We establish Drinfeld realization for the two-parameter twisted quantum affine algebras using a new method. The Hopf algebra structure for Drinfeld generators is given for both untwisted and twisted two-parameter quantum affine algebras, which include the quantum affine algebras as special cases.

Drinfeld realization8 is a loop algebra type realization of the quantum affine algebra. It was introduced in studying finite dimensional representations of quantum affine algebras, and has since played an important role in representation theory such as in vertex representations10,18 and finite dimensional representations of quantum affine algebras for quiver varieties.2,6,7,13,29

Drinfeld realization was first proved by Beck1 using Lusztig’s braid groups.28 One of us19 also gave an elementary proof of the untwisted quantum affine algebras using q-commutators. Subsequently both of these methods are generalized to twisted quantum affine algebras in Refs. 31, 23, and 24 using the Hopf algebraic structures and braid groups. Two-parameter quantum enveloping algebras were introduced as generalization of the one-parameter quantum enveloping algebras.30,3–5 It was known that the theory has analogous properties with the one-parameter counterpart such as a similar Schur-Weyl duality and Drinfeld double structure.3 Recent advances on geometric representations9 have realized the two-parameter quantum groups naturally, where the second parameter turns out to be closely associated with the Tate twist.

Two-parameter quantum enveloping algebras have a generalized root space structure where the action of generalized Lusztig’s braid groups is not closed, but a Weyl groupoid action sends Ur,s(𝔤) to Ur′,s(𝔤).14 Therefore the Drinfeld realizations of two-parameter quantum enveloping algebras cannot be studied by the braid group action. Hu, Rosso, and Zhang15 first studied the Drinfeld realization of the two-parameter quantum affine algebra in type A by constructing its quantum Lyndon basis and the vertex representation. Furthermore, vertex representations of two-parameter quantum affine algebras in other types were also considered in Refs. 11, 16, 17, 21, 22, and 32.

The goal of this paper is twofold. First, we extend the q-commutator approach19 to derive Drinfeld realizations for all twisted two-parameter quantum affine algebras using a new method, and establish the isomorphism between the Drinfeld realization and the Drinfeld-Jimbo form of the two-parameter quantum enveloping algebras. A simpler set of generators is used to replace the original full set of the Drinfeld generators in the quantized algebra, which enables us to simplify many computations involved with Drinfeld generators and Drinfeld-Jimbo generators. Thus the current work in two-parameter cases contains a brand new proof of the Drinfeld realization for quantum affine algebras as a special case.

Second, our new method gives explicit formulae for the Hopf algebra structure of Ur,s(ĝ) in terms of the Drinfeld generators. It was recently announced by Guay and Nakajima that the affine Yangian algebra Y(ĝ) has a simple Hopf algebra structure in terms of the Drinfeld generators. The special case of our result for the quantum affine algebra induces a Hopf algebra structure for the double affine Yangian algebra DY(𝔤) in view of Ref. 12.

The paper is organized as follows. After a quick introduction of preliminaries in Section II, we introduce the two-parameter twisted quantum affine algebras in the Drinfeld-Jimbo form in Section III. The loop algebra formulation of the two-parameter twisted quantum affine algebras was given in Section IV. We obtain in Section V that the Drinfeld realization is isomorphic to the Drinfeld-Jimbo form as associative algebras. This isomorphism is proved using a new method in Sections VI and VII, which is based on a set of simple generators. In Section VIII, we define the actions of a comultiplication on the simple generators of Drinfeld realization, thus we can obtain the Hopf algebra structure of Drinfeld realization. Moreover, we show that there exists a Hopf algebra isomorphism between the above two realizations.

We begin with a brief review of basic terminologies and notations of twisted affine Lie algebras following,26 in particular, on finite order automorphisms of the finite dimensional simple Lie algebra.

Let 𝔤 be a simple finite-dimensional Lie algebra with the Cartan matrix A = (Aij), i,  j ∈ {1,  2,  …,  N} of simply laced type AN (N ⩾ 2), DN (N > 4), E6 or D4. Let σ be an Dynkin diagram automorphism of 𝔤(A) of order k. We denote by I = {1,  2,  …,  n} the set of σ-orbits on {1,  2,  …,  N}. The action of σ on the Dynkin diagram is listed as follows:

AN:σ(i)=N+1i,DN:σ(i)=i,1iN2,σ(N1)=N,D4:σ(1,2,3,4)=(3,2,4,1),E6:σ(i)=6i,1i5,σ(6)=6.

Let ω=exp2πik be the primitive kth root of unity. Then we have the ℤ/rℤ-graded decomposition

g(A)=jZ/kZgj,

where 𝔤j is the eigenspace relative to the eigenvalue ωj. Subsequently 𝔤0 is a Lie subalgebra of 𝔤(A). Obviously, the nodes of the Dynkin diagram of 𝔤0 are indexed by I.

For a nontrivial automorphism σ of the Dynkin diagram, the twisted affine Lie algebra gˆσ is the following graded algebra:

gˆσ=jZg[j]CtjCcCd,

where c is the central element and ad(d)=tddt. Here we use [j] to denote j(modk). Thus the twisted affine Lie algebra gˆσ is the universal central extension of the twisted loop algebra. We denote by Iˆ=I{0} the nodes set of Dynkin diagram of gˆσ.

Let Aσ=(aijσ)(i,jIˆ) be the Cartan matrix of the twisted affine Lie algebra gˆσ of type XN(r). The Cartan matrix Aσ is symmetrizable, i.e., there exists a diagonal matrix D=diag(di|iIˆ) such that DAσ is symmetric. Let αi(iIˆ) be the simple roots of gˆσ, and Δ, Δˆ the root systems of 𝔤0, gˆσ respectively. Then Δ = {α1,  …,  αn} and Δˆ={α0}Δ, where α0 = δθ, δ is the simple imaginary root, and θ is the maximal root of 𝔤0. We let ( , ) be the canonical bilinear form of the Cartan subalgebra such that 2(αi,αj)(αi,αi)=aijσ, i,jIˆ. Let di=(αi,αi)2, iIˆ.

We assume that the ground field 𝕂 is ℚ(r, s), the field of rational functions in two indeterminates r, s (r≠ ± s).

Let ri=rdi,si=sdi,iIˆ. Define the (r, s)-integer by

[n]i=rinsinrisi,[n]i!=k=1n[k]i.

Definition 3.1.
The two-parameter twisted quantum affine algebra Ur,s(gˆσ) is the unital associative algebra over 𝕂 generated by the elements ej,fj,ωj±1,ωj±1(jIˆ),γ±12, γ±12, satisfying the following relations:
γ±12,γ±12are central withγ=ωδ,γ=ωδ,γγ=(rs)c,such thatωiωi1=ωiωi1=1,and
(𝒳1)
[ωi±1,ωj±1]=[ωi±1,ωj±1]=[ωi±1,ωj±1]=0,
ωjeiωj1=i,jei,ωjfiωj1=i,j1fi,
(𝒳2)
ωjeiωj1=j,i1ei,ωjfiωj1=j,ifi,
(𝒳3)
ei,fj=δijrs(ωiωi),
(𝒳4)
(r,s)Serrerelationsforei:foranyijIˆ,
(𝒳5)
(adlei)1aij(ej)=0,
(r,s)Serrerelationsforfi:foranyijIˆ,
(𝒳6)
(adrfi)1aij(fj)=0,

where the left-adjoint action adlei and right-adjoint action adrfi are given in Sec. III B to save space (cf. Proposition 3.3).

Here the structural constants 〈i,  j〉 in relations X2 and X3 are given in the two-parameter quantum Cartan matrix J defined below. In fact, the submatrix after removing the zeroth row and zeroth column of J is exactly the two-parameter quantum Cartan matrix of the finite dimensional type, while the data of the zeroth row and the zeroth column are compatible with the results of Section VI. We list the two-parameter quantum Cartan matrices of the twisted cases for future reference.

For type A2n1(2),
J=rs1(rs)1r11(rs)2rsrs1r111ssrs111111rs1r2(rs)211s2r2s2.
For type A2n(2),
J=r2s2r211rss2rs1r1111srs111111rs1r1(rs)111sr12s12.
For type Dn+1(2),
J=rs1r211rss2r2s2r2111s2r2s211111r2s2r2(rs)111s2rs1.
For type D4(3),
J=rs1r2s1(rs)3rs2rs1r3(rs)3s3r3s3.
For type E6(2),
J=rs1r2s1(rs)1(rs)2(rs)2rs2rs1r111rssrs1r21(rs)21s2r2s2r2(rs)211s2r2s2.

Remark 3.2.

For later purpose, we write down explicitly relation (X5) case by case as follows.

Case (I): In the case of A2n1(2), for i = 1,  2, …, n − 1 and j,kIˆ such that ajkσ=0, we have the (r, s)-Serre relations,
ejekk,jekej=0,
ei2ei+1(ri+si)eiei+1ei+(risi)ei+1ei2=0,
eiei+12(ri+1+si+1)ei+1eiei+1+(ri+1si+1)ei+12ei=0,
e02e2(r+s)e0e2e0+(rs)e2e02=0,
e0e22(r+s)e2e0e2+(rs)e22e0=0.
Case (II): In the case of A2n(2), for i = 1,  2, …, n − 1 and j,kIˆ such that ajkσ=0, we have the (r, s)-Serre relations,
ejekk,jekej=0,
ei2ei+1(ri+si)eiei+1ei+(risi)ei+1ei2=0,
eiei+12(ri+1+si+1)ei+1eiei+1+(ri+1si+1)ei+12ei=0,
e02e1(r2+s2)e0e1e0+(rs)2e1e02=0,
e0e13(rs)[3]e1e0e12+[3]e12e0e1(rs)3e13e0=0,
en12en(r+s)en1enen1+(rs)enen12=0,
en1en3(rs)12[3]12enen1en2+[3]12en2en1en(rs)32en3en1=0.
Case (III): In the case of Dn+1(2), for i = 0,  1, …, n − 1 and j,kIˆ such that ajkσ=0, we have the (r, s)-Serre relations,
ejekk,jekej=0,
ei2ei+1(ri+si)eiei+1ei+(risi)ei+1ei2=0,
eiei+12(ri+1+si+1)ei+1eiei+1+(ri+1si+1)ei+12ei=0,
en12en(r2+s2)en1enen1+(rs)2enen12=0,
en1en3(rs)[3]enen1en2+[3]en2en1en(rs)3en3en1=0.
Case (IV): In the case of D4(3), for i,jIˆ such that aijσ=0, we have the (r, s)-Serre relations,
ejekk,jekej=0,
e02e1(rs)[2]e0e1e0+(rs)3e1e02=0,
e0e12(rs)[2]e1e0e1+(rs)3e12e0=0,
e1e22(r3+s3)e2e1e2+(rs)3e22e1=0,
e14e2[4]e13e2e1+(rs)[4][3][2]e12e2e12(rs)3[4]e1e2e13+(rs)6e2e14=0.
Case (V): In the case of E6(2), for i,jIˆ such that aijσ=0, we have the (r, s)-Serre relations,
eiejj,iejei=0,
e02e1rs(r+s)e0e1e0+(rs)2e1e02=0,
e0e12rs(r+s)e1e0e1+(rs)2e12e0=0,
e12e2(r+s)e1e2e1+(rs)e2e12=0,
e1e22(r+s)e2e1e2+(rs)e22e1=0,
e32e4(r2+s2)e3e4e3+(rs)2e4e32=0,
e3e42(r2+s2)e4e3e4+(rs)2e42e3=0,
e2e32(r2+s2)e3e2e3+(rs)2e32e2=0,
e23e3[3]e22e3e2+(rs)[3]e2e3e22(rs)3e3e23=0.

One can check the following fact directly.

Proposition 3.3.
The two-parameter twisted quantum affine algebraUr,s(gˆσ)is a Hopf algebra27with the comultiplication Δ, the counit ε, and antipodeSdefined below: foriIˆ, we have
Δ(γ±12)=γ±12γ±12,Δ(γ±12)=γ±12γ±12,
Δ(ωi)=ωiωi,Δ(ωi)=ωiωi,
Δ(ei)=ei1+ωiei,Δ(fi)=fiωi+1fi,
ε(ei)=ε(fi)=0,ε(γ±12)=ε(γ±12)=ε(ωi)=ε(ωi)=1,
S(γ±12)=γ12,S(γ±12)=γ12,
S(ei)=ωi1ei,S(fi)=fiωi1,S(ωi)=ωi1,S(ωi)=ωi1.

Remark 3.4.

(1) When r = q = s−1, the quotient Hopf algebra of U=Ur,s(gˆσ) modulo, the Hopf ideal generated by the elements ωiωi1(iIˆ) and γ12γ12 is the classical twisted quantum affine algebra Uq(gˆσ) in Drinfeld-Jimbo type.

(2) In the Hopf algebra Ur,s(gˆσ), the left-adjoint and right-adjoint actions are defined in the usual manner:
adla(b)=(a)a(1)bS(a(2)),adra(b)=(a)S(a(1))ba(2),
where Δ(a)=(a)a(1)a(2) for any a, bUr,s(gˆσ).

The two-parameter twisted quantum affine algebra Ur,s(gˆσ) is endowed with a Drinfeld double structure (see Proposition 3.5), which is similar to the untwisted cases. The following statement can be proved by standard arguments similarly (see Ref. 15).

Proposition 3.5.

Ur,s(gˆσ)is isomorphic to its Drinfeld double as a Hopf algebra.

Let U0=K[ω0±1,,ωn±1,ω0±1,,ωn±1,γ,γ] denote the Cartan subalgebra of Ur,s(gˆσ).

Furthermore, let Ur,s(n̂) (respectively Ur,s(n̂)) be the subalgebra of Ur,s(gˆσ) generated by ei (respectively fi) for all iIˆ. Then, we get the standard triangular decomposition of Ur,s(gˆσ).

Corollary 3.6.

Ur,s(gˆσ)Ur,s(n̂)U0Ur,s(n̂) as vector spaces.□

Definition 3.7
cf. Ref. 15 
Let τ be the ℚ-algebra anti-automorphism of Ur,s(gˆσ) such that τ(r) = s, τ(s) = r, τ(ωi,ωj±1)=ωj,ωi1, and
τ(ei)=fi,τ(fi)=ei,τ(ωi)=ωi,τ(ωi)=ωi,
τ(γ)=γ,τ(γ)=γ.

In fact τ is an analog of the Chevalley anti-involution on Ur,s(gˆσ).

In this section, we will generalize the result from the untwisted cases17 to the twisted ones and state the Drinfeld realization in the general form.

For convenience, if two roots α = αi1 + ⋯ + αim,  β = αj1 + ⋯ + αjm are decompositions into simple roots, we denote α,β=k=1ml=1nik,jl

To state Drinfeld realization of two-parameter quantum affine algebra Ur,s(gˆσ), we need to define some functions gij±(z).

For i,  j = 1,  …,  n, let

Fij±(z,w)=lZ/kZ(zωl(i,σl(j)σl(j),i)±12w),Gij±(z,w)=lZ/kZ(σl(j),i±1z(σl(j),ii,σl(j)1)±12w).

For simple roots αi,  αj ∈ Δ, we set gij±(z)=nZ+cαi,αj,n±zn=nZ+cijn±zn, where the coefficients cijn± are determined from the Taylor series expansion at ξ = 0 of the function

nZ+cijn±ξn=gij±(ξ)=Gij±(ξ,1)Fij±(ξ,1).

To write the relations in a compact form, we need the formal distribution

δ(z)=kZzk.

Definition 4.1.
The two-parameter twisted quantum affine algebra Ur,s(gˆσ) is the unital associative algebra with the generators
{xi±(k),ϕi(m),φi(m),γ12,γ12|i=1,,n,kZ,mZ+}
satisfying the defining relations in terms of the generating functions,
xi±(z)=kZxi±(k)zk,ϕi(z)=mZ+ϕi(m)zm,
φi(z)=mZ+φi(m)zm.
The relations are given as follows:
γ±12,γ±12are central and invertible such thatγγ=(rs)c,
(4.1)
xσ(i)±(k)=ωkxi±(k),φσ(i)(m)=ωmφi(m),ϕσ(i)(n)=ωnϕi(n),
(4.2)
φi(0)ϕj(0)=ϕj(0)φi(0),
(4.3)
[φi(z),φj(w)]=[ϕi(z),ϕj(w)]=0,
(4.4)
φi(z)ϕj(w)=ϕj(w)φi(z)gij(zw1(γγ)12γ)gij(zw1(γγ)12γ),
(4.5)
φi(z)xj±(w)φi(z)1=gij(zw(γγ)12γ12)±1xj±(w),
(4.6)
ϕi(z)xj±(w)ϕi(z)1=gji(wz(γγ)12γ±12)1xj±(w),
(4.7)
[xi+(z),xj(w)]=δijrisiδ(zw1γ)ϕi(wγ12)δ(zw1γ)φi(zγ12),
(4.8)
Fij±(z,w)xi±(z)xj±(w)=Gij±(z,w)xj±(w)xi±(z),
(4.9)
xi±(z)xj±(w)=j,i±1xj±(w)xi±(z),foraijσ=0,
(4.10)
Symz1,z2,z3((rs2)k4z1(rk4+sk4)z2+(r2s)k4z3)xi±(z1)xi±(z2)xi±(z3)=0,
(4.11)
forAi,σ(i)=1,
Symz1,z2Pij±(z1,z2)t=0t=2(1)t(risi)±t(t1)22t±ixi±(z1)xi±(zt)
(4.12)
×xj±(w)xi±(zt+1)xi±(z2)=0,forAi,j=1,and1j<iNsuch thatσ(i)j,Symz1,z2Pij±(z1,z2)t=0t=2(1)t(risi)t(t1)22tixi±(z1)xi±(zt)
(4.13)
×xj±(w)xi±(zt+1)xi±(z2)=0,
forAi,j=1,and1i<jNsuch thatσ(i)j,
where [l]±i=ri±lsi±lri±1si±1,  [l]i=rilsilri1si1, Symz1,…,zk means the symmetrization over the variables z1, …, zk, and Pij± are given by
ifσ(i)=i,thenPij±(z,w)=1,dij=k,
ifAi,σ(i)=0,σ(j)=j,thenPij±(z,w)=zr(rs1)±kwrz(rs1)±1w,dij=k,
ifAi,σ(i)=0,σ(j)j,thenPij±(z,w)=1,dij=1/2,
ifAi,σ(i)=1,thenPij±(z,w)=z(rs1)±k/4+w,dij=k/2.

Remark 4.2.

Note that in relation (4.1), γ and γ′ are related by the central element c. In the one-parameter case, the central element c is absent in the relation, since γ and γ′ are inverse to each other.

We now give the component form of the two-parameter Drinfeld realization which is equivalent to the earlier formulation.

Definition 4.3.
The unital associative algebra Ur,s(gσˆ) over 𝕂 is generated by the elements xi±(k), ai(ℓ), ωi±1, ωi±1, γ±12, γ±12, (iI, k,  k′ ∈ ℤ, ℓ,  ℓ′ ∈ ℤ∖{0}), subject to the following defining relations:
γ±12,γ±12are central such thatγγ=(rs)c,ωiωi1=ωiωi1=1(iI),and
(D1)
fori,jI,one has
[ωi±1,ωj±1]=[ωi±1,ωj±1]=[ωi±1,ωj±1]=0,
xσ(i)±(l)=ωlxi±(l),aσ(i)(m)=ωmai(m),
(D2)
[ai(),aj()]=δ+,0t=0k1(γγ)2(risi)Ai,σt(j)2[Ai,σt(j)]iγγrs,
(D3)
[ai(),ωj±1]=[ai(),ωj±1]=0,
(D4)
ωixj±(k)ωi1=t=0k1σt(j),i±1xj±(k),
(D5)
ωixj±(k)ωi1=t=0k1i,σt(j)1xj±(k),[ai(),xj±(k)]=±t=0k1(i,iAi,σt(j)2i,iAi,σt(j)2)(risi)(γγ)2γ±2xj±(+k),for>0,
(D61)
[ai(),xj±(k)]=±t=0k1(i,iAi,σt(j)2i,iAi,σt(j)2)(risi)(γγ)2γ±2xj±(+k),for<0,
(D62)
Fij±(z,w)xi±(z)xj±(w)=Gij±(z,w)xj±(w)xi±(z),
(D7)
[xi+(k),xj(k)]=δijrisiγkγk+k2ϕi(k+k)γkγk+k2φi(k+k),
(D8)
where ϕi(0) = ωi, φi(0)=ωi, and ϕi(m), φi(−m) (m ∈ ℤ≥0) are defined as follows:
m=0ϕi(m)zm=ωiexp(risi)=1ai()z,
m=0φi(m)zm=ωiexp(risi)=1ai()z,
xi±(m)xj±(k)=j,i±1xj±(k)xi±(m),foraijσ=0,
(D91)
Symz1,z2,z3((rs2)k4z1(rk4+sk4)z2+(r2s)k4z3)xi±(z1)xi±(z2)xi±(z3)=0,forAi,σ(i)=1,
(D92)
Symz1,z2Pij±(z1,z2)t=0t=2(1)t(risi)±t(t1)22t±ixi±(z1)xi±(zt)×xj±(w)xi±(zt+1)xi±(z2)=0,forAi,j=1,and1j<iNsuch thatσ(i)j,
(D93)
Symz1,z2Pij±(z1,z2)t=0t=2(1)t(risi)t(t1)22tixi±(z1)xi±(zt)×xj±(w)xi±(zt+1)xi±(z2)=0,forAi,j=1,and1i<jNsuch thatσ(i)j,
(D94)
where [l]±i=ri±lsi±lri±1si±1,  [l]i=rilsilri1si1, and Pij± are given by
ifσ(i)=i,thenPij±(z,w)=1,dij=k,
ifAi,σ(i)=0,σ(j)=j,thenPij±(z,w)=zr(rs1)±kwrz(rs1)±1w,dij=k,
ifAi,σ(i)=0,σ(j)j,thenPij±(z,w)=1,dij=1/2,
ifAi,σ(i)=1,thenPij±(z,w)=z(rs1)±k/4+w,dij=k/2.

The following analog of Chevalley anti-homomorphism can be checked directly.

Proposition 4.4.
The following-linear and multiplicative mappingτdefines an anti-automorphism ofUr,s(gˆσ):τ(r) = s,τ(s) = r,τ(ωi,ωj±1)=ωj,ωi1, and
τ(ωi)=ωi,τ(ωi)=ωi,
τ(γ)=γ,τ(γ)=γ,
τ(ai())=ai(),
τ(xi±(m))=xi(m),
τ(ϕi(m))=φi(m),τ(φi(m))=ϕi(m).

Let Ur,s(𝔤σ) be the subalgebra of Ur,s(gˆσ) generated by xi±(0),ωi,ωi, (iI). Clearly Ur,s(𝔤σ) ≅ Ur,s(𝔤σ), the subalgebra of Ur,s(𝔤σ) generated by ei,fi,ωi,ωi (iI).

Using defining relations (D1)–(D9), one can easily show that Ur,s(gˆσ) has a triangular decomposition

Ur,s(gˆσ)=Ur,s(n˜)Ur,s0(ĝ)Ur,s(n˜+),

where Ur,s(n˜±)=αQ̇±Ur,s(n˜±)α is generated respectively by xi±(k) (iI), and Ur,s0(ĝ) is the subalgebra generated by ωi±1, ωi±1, γ±12, γ±12, and ai(±ℓ) for iI, ℓ ∈ ℕ. Namely, Ur,s0(ĝ) is generated by the subalgebra Ur,s(ĝ)0 and the quantum Heisenberg subalgebra Hr,s(ĝ), which is generated by the quantum imaginary root vectors ai(±ℓ) (iI, ℓ ∈ ℕ).

We recall the quantum Lie bracket from Ref. 19.

Definition 5.1.
For qi ∈ 𝕂 = 𝕂∖{0} and i = 1, 2, …, s − 1, the Lie q-brackets [ a1, a2, …, as ](q1, q2, …, qs−1) and [ a1, a2, …, as ]q1, q2, …, qs−1 are defined inductively by
[a1,a2]v1=a1a2v1a2a1,
[a1,a2,,as](v1,v2,,vs1)=[a1,,[as1,as]v1](v2,,vs1),
[a1,a2,,as]v1,v2,,vs1=[[a1,a2]v1,as1]v2,,vs2.
It follows from the definition that the quantum brackets satisfy the following identities:

[a,bc]v=[a,b]qc+qb[a,c]vq,
(5.1)
[ab,c]v=a[b,c]q+q[a,c]vqb,
(5.2)
[a,[b,c]u]v=[[a,b]q,c]uvq+q[b,[a,c]vq]uq,
(5.3)
[[a,b]u,c]v=[a,[b,c]q]uvq+q[[a,c]vq,b]uq.
(5.4)

In particular, we have that

[a,[b1,,bs](v1,,vs1)]=i[b1,,[a,bi],,bs](v1,,vs1),
(5.5)
[a,a,b](u,v)=a2b(u+v)aba+(uv)ba2=(uv)[b,a,a]u1,v1,
(5.6)
[a,a,a,b](u2,uv,v2)=a3b[3]u,va2ba+(uv)[3]u,vaba2(uv)3ba3,
(5.7)
[a,a,a,a,b](u3,u2v,uv2,v3)=a4b[4]u,va3ba+uv42u,va2ba2
(5.8)
(uv)3[4]u,vaba3+(uv)6ba4,

where [n]u,v=unvnuv, [n]u,v! = [n]u,v⋯[2]u,v[1]u,v, nmu,v=[n]u,v![m]u,v![nm]u,v!.

In this paragraph, we define the quantum root vectors using the q-bracket. For our purpose, we need to fix a particular path to realize the maximum root of 𝔤0.

Let θ = αih−1 + ⋯ + αi2 + αi1 be the maximum root and let

Xθ=[eih1,[eih2,,[ei2,ei1]]
(5.9)

be the corresponding root vector in the Lie algebra 𝔤0, which gives rise to a sequence from {1, …, n}: i1, i2, …, ih−1. We call such a sequence a root chain to the maximum root. Clearly root chains are not unique.

From now on, we fix such a sequence or root chain to the maximum root: i1, i2, …, ih−1. We define for 2 ⩽ kh − 1,

(αi1++αik1,αik)=ϵk0.
(5.10)

Remark 5.2.

Using the above fact, we fix root chains to the maximal root θ for our five twisted cases as follows.

(1) For the case of A2n1(2), the root chain is
α1α2αn1αnαn1α2.
(2) For the case of A2n(2), the root chain is
α1α2αnαnαn1α1.
(3) For the case of Dn+1(2), the root chain is
αnαn1α1.
(4) For the case of E6(2), the root chain is
α1α2α3α4α2α3α2α1.
(5) For the case of D4(3), the root chain is
α1α2α1.

Note that the root α0 = δθ in the affine Lie algebras. In the following we list the quantum root vectors xθ(1) and xθ+(1) for root vectors e0 and f0 respectively, corresponding to the root chains given above.

Case(I):   For A2n1(2), if α1t = α1 + ⋯ + αt (2 ⩽ tn) and α11 = α1, we define the quantum root vectors associated to the roots δα1t and −δ + α1t inductively as follows:

x1t(1)=xα1t(1)=[xt(0),x1(t1)(1)]t,t1t,1,
x1t+(1)=xα1t+(1)=[x1(t1)+(1),xt+(0)]t1,t11,t1.

Denote β1t = α1 + ⋯ + αn + αn−1 + ⋯ + αt (2 ⩽ tn − 1), so β1(n−1) = α1n + αn−1, and θ = β12. We define the quantum root vectors associated to the roots δβ1t and −δ + β1t inductively as follows:

y1t(1)=xβ1t(1)=[xt(0),y1(t1)(1)]t,t+1t,nt,n1t,1,y1t+(1)=xβ1t+(1)=[y1(t1)+(1),xt+(0)]t+1,t1n,t1n1,t11,t1.

In particular,

xθ(1)=y12(1)=[x2(0),,xn1(0),xn(0),,x1(1)](s,,s,s2,r1,,r1),xθ+(1)=y12+(1)=[x1+(1),,xn+(0),xn1+(0),,x2+(0)]r,,r,r2,s1,,s1.

Case(II):  For A2n(2), if α1t = α1 + ⋯ + αt (2 ⩽ tn) and α11 = α1, we define the quantum root vectors associated to the roots δα1t and −δ + α1t inductively as follows:

x1t(1)=xα1t(1)=[xt(0),x1(t1)(1)]t,t1t,1,x1t+(1)=xα1t+(1)=[x1(t1)+(1),xt+(0)]t1,t11,t1.

Denote β1t = α1 + ⋯ + αn + αn + αn−1 + αn−2 + ⋯ + αt (1 ⩽ tn), so β1n = α1n + αn, and β11 = θ. We define the quantum root vectors associated to the roots δβ1t and −δ + β1t inductively as follows:

y1t(1)=xβ1t(1)=[xt(0),y1(t1)(1)]t,t+1t,n2t,n1t,1,

where the initial one is xθ(1)=y11(1),

y1t+(1)=xβ1t+(1)=[y1(t1)+(1),xt+(0)]t+1,t1n,t2n1,t11,t1,

where the first one is xθ+(1)=y11+(1).

In particular,

xθ(1)=y1n(1)=[x1(0),,xn(0),xn(0),,x1(1)](s,,s,(rs)12,r1,,r1,(rs)1),xθ+(1)=y1n+(1)=[x1+(1),,xn+(0),x1+(0),,xn+(0)]r,,r,(rs)12,s1,,s1,(rs)1.

Case(III):   For Dn+1(2), if αnt = αn + αn−1 + ⋯ + αt (1 ⩽ tn) and αnn = αn, so θ = αn1. We define the quantum root vectors associated to the roots δαnt and −δ + αnt inductively as follows:

xnt(1)=xαnt(1)=[xt(0),xn(t1)(1)]t,t+1t,n,xnt+(1)=xαnt+(1)=[xn(t1)+(1),xt+(0)]t+1,t1n,t1.

In particular,

xθ(1)=xn1(1)=[x1(0),,xn(1)](r2,,r2),xθ+(1)=xn1+(1)=[xn+(1),,x1+(0)]s2,,s2.

Case(IV):   For E6(2), if ηj = αi1 + αi2 + ⋯ + αij, where (ik|1 ≤ k ≤ 8) = (1, 2, 3, 4, 2, 3, 2, 1), so θ = η8. We define the quantum root vectors associated to the roots δηj and −δ + ηj inductively as follows:

zj(1)=xηj(1)=[xij(0),zj1(1)]ij,ij1ij,i1,zj+(1)=xηj+(1)=[zj1+(1),xij+(0)]ij1,ij1i1,ij1.

In particular,

xθ(1)=z8(1)=[x1(0),x2(0),x3(0),x2(0),x4(0),,x1(1)](s,s2,s2,r1,s2,r2s1,r2s1),xθ+(1)=z8+(1)=[x1+(1),,x4+(0),x2+(0),x3+(0),x2+(0),x1+(0)]r,r2,r2,s1,r2,r1s2,r1s2.

Case(V):   For D4(3), we only consider the quantum root vectors associated to the roots δα1α2,  δθ and −δ + α1 + α2,   − δ + θ, where θ = 2α1 + α2 is the maximal root of G2,

x12(1)=[x2(0),x1(1)]s3,xθ(1)=[x1(0),x2(0),x1(1)](s3,r2s1).

On the other hand,

x12+(1)=[x1+(1),x2+(0)]r3

and

xθ+(1)=[x1+(1),x2+(0),x1+(0)]r3,r1s2.

In this subsection, we establish the isomorphism between the two-parameter quantum affine algebra Ur,s(gˆσ) and the (r, s)-analogue of Drinfeld quantum affinization of Ur,s(gˆσ). The identification of these two forms has been proved for the case of sl̂n in Ref. 15. We will give a new proof for the most general case in Secs. VI and VII.

We keep the same notations and assumptions as above. In particular, i1,  …,  ih−1 is the fixed sequence associated with the maximum root given in Eq. (5.9).

For simplicity, we denote

ij,ij1i2i1=ij,ij1ij,i1

and

i1i2ij1,ij1=i1,ij1ij1,ij1.

For j = 2, …, h − 1, let tij=qijpijrijsij,

pij=ij,ij1i2i1

and

qij=i1i2ij1,ij1.

Now we state our main theorem as follows.

Theorem 5.3.

Letθ = αi1 + ⋯ + αih−1be the maximal positive root of a simple Lie algebra 𝔤 and fix the associated root chain. Then there exists an algebra isomorphismΨ:Ur,s(gˆσ)Ur,s(gˆσ)given as follows. For eachiI,

ωiωi,
ωiωi,
ω0γ1ωθ1,
ω0γ1ωθ1,
γ±12γ±12,
γ±12γ±12,
eixi+(0),
fi1pixi(0),
e0axθ(1)(γ1ωθ1),
f0(γ1ωθ1)xθ+(1),
whereωθ=ωi1ωih1,ωθ=ωi1ωih1and
pi={r,ifσ(i)=i1,otherwise.
The constanta ∈ 𝕂 is given as follows:
a=(rs)n2,forA2n1(2),(rs)n2[2]n2,forA2n(2),(rs)2(n1),forDn+1(2),(rs)2,forD4(3),(rs)5[2]31,forE6(2).

We divide the proof into three steps corresponding to three theorems (Theorems 𝒜, , 𝒞), which will be proved in Secs. VI and VII.

In this section, we show that Ψ is an algebra homomorphism (Theorem 𝒜). The proof will be divided into five cases.

Theorem 𝒜.The map Ψ defined above is an algebra homomorphism fromUr,s(gˆσ)toUr,s(gˆσ).

Let Ei,  Fi, ωi, ωi denote the images of ei,  fi, ωi, ωi (iIˆ) in the algebra Ur,s(gˆσ) under the map Ψ, respectively. We shall check that the elements Ei,Fi,ωi,ωi(iIˆ),γ±12,γ±12 satisfy the defining relations X1X6, where (X = A, B, C, D, E) are given in Definition 3.1. First of all, the defining relations X1X3 can be verified directly as in the untwisted case,15 so we are left to check relations X4X6 involved with i = 0 case by case.

For relation (A4), when i ≠ 0, it follows from definition that

[E0,Fi]=a[xθ(1)(γ1ωθ1),1pixi(0)]=api[xi(0),xθ(1)]i,01(γ1ωθ1).

To prove this, we need the following technical lemma, which is proved similarly as untwisted types (see Ref. 17).

Lemma 6.1.
The following identities are true:
[xi1(0),yi1i+1(1)]s1=0,1<i<n,
(6.1)
[xi(0),yii+1(1)](rs)1=0,1in1,
(6.2)
[x2(0),y14(1)]=0,
(6.3)
[xi(0),y1i+2(1)]=0,1in2,
(6.4)
[xi1(0),y1i+2(1)]=0,2in2.
(6.5)
Proof.
For (6.1), it is easy to get that
yi1i+1(1)=xi+1(0),,xn(0),xn2(0),,xi+1(0),[xi(0),xi1(1)]s(s,,s,r1,,r1)=(rs)12xi+1(0),,xn(0),xn2(0),,xi+1(0),[xi1(0),xi(1)]r1(s,,s,r1,,r1).
Then we have that
[xi1(0),yi1i+1(1)]s1=(rs)12xi1(0),xi+1(0),,xn(0),xn2(0),,xi+1(0),[xi1(0),xi(1)]r1(s,,s,r1,,r1)s1(by(5.3)and(D91))=(rs)12xi+1(0),,xn(0),xn2(0),,xi+1(0),[xi1(0),xi1(0),xi(1)](r1,s1)(s,,s,r1,,r1)(=0by(D93))=0.
For (6.2), we argue inductively on i. The case i = n − 1 follows from definition,
[xn1(0),yn1n(1)](rs)1=[xn1(0),xn(1)](rs)1=0.
Suppose (6.2) holds for the case of i, then we have for the case of i − 1 that
yi1i(1)=[xi(0),,xn(0),xn2(0),,xi(0),xi1(1)](s,,s,r1,,r1)=r1sxi(0),,xn(0),xn2(0),,[xi(1),xi1(0)]r(s,,s,r1,,r1)(by(5.3)and(D91))==r1sxi(0),[xi+1(0),,xn(0),xn2(0),,xi(1)](s,,s,r1,,r1),xi1(0)(r,r1)(by definition)=r1s[xi(0),yii+1(1),xi1(0)](r,r1)(by(5.3))
=r1s[[xi(0),yii+1(1)](rs)1,xi1(0)]rs(=0by inductive hypothesis)+r2[yii+1(1),[xi(0),xi1(0)]s]r2s=r2[yii+1(1),[xi(0),xi1(0)]s]r2s.
Using the above identity, we have
[xi1(0),yi1i(1)]r2=r2xi1(0),yii+1(1),[xi(0),xi1(0)]sr2sr2(by(5.3))=r2[xi1(0),yii+1(1)]r1,[xi(0),xi1(0)]srs+r3yii+1(1),[xi1(0),xi(0),xi1(0)](s,r1)(rs)2(=0by(D93))=r2[xi1(0),yii+1(1)]r1,[xi(0),xi1(0)]srs.
At the same time, we also get
[xi1(0),yii+1(1)]r1=xi1(0),xi+1(0),,xn(0),xn2(0),,xi+1(0),xi(1)(s,,s,r1,,r1)r1(by(D91))=xi+1(0),,xn(0),xn2(0),,xi+1(0),[xi1(0),xi(1)]r1(s,,s,r1,,r1)]r1(by the definition)=(rs)12xi+1(0),,xn(0),xn2(0),,xi+1(0),[xi(0),xi1(1)]s(s,,s,r1,,r1)]r1(by definition)=(rs)12yi1i+1(1).
Therefore we get
[xi1(0),yi1i(1)]r2=r52s12[yi1i+1(1),[xi(0),xi1(0)]s]rs(by(5.3))=r52s12[[yi1i+1(1),xi(0)]r,xi1(0)]s2(by definition)r32s12[xi(0),[yi1i+1(1),xi1(0)]s]r1s=(rs)1[yi1i(1),xi1(0)]s2.
Expanding the two sides of the above identity, one gets
(1+r1s)[xi1(0),yi1i(1)](rs)1=0,
which implies that if r≠ − s, then [xi1(0),yi1,i(1)](rs)1=0. Thus we have checked (6.2) for the case of i − 1. Consequently, (6.2) is true by induction.
For (6.3), we first note that
[x2(0),y14(1)](by definition)=[x2(0),[x4(0),,xn(0),xn2(0),,x4(0),x14(1)](s,,s,r1,,r1)](by(5.3)and(D91))=[x4(0),,xn(0),xn2(0),,x4(0),[x2(0),x14(1)]](s,,s,r1,,r1).
So it suffices to check the relation [x2(0),x14(1)]=0.
In fact, it is easy to see that
[x2(0),x14(1)]r1s(by definition)=[x2(0),[x3(0),x2(0),x1(1)](s,s)]r1s(by(5.3))=[x2(0),[x3(0),x2(0)]s,x1(1)]](s,r1s)(by (5.3))+s[x2(0),x2(0),[x3(0),x1(1)]](1,r1s)(=0by(D91))=[[x2(0),[x3(0),x2(0)]s]r1,x1(1)]s2(=0by(D93))+r1[[x3(0),x2(0)]s,[x2(0),x1(1)]s]rs(by definitionand(5.4))=r1[x3(0),[x2(0),[x2(0),x1(1)]s]r]s2(=0by(D92))+[[x3(0),x2(0),x1(1)](s,s),x2(0)]r1s(by definition)=[x14(1),x2(0)]r1s.
Then, we obtain (1+r1s)[x2(0),x14(1)]=0. When r ≠ − s, we arrive at our required conclusion [x2(0),x14(1)]=0.

Eqs. (6.4) and (6.5) can be verified similarly, which are left to the readers.□

The following three lemmas are needed for later purpose.

Lemma 6.2.

One has that[xi(0),xi1(0),xi(0),xi+1(0)](r1,r1,1)=0.

Proof.
Using (5.3) and the Serre relations, we have
[xi(0),[xi1(0),xi(0),xi+1(0)](r1,r1)]r1s(using (5.3))
=[xi(0),[xi1(0),xi(0)]r1,xi+1(0)](r1,r1s)(using (5.3))
+r1[xi(0),xi(0),[xi1(0),xi+1(0)]](1,r1s)(=0 by the Serre relation)
=[[xi(0),[xi1(0),xi(0)]r1]s,xi+1(0)]r2(=0 by the Serre relation)
+s[[xi1(0),xi(0)]r1,[xi(0),xi+1(0)]r1](rs)1(using (5.4))
=s[xi1(0),[xi(0),xi(0),xi+1(0)](r1,s1)]r2(=0 by the Serre relation)
+[[xi1(0),xi(0),xi+1(0)](r1,r1),xi(0)]r1s,
which implies that (1+r1s)[xi(0),[xi1(0),xi(0),xi+1(0)](r1,r1)]=0. Thus if r≠ − s, then
[xi(0),[xi1(0),xi(0),xi+1(0)](r1,r1)]=0.

Lemma 6.3.

Using the same notations, we have[xn1(0),x1n1(0)]r=0.

Proof.
Combining (5.3) with the Serre relations, we get
[xn1(0),x1n1(0)]r(by definition)
=[xn1(0),[xn1(0),xn2(0),x1n3(1)](s,s)]r(using (5.3))
=[xn1(0),[xn1(0),xn2(0)]s,x1n3(1)](s,r)(using (5.3))
+s[xn1(0),xn2(0),[xn1(0),x1n3(1)]1](1,r)(=0 by the Serre relation)
=[[xn1(0),xn1(0),xn2(0)](s,r),x1n3(1)]s(=0 by the Serre relation)
+r[[xn1(0),xn2(0)]s,[xn1(0),x1n3(1)]1]r1s(=0 by the Serre relation)
=0.

Lemma 6.4.

We have that[xn(0),x1n(1)]r2=0.

Proof.
Using (5.3) and the Serre relations, we obtain
[xn(0),x1n(1)]r2(by definition)
=[xn1(0),[xn(0),xn1(0),x1n2(1)](s,s2)]r(using (5.3))
=[xn(0),[xn(0),xn1(0)]s2,x1n2(1)](s,r2)(using (5.3))
+s2[xn(0),xn1(0),[xn(0),x1n2(1)]1](s1,r2)(=0 by the Serre relation)
=[[xn(0),xn(0),xn1(0)](s2,r2),x1n3(1)]s(=0 by the Serre relation)
+r2[[xn(0),xn1(0)]s2,[xn(0),x1n2(1)]1]r2s(=0 by the Serre relation)
=0.

Now we turn to relation (A4).

Proposition 6.5.

[xi(0),xθ(1)]i,01=0, for iI.

Proof.
(I) If i = 1, 〈1, 0〉 = rs. By Lemma 6.2, fix i = 1, we immediately have
[x1(0),xθ(1)](rs)1=[xi(0),yii+1(1)](rs)1=0.
(II) When i = 2, 〈2,  0〉 = s. By the definition of quantum root vectors, we get easily
[x2(0),xθ(1)]s1(by definition)
=[x2(0),[x2(0),x3(0),y14(1)](r1,r1)]s1(using (5.3))
=[x2(0),[[x2(0),x3(0)]r1,y14(1)]r1]s1(using (5.3))
+r1[x2(0),x3(0),[x2(0),y14(1)]1](1,r1)(=0 by (6.4))
=[[x2(0),[x2(0),x3(0)]r1]s1,y14(1)]r1(=0 by the Serre relation)
+s1[[x2(0),x3(0)]r1,[x2(0),y14(1)]1]r1s(=0 by (6.4))
=0.
(III) When 2 < i < n, 〈i,  0〉 = 1. It follows from (5.3) and definition that
xθ(1)=[x2(0),,xi1(0),xi(0),xi+1(0),y1i+2(1)](r1,,r1)(using (5.3))=[x2(0),,xi1(0),[xi(0),xi+1(0)]r1,y1i+2(1)](r1,,r1)(using (5.3))+r1[x2(0),,xi1(0),xi+1(0),[xi(0),y1i+2(1)]1](1,r1,,r1)(=0 by (6.4))=[x2(0),,[xi1(0),xi(0),xi+1(0)](r1,r1),y1i+2(1)](r1,,r1)+r1[x2(0),,[xi(0),xi+1(0)]r1,[xi1(0),y1i+2(1)]1](1,r1,,r1)(=0 by (6.5))=[x2(0),,[xi1(0),xi(0),xi+1(0)](r1,r1),y1i+2(1)](r1,,r1).
The above result implies that
[xi(0),xθ(1)]
=[xi(0),[x2(0),,[xi1(0),xi(0),xi+1(0)](r1,r1),y1i+2(1)](r1,,r1)]
(by (5.3) and the Serre relation)
=[x2(0),,[xi(0),[xi1(0),xi(0),xi+1(0)](r1,r1),y1i+2(1)](r1,1)](r1,,r1).
Therefore it suffices to check
[xi(0),[xi1(0),xi(0),xi+1(0)](r1,r1),y1i+2(1)](r1,1)=0.
Actually, it is obvious that
[xi(0),[xi1(0),xi(0),xi+1(0)](r1,r1),y1i+2(1)](r1,1)(using (5.3))
=[[xi(0),xi1(0),xi(0),xi+1(0)](r1,r1,1),y1i+2(1)]r1(=0 by Lemma 6.2)
+[[xi1(0),xi(0),xi+1(0)](r1,r1),[xi(0),y1i+2(1)]1]r1(=0 by (6.4))
=0.
(IV) When i = n, 〈n,  0〉 = (rs)−2. Note that
y1n1(1)=[xn1(0),xn(0),x1n1(1)](s2,r1)(using (5.3))=[[xn1(0),xn(0)]r2,x1n1(1)]rs2+r2[xn(0),[xn1(0),x1n1(1)]r](rs)2(=0 by Lemma 6.3)=[[xn1(0),xn(0)]r2,x1n1(1)]rs2.
Applying the above result, it is easy to get
[xn(0),y1n1(1)]s4
=[xn(0),[xn1(0),xn(0)]r2,x1n1(1)](rs2,s4)(using (5.3))
=[[xn(0),[xn1(0),xn(0)]r2]s2,x1n1(1)]rs3(=0 by the Serre relation)
+s2[[xn1(0),xn(0)]r2,[xn(0),x1n1(1)]s2]r(using (5.4))
=s2[xn1(0),[xn(0),x1n(1)]r2]r3(=0 by Lemma 6.4)
+r2s2[[xn1(0),x1n(1)]r1,xn(0)]r4
=r2s2[[xn1(0),x1n(1)]r1,xn(0)]r4.
Expanding two sides of the above relation, we have that
(1+r2s2)[xn(0),y1n1(1)](rs)2=0.
So, if r≠ − s, it holds that
[xn(0),y1n1(1)](rs)2=0.
Consequently, using (5.3) and Serre relation repeatedly, our previous result implies immediately that
[xn(0),xθ(1)]rs(by definition)
=[xn(0),[x2(0),,xn2(0),y1n1(1)](r1,,r1)](rs)2
=[x2(0),,xn2(0),[xn(0),y1n1(1)](rs)2](r1,,r1)]rs
=0.
Hence Proposition 6.5 is proved.□

Next, we turn to the commutation relation in (A4) involved with i = j = 0.

Proposition 6.6.

[E0,F0]=γ1ωθ1γ1ωθ1rs.

Proof.
First we consider
[E0,F0]=(rs)n2[xθ(1)γ1ωθ1,γ1ωθ1xθ+(1)]=(rs)n2[xθ(1),xθ+(1)](γ1γ1ωθ1ωθ1).
Recall the constructions of xθ(1) and xθ+(1) in the present case,
xθ(1)=y12(1)=[x2(0),,xn(0),x1n1(1)](s2,r1,,r1),xθ+(1)=y12+(1)=[[[x1n1+(1),xn+(0)]r2,xn1+(0)]s1,,x2+(0)]s1.
Using the result for An1(1),15 one has
[x1n1(1),x1n1+(1)]=γωα1n1γωα1n1rs.
Using a similar calculation, we have that
[x1n(1),x1n+(1)](by definition)=[[xn(0),x1n1(1)]s2,[x1n1+(1),xn+(0)]r2](using (5.3))=[[[xn(0),x1n1+(1)],x1n1(1)]s2,xn+(0)]r2(=0 by (5.3) and (D9))+[[xn(0),[x1n1(1),x1n1+(1)]]s2,xn+(0)]r2(using (D6) and (D9))+[x1n1+(1),[[xn(0),xn+(0)],x1n1(1)]s2]r2(using (D9) and (D6))+[x1n1+(1),[xn(0),[x1n1(1),xn+(0)]]s2]r2(=0 by (5.3) and (D9))=γωα1n1ωnωnrs+γωα1n1γωα1n1rsωn=γωα1nγωα1nrs.
We can proceed in the same way to obtain
[y1n1(1),y1n1+(1)](by definition)=[[xn1(0),x1n(1)]r1,[x1n+(1),xn+(0)]s1](using (5.3))=[[[xn1(0),x1n+(1)],x1n(1)]r1,xn1+(0)]s1(using (5.3), (D9) and (5.5))+[[xn1(0),[x1n(1),x1n+(1)]]r1,xn1+(0)]s1(=0 by the above result and (D6))+[x1n+(1),[[xn1(0),xn1+(0)],x1n(1)]r1]s1(=0 by (D9) and (D6))+[x1n+(1),[xn1(0),[x1n(1),xn1+(0)]]r1]s1(using (5.3), (D9) and (5.5))=(rs)1γωα1nωn1ωn1rs+(rs)1γωα1nγωα1nrsωn1=(rs)1γωβ1n1γωβ1n1rs.
By the inductive step, it is obvious that
[y12(1),y12+(1)]=(rs)2nγωβ12γωβ12rs.
Hence we arrive at the required relation.□

We pause to recall the following fact, which will be used in the sequel.

Lemma 6.7.

IfXUq(𝔤0)+, and [ A,  Fk ] = 0, ∀kI, thenA = 0. On the other hand, IfXUq(𝔤0), and [ A,  Ek ] = 0, ∀kI, thenA = 0.

We now return to check relations (A5) and (A6). Indeed, relations (A5) or (A6) can be obtained from the other one by applying τ. The following three relations are the main statements.

Lemma 6.8.

Using the above notations, we have the following relations:

(1) [ E0,  En ](rs)−2 = 0,

(2) [ E2,  E2,  E0 ](s−1, r−1) = 0,

(3) [ Fn−1,  Fn−1,  Fn−1,  Fn ](r−2, (rs)−1, s−2) = 0.

Proof.
(1) Combining the definitions and Drinfeld relations, we see that
[E0,En,](rs)2=a[xθ(1),xn+(0)]γ1ωθ1=a[x2(0),,xn1(0),[xn(0),xn+(0)],x1n1(1)](s2,r1,,r1)γ1ωθ1=a[x2(0),,[xn1(0),x1n1(1)]rωn](r1,,r1)γ1ωθ1=0,
where the last step follows from the following calculations:
[xn1(0),x1n1(1)]r
=[xn1(0),[xn1(0),xn2(0),x1n3(1)](s,s)]r(using (5.3))
=[xn1(0),[xn1(0),xn2(0)]s,x1n3(1)](s,r)(using (5.3))
+s[xn1(0),xn2(0),[xn1(0),x1n3(1)]1](1,r)(=0by(D93))
=[[xn1(0),xn1(0),xn2(0)](s,r),x1n3(1)]s(using (5.8))
+r[[xn1(0),xn2(0)]s,[xn1(0),x1n3(1)]1]r1s
=0.
(2) Relation (D9) yields directly that
[x2+(0),xθ(1)]=[[x2+(0),x2(0)],y13(1)]r1+[x2(0),,xn(0),,x3(0),[x2+(0),x2(0)],x1(1)](s,,s,s2,r1,,r1)=(rs)1y13(1)ω2,

The statement (2) follows from the above results and [x2+(0),y13(1)]=0, which holds by direct calculation.

(3) Denote that X = [ Fn−1,  Fn−1,  Fn−1,  Fn ](r−2, (rs)−1, s−2). It is obvious that XUr,s(𝔤σ). Therefore, by Lemma 6.7, in order to prove X = 0, it suffices to check [xi+(0),X]=0 for iI. For i not equal to n − 1 or n, the claim is obvious. So we only need to verify the cases of i = n − 1 and i = n.

First we get from relation (D9) in the case of i = n,
[xn+(0),X]=12[xn+(0),[xn1(0),xn1(0),xn1(0),xn(0)](r2,(rs)1,s2)]=12[xn1(0),xn1(0),xn1(0),[xn+(0),xn(0)]](r2,(rs)1,s2)=r2s62ωn[xn1(0),xn1(0),xn1(0)](r1s,1)=0.
One sees that the same is true for i = n − 1,
[xn1+(0),X]=12[xn1+(0),[xn1(0),xn1(0),xn1(0),xn(0)](r2,(rs)1,s2)]=12[[xn1+(0),xn1(0)],xn1(0),xn1(0),xn(0)](r2,(rs)1,s2)+[xn1(0),[xn1+(0),xn1(0)],xn1(0),xn(0)](r2,(rs)1,s2)+[xn1(0),xn1(0),[xn1+(0),xn1(0)],xn(0)](r2,(rs)1,s2)
=(rs)2(r+s)2[xn1(0),xn1(0),xn(0)](r2,(rs)1)ωn1(rs)2(r+s)2[xn1(0),xn1(0),xn(0)](r2,(rs)1)ωn1=0.
Consequently, theorem 𝒜 is proved for the case of A2n1(2).□

Let us turn to the case of A2n(2). Similarly we only show some key relations (B4)–(B6) involving i = 0.

When i ≠ 0, observe that

[E0,Fi]=a[xθ(1)(γ1ωθ1),1pixi(0)]=api[xi(0),xθ(1)]i,01(γ1ωθ1).

Hence, in order to verify relation (B4), it is enough to check the following result.

Proposition 6.9.

[xi(0),xθ(1)]i,01=0, for iI.

Before giving the proof of Proposition 6.9, we need the following crucial lemma which can be proved directly.

Lemma 6.10.
One has that
[xi1(0),y1i+1(1)]=0,1<i<n
(6.6)
[xi(0),y1i(1)]s1=0,1in1
(6.7)
[xn1(0),x1n(1)]=0,
(6.8)
[xn1(0),x1n1(1)]r=0,
(6.9)
[xn(0),y1n(1)]r=0.
(6.10)

Proof of Proposition 6.9.
(I) When i = 1, 〈1,  0〉 = s2, one gets from (6.6),
[x1(0),xθ(1)]s2(by definition)
=[x1(0),[x1(0),x2(0),y13(1)](r1,(rs)1)]s2(using (5.3))
=[x1(0),[[x1(0),x2(0)]r1,y13(1)](rs)1]s2(using (5.3))
+r1[x1(0),x2(0),[x1(0),y13(1)]s1](1,s2)(=0 by (6.6))
=[[x1(0),[x1(0),x2(0)]r1]s1,y13(1)]r1s2(=0 by the Serre relation)
+s1[[x1(0),x2(0)]r1,[x1(0),y13(1)]s1]r1(=0 by (6.6))
=0.
(II) When 1 < i < n, 〈i,  0〉 = 1. In this case, it holds by the following direct calculations:
[xi(0),y1i1(1)]r1s(by definition)
=[xi(0),[xi1(0),xi(0),y1i+1(1)](r1,r1)]r1s(using (5.3))
=[xi(0),[[xi1(0),xi(0)]r1,y1i+1(1)]r1]r1s(using (5.3))
+r1[xi(0),xi(0),[xi1(0),y1i+1(1)]1](1,r1s)(=0by(6.6))
=[[xi(0),[xi1(0),xi(0)]r1]s,y1i+1(1)]r2(=0 by the Serre relation)
+s[[xi1(0),xi(0)]r1,[xi(0),y1i+1(1)]r1](rs)1(by the definition)
=s[[xi1(0),xi(0)]r1,y1i(1)](rs)1(using (5.3))
=s[xi1(0),[xi(0),y1i(1)]s1]r2(=0 by (6.7))
+[[xi1(0),y1i(1)]r1,xi(0)]r1s(by the definition)
=[y1i1(1),xi(0)]r1s.
As a consequence of above result, it yields that (1+r1s)[xi(0),y1i1(1)]=0. Under the condition r ≠ − s, it follows that
[xi(0),y1i1(1)]=0.
Using the above result, it yields from an immediate calculation that
[xi(0),xθ(1)](by the definition)
=[x1(0),,xi2(0),[xi(0),y1i1(1)]](r1,,r1)
=0.
(III) When i = n, 〈n,  0〉 = (rs)−1. Observe that
y1n1(1)=[xn1(0),xn(0),xn(0),x1n1(1)](s,(rs)12,r1)(using (5.3))=[[xn1(0),xn(0)]r1,[xn(0),x1n1(1)]s](rs)12(using (5.3))+r1[xn(0),[xn1(0),xn(0),x1n1(1)](s,1)]r32s12(=0 by (6.8))=[[[xn1(0),xn(0)]r1,xn(0)](rs)12,x1n1(1)]rs2+(rs)12[xn(0),[[xn1(0),xn(0)]r1,x1n1(1)]rs]r12s32(using (5.3))=[[[xn1(0),xn(0)]r1,xn(0)](rs)12,x1n1(1)]rs2+(rs)12[xn(0),[xn1(0),[xn(0),x1n1(1)]s]1](r12s32)(=0 by (6.8))+(r1s)12[xn(0),[xn1(0),x1n1(1)]r,xn(0)]((rs)1,r12s32)(=0 by (6.9))=[[[xn1(0),xn(0)]r1,xn(0)](rs)12,x1n1(1)]rs2.
Applying the above result, one sees that
[xn(0),y1n1(1)]s2
=[xn(0),[[xn1(0),xn(0)]r1,xn(0)](rs)12,x1n1(1)](rs2,s2)(using (5.3))
=[[xn(0),[xn1(0),xn(0)]r1,xn(0)]((rs)12,s),x1n1(1)]rs3
+s[[[xn1(0),xn(0)]r1,xn(0)](rs)12,[xn(0),x1n1(1)]s]rs(using (5.3))
=s[[xn1(0),xn(0)]r1,[xn(0),x1n(1)](rs)12]1(using (5.3))
+r12s32[[[xn1(0),xn(0)]r1,x1n(1)](rs)12,xn(0)](rs)1(using (5.3))
=s[xn1(0),[xn(0),y1n(1)]r]r2(=0 by (6.10))
+rs[[xn1(0),y1n(1)]r1,xn(0)]r2(by the definition)
+r12s32[[xn1(0),y1n1(1)]r1,xn(0)](rs)1(by the definition)
+rs2[[[xn1(0),x1n(1)]1,xn(1)]r32s12,xn(0)](rs)1(=0 by (6.8))
=rs[y1n1(1),xn(0)]r2+r12s32[y1n1(1),xn(0)](rs)1.
The above equation means that
(1+r12s12+r1s)[xn(0),y1n1(1)]rs=0.
So, if (r/s)3/2≠1, it follows that
[xn(0),y1n1(1)]rs=0.
By this result it follows from (5.3) and the Serre relation that
[xn(0),xθ(1)]rs(by definition)
=[xn(0),[x2(0),,xn2(0),y1n1(1)](r1,,r1)]rs
=[x2(0),,xn2(0),[xn(0),y1n1(1)]rs](r1,,r1)]rs
=0.

Thus Proposition 6.9 has been proved.

Now, we are ready to check the commutation relation (B4).

Proposition 6.11.

[E0,F0]=γ1ωθ1γ1ωθ1rs.

Proof.
First we observe that
[E0,F0]=(rs)n2[xθ(1)γ1ωθ1,γ1ωθ1xθ+(1)]=(rs)n2[xθ(1),xθ+(1)](γ1γ1ωθ1ωθ1).
Hence we have to compute the bracket [xθ(1),xθ+(1)]. Recalling the construction of xθ(1) and xθ+(1), we have by the inductive step that
xθ(1)=y12(1)=[x2(0),,xn(0),x1n1(1)](s2,r1,,r1),
xθ+(1)=τ(xθ(1))=y12+(1)
=[[[x1n1+(1),xn+(0)]r2,xn1+(0)]s1,,x2+(0)]s1.
As a consequence of the case An1(1)15 it follows that
[x1n1(1),x1n1+(1)]=γωα1n1γωα1n1rs.
Next, we consider
[x1n(1),x1n+(1)](by definition)=[[xn(0),x1n1(1)]s,[x1n1+(1),xn+(0)]r](using (5.3))=[[[xn(0),x1n1+(1)],x1n1(1)]s,xn+(0)]r(=0 by (5.3) and (D9))+[[xn(0),[x1n1(1),x1n1+(1)]]s,xn+<