Abstract
The Drinfeld 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 braid group action.
1. Introduction
In studying finite-dimensional representations of Yangian algebras and quantum affine algebras, Drinfeld gave a new realization of the Drinfeld-Jimbo quantum enveloping algebras of the affine types. Drinfeld realization is a 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 the quantum conformal field theory. For example, the basic representations of quantum affine algebras were constructed based on Drinfeld realization [1, 2] and the quantum Knizhnik-Zamolodchikov equation [3] was also formulated using this realization.
The first proof of the Drinfeld realization for the untwisted types was given by Beck [4] using Lusztig’s braid group actions (see also [5] for ). For other approaches see [6–8]. By directly quantizing the classical isomorphism of the Kac realization to the affine Lie algebras, the first author later gave an elementary proof [9] of the Drinfeld automorphism using -brackets starting from Drinfeld’s quantum loop algebras. In this elementary approach a general strategy and algorithm was formulated to prove the Serre relations. In particular, all type relations were verified in detail including the exceptional type .
In [10, 11], we gave an elementary proof of the twisted Drinfeld realization starting from Drinfeld’s quantum loop algebras. In particular in [11] we have shown that twisted quantum affine algebras obey some simplified Serre relations in certain types as in the classical cases and from which twisted Serre relations are consequences.
The purpose of this work is to give a conceptual proof of twisted Drinfeld realizations using braid group actions starting from the Drinfeld-Jimbo definition. Braid group actions have been very useful in Lusztig’s construction of canonical bases [12, 13] 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. We also give a complete proof of all Serre relations for both untwisted and twisted cases using -bracket techniques [2]. 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.
The second goal of our paper is to provide a different proof that the Drinfeld-Jimbo algebra is indeed isomorphic to the Drinfeld algebra of the quantum affine algebra without passing to case. We achieve this by combining our previous work identifying the Drinfeld-Jimbo quantum affine algebra inside the Drinfeld realization and the explicit knowledge of the -bracket computations developed in [9, 10], 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 2, 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 3 shows how to construct the quantum affine algebra (or ) inside the twisted quantum affine algebra (or ). In Section 4, 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 -bracket techniques.
2. Definitions and Preliminaries
2.1. Finite Order Automorphisms of
In this paragraph some basic notations of Kac-Moody Lie algebras are recalled [14]. Let be a simple finite-dimensional Lie algebra, and let be an (outer) automorphism of of order . Then induces an automorphism of the Dynkin diagram of with the same order. Fix a primitive th root of unity . Since is diagonalizable, it follows that where is -eigenspace with eigenvalue . Clearly, the decomposition is - gradation of , and then is a Lie subalgebra of .
Let be one of the simply laced Cartan matrices. It is well-known that the Dynkin diagram has a diagram automorphism of order or . Explicitly is one of the following types: , , , and with the canonical action of :
Let be the set of -orbits on , where we use representatives to denote the orbits. For example, is simply denoted by in type . We can write . Consequently the nodes of the Dynkin diagram are indexed by .
2.2. Twisted Affine Lie Algebras
For a nontrivial automorphism of the Dynkin diagram, the twisted affine Lie algebra is the central extension of the twisted loop algebra: where is the central element and . Denote by the node set of the Dynkin diagram of . Let be the Heisenberg subalgebra of .
Let us use to denote the Cartan matrix of the twisted affine Lie algebra of type . The Cartan matrix is symmetrizable; that is, there exists a diagonal matrix such that is symmetric. Let be the simple roots and let be the simple coroots of such that . Let be the affine root lattice defined by Subsequently will denote the finite root lattice of the subalgebra . Indeed, . Let ; then .
Introduce the nondegenerate symmetric bilinear form on determined by . Let be the canonical imaginary root of minimal height such that and , . Here the coefficients are unique such that is always , and we have chosen the labels of different from [14].
Let be the coweight lattice over , and define the fundamental coweights such that , , . So . Additionally, we are going to introduce another important sublattice defined by , where for and , and Let us denote by and the finite coweight and weight lattice of the subalgebra , respectively. Then and .
Recall that the (affine) Weyl group is generated by , where ; thus . The Weyl group acts on by . For any , if is a reduced expression of , then we define the length . Let be the subgroup generated by ; thus is the finite Weyl group of .
Let be the group of automorphisms of the Dynkin diagram associated with . Then acts on by , , . We denote . We further introduce the extended affine Weyl group , which can be written as for twisted cases. The length function is extended to by , for , .
2.3. Twisted Quantum Affine Algebras
In this paragraph let us review the definition of the twisted quantum affine algebra . Set , . Introduce -integer by
Definition 1. The twisted quantum affine algebra is an associative -algebra generated by for , satisfying the following relations:
Remark 2. (1) There exists a unique Hopf algebra structure on with the comultiplication , counit , and antipode defined by (): (2) Let (resp., ) be the subalgebra of generated by the elements (resp., ) for , and let be the subalgebra of generated by and . The twisted quantum affine algebra has the triangular decomposition: (3) When , the quantum Serre relation becomes Similarly, if , we have the following equality: where .
From the above identities, it is not hard to see that the higher degree Serre relations can be derived from the lower degree ones. Also it is expected that the nonsimply laced Dynkin diagrams can be obtained from simply laced ones by the action of .
The following proposition can be checked easily.
Proposition 3. There exist an antiautomorphism (over ) and an automorphism (over ) of defined as follows:
We recall Lusztig’s braid group associated with , which acts as an automorphism group of the twisted quantum affine algebra ([12]). For simplicity, we denote , . The actions of and on Chevalley generators are defined as follows:where and .
We extent the braid group action to the extended Weyl group by defining as
Remark 4. The braid group action commutes with ; that is, . Furthermore, one has .
Note that ; some properties about will be discussed which are almost the same to nontwisted cases, where .
Finally for each , we define inductively the twisted derivation of by , and Here is the -graded subspace of with the weight . The following result is from Lusztig [12].
Lemma 5. (1) If and for all , then .
(2) One has .
3. Vertex Subalgebras
There are three different lengths of real roots for the type of , which requires a different treatment from other twisted types. For each we construct a copy of inside for . For the case of we will construct a subalgebra isomorphic to .
3.1. Root System
The root system of the twisted affine algebra is given by and , where and are listed as follows (see [14]): For type , for other types, where is the positive roots system of and . Here we call the set of positive roots of .
3.2. Quantum Root Vectors
In order to define the quantum root vectors, we review some notations from Beck-Nakajima [13]. For , , we choose such that . Choose a reduced expression of for each . We fix a reduced expression of as follows: , where .
We define a doubly infinite sequence by setting for .
Then we have
Set Define a total order on by setting where denotes .
The root vectors for each element of can be defined as follows: It follows from [12] that the elements .
Remark 6. For a positive root in the root system of , there exists another presentation , where are in the Weyl group and are simple roots. Then the quantum root vector can be defined by two ways: Actually, the two definitions agree up to a constant, because there exists such that , which means that and ; then we have where are functions in .
Furthermore, if for some , then for a function . As usual, define for all .
Now we introduce the root vectors of Drinfeld generators [15]:
Note that the definitions are not so different from those of nontwisted cases up to some slight adjustments because of the difference between their root systems.
3.3. Vertex Subalgebra
Let subalgebra of twisted quantum affine algebra be generated by and for .
It is clear that the vertex subalgebra is stable under the actions of , and . Additionally, is pointwise fixed by for .
We list the following result which is already proved in [16] (also see [4]).
Lemma 7. Let ; one has that
The following statements are based on the construction of [4].
Lemma 8. One has
Proposition 9. For , there exists an algebra isomorphism , defined by
3.4. Relations of Imaginary Root Vectors
Now we define the positive imaginary root vectors. For and let then we use to denote .
Define the elements by the functional equation
Similarly, we introduce for . Then we have the following lemma.
Lemma 10. For , , one has
Define a map such that for . For and , let
Similarly, we can define and for .
The following was given in [17], which is an application of Beck’s work [4].
Proposition 11. For , for , , the following relations hold:
3.5. Relations between and
We recall the commutation relations among the root vectors and remark that the argument also works in the case of .
Lemma 12 (see [4, 16]). One has that, for ,
We also let . The following is immediate.
Lemma 13. If for , then
Lemma 14. For and ,
If such that , it is easy to see that and and , . Thus one has the following lemma.
Lemma 15. If such that , then for , one has the following relations:
If such that and , it is easy to see that and and , . Then the next result follows.
Lemma 16. Suppose and for . Then for , , one has
The following statements follow directly from Lemmas 18 and 16. Note that for , which will be used later.
Lemma 17. Let such that . For , one has
Lemma 18. Let such that and . Then for , ,
3.6. Copies of in
For the case of , the definition of the quantum root vectors differs from other twisted cases because of its slightly complicated root system. This situation has been discussed in detail in [17]; here we will review some definitions and results. The case of has been dealt with care in [18].
In this paragraph we discuss the remaining cases and show that, for , there exists a copy of in .
Let us introduce the longest element of the Weyl group : then , where .
In particular, define and .
Proposition 19. Let be the subalgebra generated by , . There exists an algebra isomorphism 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 20. For , define
Definition 21. For , define
Remark 22. From the above definitions it follows that if and is not divisible by .
For convenience we extend the indices from to . For and , , we define that
4.2. 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 [19]. In the following we will set out to prove the remaining Serre relations using braid groups and other techniques developed in [10, 11].
Theorem 23. The twisted quantum affine algebra is generated by the elements , , , and , where , , and , satisfying the following relations: where and are defined bywhere means the symmetrization over , , and and are defined as follows:
4.3. Proof of the Main Theorem
We need to verify that the above Drinfeld generators satisfy all relations . Relations are already checked in the previous paragraphs. We are going to show the last three relations.
We first proceed to check relation (8).
Proposition 24. For all one has that where .
The relation holds if , so we only consider the case of . The proof is divided into several cases.
Case a (). The required relations are generating functions of the following component relations: On the other hand, the following relations hold in similar to those of the untwisted cases:Hence the required relation follows by recalling the definition of .
Case b ( such that ). First for we define
Lemma 25. For such that and ,
Proof. Applying and to and invoking Lemma 12, we can pull out the action of to arrive at which was essentially proved by Beck [4] since .
The following well-known fact will be used to prove the remaining relations.
Lemma 26. If and , then .
We concentrate mainly on relation and divide it into four cases. The Serre relation in the case of can be derived from that of nontwisted case. Moreover, the proof will explain why the Serre relation with the lower power works by the action of diagram automorphism in the twisted case.
Proposition 27. For and , one has
Proof. This is proved case by case.
Case i ( and ). In this case and , it is also clear that , and then the relation is exactly like the Serre relation in the nontwisted case. For completeness we provide a proof for this Serre relation. That is, we will show that for any integers Note that . Lemma 12 says that one can pull out any factor of or common factors of from the left-hand side (LHS). This means that for any natural number the following relation is equivalent to the Serre relation: We prove this last relation by induction on . First note that when , the relation is essentially relation . We assume that the above relation holds when the parameter is less than , and we would like to show it for . The remark above further says that once we have made the inductive assumption then all Serre relations with and arbitrary are also assumed to be true. Using relation (6) in Theorem 23 yields Plugging this into LHS of the Serre relation we get that Then we repeatedly use to move to the extreme left to get an expression of the form where “” only involves LHS of Serre relations with . So the whole expression is zero by the inductive assumption. Thus we have finished the proof of Serre relation (9) in this case.
Case ii ( and , ). For , without loss of generality we take ; for example, the other cases are treated similarly. In this case we only need to consider the situation when and ; then and . So we need to prove the following relations: Using the definition of and collecting the action of , we are left to show thatTo see this we use Lemma 26 and compute all the commutators for . First we consider the case of . Note that whenever . On the other hand we claim that . To see this we check that by using the twisted derivation . By Lemma 5 the last equation is equivalent to , which can be easily seen as when . Consequently it implies that .
Since is expressed by and , for any . Using Drinfeld relation (7), we compute that where Drinfeld relation (5) has been used:where we have used Drinfeld relation (7), (5), and (6) for the last step. Collecting common terms, we arrive at For , the exceptional type should also be checked. In this case we know that and . More specifically we have in this case , and , , and the relation is reduced to the following equivalent one: By definition of and collecting the action of , we can rewrite the LHS of the above relation as follows:Let us use Lemma 26 to show that by checking that for all , when is clear as is only expressed in terms of and and . This also implies that for or .
For , we use Drinfeld relation (4) and the commutation relation to get that Next we calculate that To show that the above is actually zero, we collect similar terms into five summands:Their total sum is zero; thus we have shown that and subsequently the Serre relations hold in this case.
Case iii ( and , ). The required relation follows from that of the untwisted case verified in Case i.
Case iv ( and ). This only happens for type . Here and , which is exactly the same as that of Case ii in type . Thus the Serre relation is proved by repeating the argument of Case ii.
By now we have proved all cases of Serre relation (9).
The last Serre relation (10) only exists for type .
Proposition 28. For ,
Proof. By the same translation property of the Drinfeld generators, this relation can be replaced by By definition of , the above relation is rewritten as Using the same trick of Lemma 26, we must show that for all :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 [9–11] together with Beck’s idea of braid groups 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 that there exists an isomorphism between the above two structures, we recall the inverse map of from Drinfeld realization to twisted quantum affine algebra developed in [10], where we denoted by the Drinfeld realization, which is the associative algebra over the complex field generated by the elements , and , where , , and , satisfying relations .
First of all, we review the notation of quantum Lie brackets from [9].
Definition 29. Let be a field for and . The quantum Lie brackets are defined inductively by
To state the inverse homomorphism, we need to fix a particular path to realize the maximum root of .
Let be the maximum root and letbe the corresponding root vector in the Lie algebra , which gives rise to a sequence from . We call such a sequence a root chain to the maximum root, which is not unique.
From now on we fix a particular path to realize the maximum root of and the associated sequence . We define for
Theorem 30. Let be the sequence of indices in the particular path realizing the maximum root given in (76); then there is an algebra homomorphism defined by where , , and , are defined by quantum Lie bracket as follows:
It is not difficult to check that the algebra is actually generated by , , and ; therefore is an epimorphism.
Remark 31. In [10, 11], we have checked that is an algebra homomorphism using quantum Lie brackets. In the sequel we show that the map is in fact the inverse of the action of the braid group.
5.2. Isomorphism between Two Presentations
Theorem 23 induces that the homomorphism from Drinfeld realization to Drinfeld-Jimbo algebra is surjective. We now show its injectivity by checking that the products of two maps are identity. We start with the following proposition.
Proposition 32. With homomorphisms and defined as above, one has .
Proof. Note that Note that, for (see [20]), So the above bracket can be written as , when is not equal to for all .
Thus we need to consider the case : where we have used (see [4]).
Therefore the above -bracket also can be written as where are in the set such that every two elements are different and is a polynomial of .
In fact, we have Recall that we have defined as the quantum root vector independently from the sequence (by Remark 6). Thus for , where is a polynomial of . Then we can adjust the map such that the action of on is identity.
Similarly we can show that . Note that by definition and fix all and for , and also the homomorphisms and are surjective by construction; therefore . So we have shown the following.
Corollary 33. The homomorphisms and are two algebra isomorphisms. In particular .
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
Naihuan Jing would like to thank the partial support of Simons Foundation Grant 198129, NSFC Grant (11271138 and 11531004), and NSF Grant (1014554 and 1137837). Honglian Zhang would like to thank the support of NSFC Grant (11371238 and 11101258).