Drinfeld realization of quantum twisted affine algebras via braid group

The Drinfled realization of quantum affine algebras has been tremendously useful since its discovery. Combining techniques of Beck and Nakajima with our previous approach, we give a complete and conceptual proof of the Drinfeld realization for the twisted quantum affine algebras using Lusztig's twisted braid group action.


Introduction
In studying finite dimensional representations of Yangian algebras and quantum affine algebras, Drinfeld stated a new realization of the Drinfeld-Jimbo quantum enveloping algebras of the affine types. Drinfeld realization is the quantum analog of the loop algebra realization of the affine Kac-Moody Lie algebras, and has played a pivotal role in later developments of quantum affine algebras and quantum conformal theory. For example the basic representations of quantum affine algebras were constructed based on Drinfeld realization [FJ,J1] and the quantum Knizhnik-Zamoldchikov equation [FR] was also formulated using this realization.
The first proof of the Drinfeld realization for the untwisted types was given by Beck [B] using Lusztig's braid group actions and the earlier computation for U q ( sl 2 ) [Da1,LSS]. By directly quantizing the classical isomorphism of the Kac realization to the affine Lie algebras, the first author also gave an elementary proof [J2] of the Drinfeld automorphism using q-brackets starting from Drinfeld quantum loop algebras. In this elementary approach a general strategy and algorithm was formulated to prove all the Serre relations. In particular all type A relations were verified in details including the exceptional type D 4 . The proof of Drinfeld realizations of the twisted types has taken longer. In [ZJ] and [JZ] we gave an elementary proof of the twisted Drinfeld realization starting from Drinfeld quantum loop algebras. In particular in [JZ] we have shown that twisted quantum affine algebras obey some simplified Serre relations of lower degrees in certain types as in the classical cases, and from which twisted Serre relations are consequences. Moreover this same direction has also been taken [Da3] to check all twisted quantum cases, where a homomorphism from the Drinfeld's quantum loop algebra to the Drinfeld-Jimbo quantum enveloping algebra was given just as our earlier elementary approach.
The purpose of this work is to give a conceptual proof of twisted Drinfeld realizations using braid group actions starting from Drinfeld-Jimbo definition. Braid group actions have been very useful in Lusztig's construction of canonical bases [L1,BN] and are particularly useful in Beck's proof of untwisted cases. We use the extended braid group action to define root vectors in the twisted case as in the untwisted cases. Some of the root vector computations can be done in a similar way as in untwisted cases (cf. [Da2]). We also give a complete proof of all Serre relations for the first time for both untwisted and twisted cases using q-bracket techniques [J1]. One notable feature of our work is that we directly prove the realization using the braid group action and verify the unchecked relations for all cases.
It has been observed (cf. [Da3]) that the usual argument of identifying of two quantum algebras by passing to q = 1 should be studied with care, and in strict sense all previous works on Drinfeld realization may be viewed as establishing a homomorphism from one form of the quantum affine algebra to the other form. The second goal of our paper is to provide a detailed proof that the Drinfeld-Jimbo algebra is indeed isomorphic to the Drinfeld algebra of the quantum affine algebra. We achieve this by combining our previous work identifying of the Drinfeld-Jimbo quantum affine algebra inside the Drinfeld realization [ZJ, JZ] and the explicit knowledge of the q-bracket computations developed in [J2,ZJ], which also established the epimorphism from the Drinfeld-Jimbo form to the Drinfeld form of the quantum affine algebra.
The paper is organized in the following manner. In Section two we first recall the Drinfeld-Jimbo quantum affine enveloping algebras and define Lusztig's braid group action as well as the extended braid group action, and then use this to define quantum root vectors. Section three shows how to construct the quantum affine algebra U q (A 2 )) inside the twisted quantum affine algebra U q (X (r) ) (or U q (A (2) 2n ) ). In Section four we check all Drinfeld relations, in particular all Serre relations for twisted quantum affine algebras. Finally we prove the isomorphism of two forms of quantum affine algebras using our previous work on Drinfeld realization and q-bracket techniques.

Definitions and Preliminaries
2.1. Finite order automorphisms of g. In this paragraph some basic notations of Kac-Moody Lie algebras are recalled [K] . Let g be a simple finitedimensional Lie algebra, and let σ be an automorphism of g of order r. Then σ induces an automorphism of the Dynkin diagram of g with the same order. Fix a primitive rth root of unity ω = exp 2πi r . Since σ is diagonalizable, it follows that where g j is the eigenspace relative to ω j . Clearly, the decomposition is a Z/rZ− gradation of g, then g 0 is a Lie subalgebra of g. Let A = (A ij ), (i, j ∈ {1, 2, · · · , N }) be one of simply laced Cartan matrices It is well-known that the Dynkin diagram D(A) has a diagram automorphism σ of order r = 2 or 3. Explicitly A is one of the following types: A N (N 2), D N (N > 4), E 6 and D 4 with the canonical action of σ: Let I = {1, 2, · · · , n} be the set of σ−orbits on {1, 2, · · · , N }, where we use representatives to denote the orbits. For example {i, N + 1 − i} is simply denoted by i in type A. We can write {1, . . . , N } = I ∪ σ(I). Consequently the nodes of the Dynkin diagram g 0 are indexed by I.
2.2. Twisted affine Lie algebras. For a non trivial automorphism σ of the Dynkin diagram, the twisted affine Lie algebra g σ is the central extension of the twisted loop algebra: Let us use A σ = (a ij ) (i, j ∈Î) to denote the Cartan matrix of the twisted affine Lie algebra g σ of type X (r) N . The Cartan matrix A σ is symmetrizable, that is, there exists a diagonal matrix D = diag(d i |i ∈Î) such that DA σ is symmetric. Let α i (i ∈Î) ⊂ h σ * be the simple roots and let α ∨ i (i ∈Î) ⊂ h σ be the simple coroots of g σ such that α j , α ∨ i = a ij . Let Q be the affine root lattice defined by: Subsequently Q 0 will denote the finite root lattice of the subalgebra g 0 . Indeed, Introduce the non-degenerate symmetric bilinear form ( , ) on Q determined by (α i , α j ) = d i a ij . Let δ = i∈Î r i α i ∈ Q + be the canonical imaginary root of minimal height such that (δ, δ) = 0 and (δ, α i ) = 0, ∀i ∈Î. Here the coefficients r i are unique such that r 0 is always 1, and we have chosen the labels of A (2) 2n different from [K].
Recall that the (affine) Weyl group W is generated by . . s i l is a reduced expression of w, then we define the length l(w) = l. Let W 0 be the subgroup generated by {s i |i ∈ I}, thus W 0 is the finite Weyl group of g 0 .
Finally for each i ∈Î, we define inductively the twisted derivation r i of U + q by r i (E j ) = δ ij , and r i (xy) = q (|y|,αi) r i (x)y + xr i (y), x ∈ U + q,|x| , y ∈ U + q,|y| . Here U + q,β is the Q + -graded subspace of U + q with the weight β. The following result is from Lusztig [L1].
There are three different lengths of real roots for the type of A (2) 2n , which requires a different treatment from other twisted types. For each i ∈ I we construct a copy of U q ( sl 2 ) inside U q ((X (r) N ) for X n = A 2n . For the case of (A (2) 2n , n) we will construct a subalgebra isomorphic to U q (A (2) n ).

Quantum root vectors.
In order define the quantum root vectors, we review some notations from Beck-Nakajima [BN]. For ω i ∈ P, ∀i ∈ I, we choose We fix a reduced expression of ω n ω n−1 · · · ω 1 as follows: We define a doubly infinite sequence Then we have Define a total order on ∆ + by setting The root vectors for each element of ∆ > ∪ ∆ < can be defined as follows: It follows from [L1] that the elements E β k ∈ U + q .
Remark 3.1. For a positive root α = ω 1 (α i ) in the root system ∆ > ∪ ∆ < of U q (ĝ σ ), there exists another presentation α = ω 2 (α j ) where ω i , ω 2 are in the Weyl group W and α i , α j are simple roots. Then the quantum root vector E α can be defined by two ways: Actually, the two definitions agree up to a constant, because there exists ω ∈ W 0 such that α j = ω(α i ), which means that ω 1 = ω 2 ω and l(ω 1 ) = l(ω 2 ) + l(ω), then we have Now we introduce the root vectors of Drinfeld generators [BCP,Da2].
Note that the definitions are not so different from those of non twisted cases up to some slight adjustments because of the difference between their root systems.

Vertex subalgebra U
We list the following result which is already proved in [L2] (also see [B]).
The following statements are based on the construction of [B], also see [Da2].
3.4. Relations of imaginary root vectors. Now we define the positive imaginary root vectors. For k > 0 and i ∈ I let Define the elements E i kpiδ ∈ U + q by the functional equation Similarly, us introduce F i, kpiδ = Φ(E i, kpi δ ) for k > 0. Then we have the following lemma Lemma 3.5. For i, j ∈ I, k, l ≥ 0, we have Similarly, we can define a i (−p i k) and ϕ i (−p i k) for k > 0.
The following was given in [Da2], which is an application of Beck's work ( [B]).

Relations between U
q . We recall the commutation relations among the root vectors from [Da2] and remark that the argument also works in the case of A (2) 2n .
Lemma 3.9. For i = j ∈ I, and k, l ∈ Z, If i = j ∈ I such that a ij a ji = 1, it is easy to see that σ(i) = i and σ(j) = j and d i = p i = 1, d j = p j = 1. Thus one has the following lemma.
Lemma 3.10. If i = j ∈ I such that a ij a ji = 1, then for k > 1, l ∈ Z one has following relations.
If i = j ∈ I such that a ij = −r and a ji = −1, it is easy to see that σ(i) = i and σ(j) = j and p i = d i = 1, d j = r. Then the next result follows.
Lemma 3.11. Suppose a ij = −r and a ji = −1 for i = j ∈ I. Then for k > r, l ∈ Z, we have The following statements follow directly from Lemma 3.13 and 3.11. Note that d i a ij = r−1 s=0 A i,σ s (j) for i, j ∈ I, which will be used later.
Lemma 3.12. Let i = j ∈ I such that a ij a ji = 1. For k > 1, l ∈ Z one has Lemma 3.13. Let i = j ∈ I such that a ij = −r and a ji = −1. Then for k > r, l ∈ Z, 2n . For the case of A 2n , the definition of the quantum root vectors differ from other twisted cases because of its slightly complicated root system. This situation has been discussed in detail in [Da2], here we will review some definitions and results. The case of A (2) 2 has been dealt with carefully in [A].
In this paragraph we discuss the remaining cases and show that for i = n, there exists a copy of U 2n . Let us introduce the longest element of the Weyl group W : w = s 0 s 1 s 2 · · · s n , then w n−1 (α 0 ) = δ − 2α n , where δ = α 0 + 2α 1 + · · · + 2α n .
Proposition 3.14. Let U (n) q be the subalgebra generated by E n , F n , K ±1 n , E δ−2αn , F δ−2αn , K δ−2αn . There exists an algebra isomorphism ϕ n : U q (A 2 ) → U (n) q defined as follows: 4. Drinfeld realization for twisted cases 4.1. Drinfeld generators. In order to obtain the Drinfeld realization of twisted quantum affine algebras, we introduce Drinfeld generators as follows.
Definition 4.1. For k > 0, define Definition 4.2. For k < 0, define Remark 4.3. From the above definitions it follows that a i (k) = 0 if σ(i) = i and k is not divisible by r.

Drinfeld realization for twisted cases.
In previous sections we have prepared for the relations among Drinfeld generators. The complete relations are given in the following theorem stated first in [Dr]. In the following we will set out to prove the remaining Serre relations using braid groups and other techniques developed in [ZJ, JZ].

The Proof of the main theorem.
We need to verify that the above Drinfeld generators x ± i (k), a i (l), K ±1 i satisfy all relations (1) − (10). The relations (1) − (7) are already checked in the previous paragraphs. We are going to show the last three relations.
We first proceed to check relation (8).
Proposition 4.5. For all i, j ∈ I one has that The relation holds if A ij = 0, so we only consider the case of A ij = 0. The proof is divided into several cases.
Case (a): i = j. The required relations are generating functions of the following component relations: On the other hand the following relations hold in relation in U (i) q similar as in the untwisted cases.
Hence the required relation follows by recalling the definition of x + i (k).
Proof. Applying T −k ωi and T −l ωj to E ij and invoking Lemma 3.7, we can pull out the action of T ωj to arrive at which was essentially proved by Beck [B] since d j p i = d i p j .
The following well-known fact will be used to prove the remaining relations.
We concentrate mainly on the relation (9) and divide it into four cases. The Serre relation in the case of P ± ij (z 1 , z 2 ) = 1 can be derived from that of non twisted case. Moreover, the proof will explain why the Serre relation with the lower power works by the action of diagram automorphism σ in the twisted case.
Proposition 4.8. For A ij = −1 and σ(i) = j, we have: Proof. This is proved case by case. Case (i): A ij = −1 and σ(i) = i. In this case P ± ij (z 1 , z 2 ) = 1 and d ij = r, it is also clear that d i = r, then the relation is exactly like the Serre relation in the non-twisted case. For completeness we provide a proof for this Serre relation. i.e. we will show that for any integers k 1 , k 2 , l Note that x + j (l) = T −l ωj E j . Lemma 3.7 says that one can pull out any factor of T ωj or common factors of T ωi from the left-hand side (LHS). This means that for any natural number t the following relation is equivalent to the Serre relation.
We prove this last relation by induction on t. First note that when t = 0, the relation is essentially the relation (R5). We assume that the above relation holds for ≤ t − 1. The remark above further says once we have made the inductive assumption then all Serre relations with |k 1 − k 2 | ≤ t − 1 and arbitrary l are also assumed to be true. Using relation (6) in Theorem 4.4 yieldŝ Plugging this into LHS of the Serre relation we get that Then we repeatedly use ( * ) to move a i (1) to the extreme left to get an expression of the form where · · · only involves with LHS of Serre relations with t − 2. So the whole expression is zero by the inductive assumption. Thus we have finished the proof of the Serre relation (9) in this case. Case (ii): A ij = −1 and A i,σ(i) = 0, σ(j) = j.
For r = 2, without loss generality we take A 2n−1 for an example, the other cases are treated similarly. In this case we only need to consider the situation when i = n − 1 and j = n, then P ± ij (z 1 , z 2 ) = z 1 q ±2 + z 2 and d ij = 2. So we need to prove the following relations.
Using the definition of x + i (k) and collecting the action ofT −k ωiT −l ωj , we are left to show that To see this we use Lemma 4.7 and compute all the commutators [X, F k ] = 0 for k ∈Î. First we consider the case of k = 0. Note that [E i , F 0 ] = 0 whenever i = 0. On the other hand we claim that [T −1 ωi (E i ), F 0 ] = 0. To see this we check that r 0 (T −1 ωi (E i )) = 0 by using the twisted derivation r 0 . By Lemma 2.5 the last equation is equivalent to T −1
where we have used the Drinfeld relation (7), (5) and (6) for the last step. Collecting common terms, we arrive at For r = 3, the exceptional type D should also be checked. In this case we know that P ± ij (z 1 , z 2 ) = z 2 1 q ±4 + z 1 z 2 q ±2 + z 2 2 and d ij = 3. More specifically we have in this case i = 2, j = 1 and d i = 2, d j = 1, and the relation is reduced to the following equivalent one: By definition of x + i (k) and collecting the action ofT −k ωiT −l ωj , we can rewrite the LHS of the above relation as follows.
Let us use Lemma 4.7 to show that Y = 0 by checking that [ Y, F k ] = 0 for all k ∈Î. When k = 0 is clear as Y is only expressed in terms of E i and E j and i, j ∈ I. This also implies that [Y, F k ] = 0 for k = i or j.
For k = j, we use Drinfeld relation (4) and the commutation relation Next we calculate that [ Y, F i To show the above is actually zero, we collect similar terms into five summands.
The first term The fifth term Their total sum is zero, thus we have shown that [Y, F k ] = 0 and subsequently the Serre relations hold in this case.
Case (iii): A ij = −1 and A i,σ(i) = 0, σ(j) = j. The required relation follows from that of the untwisted case verified in Case (i).
Case (iv): A ij = −1 and A i,σ(i) = −1. This only happens for type A (2) 2n . Here P ± ij (z 1 , z 2 ) = z 1 q ±1 + z 2 and d ij = 1 2 , which is exactly the same as that of Case (ii) in type A (1) 2n−1 . Thus the Serre relation is proved by repeating the argument of Case (ii).
By now we have proved all cases of the Serre relation (9).
The last Serre relation (10) only exists for type A 2n .
Proposition 4.9. For A i,σ(i) = 0, Proof. By the same translation property of the Drinfeld generators, this relation can be replaced by By definition of x + i (k), the above relation is rewritten as: Using the same trick of Lemma 4.7, we must show that [ Z, F k ] = 0 for all k ∈Î.
With this last Serre relation we have completed the verification of all Drinfeld relations.
5. Isomorphism between the two structures 5.1. The inverse homomorphism. To complete the proof of Drinfeld realization we need to establish an isomorphism between the Drinfeld-Jimbo algebra and the Drinfeld new realization. There have been several attempts to show the isomorphism in the literature, and all previous proofs only established a homomorphism from one form of the algebra into the other one. In this section we will combine our previous approach [J2,ZJ,JZ] together with Beck and Damiani's work to finally settle this long-standing problem and prove the isomorphism between the two forms of the quantum affine algebras in both untwisted and twisted cases.
In order to show there exists an isomorphism between the above two structures, we recall the inverse map of Ψ from Drinfeld realization to twisted quantum affine algebra developed in [ZJ], where we denoted by U q (ĝ σ ) the Drinfeld realization, which is the associative algebra over the complex field generated by the elements x ± i (k), a i (l), K ±1 i and γ 1 2 , where i ∈ {1, 2, . . . , N }, k ∈ Z, and l ∈ Z/{0}, satisfying the relations (1) − (10).
First of all, we review the notation of quantum Lie brackets from [J2].
In fact, we have Recall that we have defined E δ−θ as the quantum root vector ))K δ−θ independently from the sequence (by Remark 3.1). Thus ψφ(E 0 ) = AE 0 for α 0 = δ − θ, where A is a polynomial of q. Then we can adjust the map φ such that the action of ψφ on E 0 is identity.
Similarly we can show that ψφ(F 0 ) = F 0 . Note that by definition ψφ and ψφ fix all E i and F i for i = 0, and also the homomorphisms φ and ψ are surjective by construction, therefore ψφ = φψ = I. So we have shown that Corollary 5.5. The homomorphisms φ : U q (ĝ σ ) → U q (ĝ σ ) and ψ : U q (ĝ σ ) → U q (ĝ σ ) are two algebra isomorphisms. In particular φ = ψ −1 .