s − s − s − s s − s − s − s − s − s − s − s s − s − s − s − s − s − s − − − s + s − − − s + s − D (1; 1 , , 2; 2) D s − s − s − s − s s − s − s − s − s − s − s − s s − s − s − s − s − s − s − s − s − s − s − s − s − s − s − s s − s − s − s − s − s − s − s s − s − s − C.4. General case. In Appendix C.2 we gave several deformed Cartan matrices of finite type, andin Appendix C.3 several deformed Cartan matrices of affine type. In fact Appendix C.2 contains allmaximal submatrices of stable deformed affine Cartan matrices whose upper right and bottom leftcorner elements are both non zero. Appendix C.3 contains all non-stable deformed affine Cartanmatrices appearing in our construction.It is important to keep in mind that we immediately obtain many more examples by permuting s , s , s . Another set of examples is obtained by reading the data from right to left. More precisely,the matrices ˆ C corresponding to X ( c ; c , . . . , c ℓ ; c ℓ +1 ) Y and to Y ( c ℓ +1 ; c ℓ , . . . , c ; c ) X are related byconjugation by the matrix ( δ i + j,ℓ ) ℓi,j =0 .The matrices listed in Appendices C.2 and C.3 allow us to write a deformed affine Cartan matrixcorresponding to arbitrary data X ( c ; c , . . . , c ℓ ; c ℓ +1 ) Y as for larger ℓ the deformed Cartan matricesstabilize. One has to follow the following procedure.First, search the affine examples, keeping in mind the symmetries. If the matrix is found, stop. Ifthe matrix is not in the list, conclude that ˆ C is stable, that is C ,ℓ = C ℓ, = 0. Second, find the listed matrix of the finite type and of the largest size corresponding to right mostcolors: ( c ℓ − i , . . . , c ℓ ; c ℓ +1 ) Y (it is 2 × Y = D when it is 3 × C .Third, repeat for left most colors, that is, find the listed matrix of the finite type and of the largestsize corresponding to X ( c ; c , . . . , c i ). This gives the left upper submatrix.The rest nonzero entries of the matrix are recovered from matrices of type A .The final result is the superposition of matrices which are linked via diagonal entries.For example, the matrix ˆ C corresponding to B (2; 3 , 1; 3) C is a superposition of the last matrix oftype B in our list and of the second matrix of type C with necessary symmetries (the C , entry t iscommon): s / s + s − / s − − s / − s − / t t s − s − − s s − + s − s . 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Olshanskii, Twisted Yangians and infinite-dimensional classical Lie algebras , Lecture Notes in Math., , Springer, Berlin, 1992 BF: National Research University Higher School of Economics, 101000, Myasnitskaya ul. 20,Moscow, Russia, and Landau Institute for Theoretical Physics, 142432, pr. Akademika Semenova1a, Chernogolovka, Russia E-mail address : [email protected] MJ: Department of Mathematics, Rikkyo University, Toshima-ku, Tokyo 171-8501, Japan E-mail address : [email protected] EM: Department of Mathematics, Indiana University Purdue University Indianapolis, 402 N.Blackford St., LD 270, Indianapolis, IN 46202, USA E-mail address : [email protected] IV: Center for Advanced Studies, Skolkovo Institute of Science and Technology, 1 Nobel St.,143026, Moscow, Russia, and National Research University Higher School of Economics, RussianFederation, 101000, Myasnitskaya ul. 20, Moscow, Russia E-mail address ::