Quadratic Relations of the Deformed W -Algebra for the Twisted Aﬃne Lie Algebra of Type A (2)2 N

. We revisit the free ﬁeld construction of the deformed W -algebra by Frenkel and Reshetikhin [ Comm. Math. Phys. 197 (1998), 1–32], where the basic W -current has been identiﬁed. Herein, we establish a free ﬁeld construction of higher W -currents of the deformed W -algebra associated with the twisted aﬃne Lie algebra A (2) 2 N . We obtain a closed set of quadratic relations and duality, which allows us to deﬁne deformed W -algebra W x,r (cid:0) A (2)2 N (cid:1) using generators and relations.


Introduction
The deformed W -algebra W x,r (g) is a two-parameter deformation of the classical W -algebra W(g). The deformation theory of the W -algebra has been studied in papers [2,3,4,5,6,8,10,12,13,14,16,17]. For instance, free field constructions of the basic W -current T 1 (z) of W x,r (g) were suggested in the case when the underlying Lie algebra is of classical type. However, in comparison with the conformal case, the deformation theory of W -algebras is still not fully developed and understood. Moreover, finding quadratic relations of the deformed W -algebra W x,r (g) is still an unresolved problem.
In this paper, we generalize the study for W x,r A (2) 2 1 by Brazhnikov and Lukyanov [3]. They obtained a quadratic relation for the W -current T 1 (z) of the deformed W -algebra W x,r A (2) 2 with an appropriate constant c and a function f (z). This study aims to generalize the result for the cases A 2 to A 2N . We introduce higher W -currents T i (z), 1 ≤ i ≤ 2N , by fusion of the free field construction of the basic W -current T 1 (z) of W x,r A (2) 2N [8] (see formula (3.2)). We obtain a closed set of quadratic relations for the W -currents T i (z), which is completely different from 1 We use two types of symbols, Wx,r(g) and Wx,r(X (r) n ), for the deformed W -algebra associated with the affine Lie algebra g of type X (r) n .

2
T. Kojima those in the case of deformed W -algebras associated with affine Lie algebras of types A (1) N and A(M, N ) (1) (see formula (3.4)). We refer the reader to references [18,19] for the affine Lie superalgebra notation. We obtain the duality T 2N +1−i (z) = c i T i (z) with 1 ≤ i ≤ N , which is a new phenomenon that does not occur in the case of deformed W -algebras associated with affine Lie algebras of types A (2) 2 , A (1) N , and A(M, N ) (1) (see formula (3.3)). This allows us to define W x,r A (2) 2N using generators and relations. We believe that this paper presents a key step toward extending our construction for general affine Lie algebras g, because the structures of the free field construction of the basic W -current T 1 (z) for the affine algebras other than that of type A (1) N are quite similar to those of type A (2) 2N , not A (1) N . We have checked that there are similar quadratic relations as those for type A (2) 2N in the case of type B (1) N with small rank N . The remainder of this paper is organized as follows. In Section 2, we review the free field construction of the basic W -current T 1 (z) of the deformed W -algebra W x,r A (2) 2N [8]. In Section 3, we introduce higher W -currents T i (z) and present a closed set of quadratic relations and duality. We also obtain the q-Poisson algebra in the classical limit. In Section 4, we establish proofs of Proposition 3.1 and Theorem 3.2. Section 5 is devoted to discussion. In Appendices A and B, we summarize normal ordering rules.

Free field construction
In this section, we define notation and review the free field construction of the basic W -current T 1 (z) of W x,r A (2) 2N . Throughout this paper, we fix a natural number N = 1, 2, 3, . . . , a real number r > 1, and a complex number x with 0 < |x| < 1.

Notation
In this section, we use complex numbers a, w, q, and p with w ̸ = 0, q ̸ = 0, ±1, and |p| < 1. For any integer n, we define q-integers [n] q = q n − q −n q − q −1 .
We use symbols for infinite products, We use the elliptic theta function Θ p (w) and the compact notation Θ p (w 1 , w 2 , . . . , w N ), Θ p (w) = p, w, pw −1 ; p ∞ , Θ p (w 1 , w 2 , . . . , w N ) = In this section we recall the definition of the twisted affine Lie algebra of type A (2) 2N , N = 1, 2, 3, . . . , in [11]. The Dynkin diagram of type A (2) 2N is given by Quadratic relations of the deformed W -algebra W x,r ( A In this section we recall the definition of the twisted affine Lie algebra of type A (2) 2N , N = 1, 2, 3, . . . , in Ref. [16]. The Dynkin diagram of type A (2) 2N is given by The corresponding Cartan matrix 2N is given by with N = 1. We set the labels a i = 2, 0 ≤ i ≤ N − 1, a N = 1, and the co-labels . We obtain A = DB, where B is a symmetric matrix. Thus, the Cartan matrix A is symmetrizable. Let h be an N + 2-dimensional vector space over C. Let {h 0 , h 1 , . . . , h N , d} be a basis of h, and {α 0 , α 1 , . . . , α N , Λ 0 } a basis of h * = Hom C (h, C) such that we have with respect to pairing ⟨·, ·⟩ : Let g(A) be the affine Lie algebra associated with the Cartan matrix A. Since A is symmetrizable, it is defined as the Lie algebra generated by e i , f i , 0 ≤ i ≤ N , and h with the following relations: Here we used the adjoint action (ad x)y = [x, y].

The corresponding Cartan matrix
2N is given by with N = 1. We set the labels a i = 2, 0 ≤ i ≤ N − 1, a N = 1, and the co-labels is a symmetric matrix. Thus, the Cartan matrix A is symmetrizable. Let h be an (N + 2)dimensional vector space over C. Let {h 0 , h 1 , . . . , h N , d} be a basis of h, and {α 0 , α 1 , . . . , α N , Λ 0 } a basis of h * = Hom C (h, C) such that we have with respect to pairing ⟨·, ·⟩ : Let g(A) be the affine Lie algebra associated with the Cartan matrix A. Since A is symmetrizable, it is defined as the Lie algebra generated by e i , f i , 0 ≤ i ≤ N , and h with the following relations: Here we used the adjoint action (ad x)y = [x, y].

Free field construction
In this section, we recall the free field construction of the basic W -current T 1 (z) and of the screening operators S i of the deformed W -algebra W x,r A 2N introduced by Frenkel and Reshetikhin [8].
First, we define the N × N symmetric matrix B(m) = (B i,j (m)) N i,j=1 , m ∈ Z, associated with A (2) 2N , N = 1, 2, 3, . . . , as follows: We introduce the Heisenberg algebra H x,r with generators a i (m), The remaining commutators vanish. The generators a i (m), Q i are "root" type generators of H x,r . There is a unique set of "fundamental weight" type generators y i (m), Q y i , m ∈ Z, 1 ≤ i ≤ N , which satisfy the following relations The explicit formulas for y i (m) and Q y j are given in (A.7). We use the normal ordering : : on H x,r that satisfies Let |0⟩ ̸ = 0 be the Fock vacuum of the Fock space of H x,r such that a i (m)|0⟩ = 0, m ≥ 0, 1 ≤ i ≤ N . Let π λ be the Fock space of H x,r generated by |λ⟩ = e λ |0⟩, λ = N j=1 λ j Q y j . We obtain We work in the Fock space π λ of the Heisenberg algebra H x,r . Let the vertex operators A i (z), Y i (z), and S i (z), 1 ≤ i ≤ N , be Quadratic Relations of the Deformed W -Algebra for the Twisted Affine Lie Algebra The main parts of (2.2), (2.3), and (2.4) are the same as those of [8]. We corrected the misprints in the formulas for A i (z), Y i (z), and S i (z) in [8] Let k = k, k = 1, 2, . . . , N , and 0 = 0. The indices i, j ∈ J N satisfy i ≺ j if and only if j ≺ i. We define I = {i 1 , i 2 , . . . , i k } for a subset I ⊂ J N , I = {i 1 , i 2 , . . . , i k }. Let T 1 (z) be the generating series with operator valued coefficients acting on the Fock space π λ , We call T 1 (z) the basic W -current of the deformed W -algebra W x,r A 2N . Let π µ be the Fock space of H x,r generated by |µ⟩ = e µ |0⟩ with , takes values in integers on π µ . Hence, S i is well-defined on π µ . We define the screening operators S i , 1 ≤ i ≤ N , acting on the Fock space π µ as The integral in formula (2.6) means the residue at zero.

Quadratic relations
In this section, we introduce the higher W -currents T i (z) and present a set of quadratic relations between T i (z) for the deformed W -algebra W x,r A 2N .

Quadratic relations
We define the formal series ∆(z) ∈ C[[z]] and the constant c(x, r) as The formal series ∆(z) satisfies We define the structure functions f i,j (z), i, j = 0, 1, 2, . . . , as The ratio of the structure functions f 1,1 (z) is We introduce higher W -currents T i (z) as follows: Here, for a subset Ω i = {s 1 , s 2 , . . . , s i } ⊂ J N with s 1 ≺ s 2 ≺ · · · ≺ s i , we set Proposition 3.1. The W -currents T i (z) satisfy the duality Theorem 3.2. The W -currents T i (z) satisfy the set of quadratic relations In view of Proposition 3.1 and Theorem 3.2, we obtain the following definition.
Definition 3.3. Let W be the free complex associative algebra generated by elements 2N is the quotient of W by the two-sided ideal generated by the coefficients of the generating series which are the differences of the right hand sides and of the left hand sides of the relations (3.3) and (3.4), where the generating series The justification of this definition is presented later. We compare this definition of the deformed W -algebra with other definitions in Section 5.
We present the proofs of Proposition 3.1, Theorem 3.2, and Lemma 3.4 in Section 4.

Classical limit
The deformed W -algebra W x,r g yields a q-Poisson W -algebra [7,8,9,15] in the classical limit.
As an application of the quadratic relations (3.4), we obtain a q-Poisson W -algebra of type A 2N . We set parameters q = x 2r and β = (r − 1)/r. We define the q-Poisson bracket {·, ·} by taking the classical limit β → 0 with q fixed as Here, we introduce The β-expansions of the structure functions are given as As corollaries of Proposition 3.1 and Theorem 3.2 we obtain the following.
2N , the currents T PB i (z) satisfy Here, the structure functions C i,j (z) are given by Corollary 3.6. The currents T PB i (z) satisfy the duality relations 4 Proof of Theorem 3.2 In this section, we prove Proposition 3.1, Theorem 3.2, and Lemma 3.4.

Proof of Proposition 3.1
Proof . Using (A.2) and (A.8), we obtain the normal ordering rules (4.1). ■ : Proof . We show (4.6) here. From the definitions, we have Using the relation we obtain f 1,i (z) in the right hand side of the previous formula. We obtain (4.5), (4.7), (4.8), and (4.9) by straightforward calculation from the definitions. Using (4.5) and (4.6), we obtain the relations (4.10), (4.11), and (4.12). ■ Lemma 4.4. The following relation holds for A ⊂ J N : (4.13) Proof . First, we consider the case A = ∅ and J N \ A = J N . In this case, (4.13) can be rewritten as (4.14) Using (2.2), (2.3), and (2.5), the left side of (4.14) can be written as , the generators y 1 (m) in (A.7) are a j (m). Hence, we obtain (4.14). Next, we show (4.13) for A ⊂ J N . Cases (i), 0 ∈ A and (ii), 0 / ∈ A are proved separately. First, we study case (i), 0 ∈ A. Let Multiplying (4.14) by − → Λ A x L−K+1 z on the left, and using (4.1) and (4.7) yields Using (4.2), (4.3) and (4.4) yields Using the above five relations yields From (4.15) we obtain (4.13) for 0 ∈ A. Next, we study case (ii), 0 / ∈ A. The proof for this case is similar to that of case (i). Let Multiplying (4.14) by − → Λ A x L−K z on the left, and using (4.1) and (4.7) yields Using the above five relations yields From (4.16) we obtain (4.13) for 0 / ∈ A. ■ Lemma 4.5. The following relation holds for A ⊂ J N with |A| ≤ N : Proof . We define the map σ : Hence, the relation We prove (4.17) by induction on N . First, we establish the base N = 1 using case-by-case This implies that (4.17) holds for N = 1.

Conclusion and discussion
In this paper, we obtained the free field construction of higher W -currents T i (z), i ≥ 2, of the deformed W -algebra W x,r A 2N . We obtained a closed set of quadratic relations for the Wcurrents T i (z), which are completely different from those in types A (1) N and A(M, N ) (1) . The quadratic relations of W x,r A (2) 2N do not preserve "parity", though those of W x,r A (1) N and W x,r A(M, N ) (1) do. Here we define "parity" of T i (z)T j (w) as i + j. We obtained the duality which is a new structure that does not occur in types A N , and A(M, N ) (1) . This allowed us to define the deformed W -algebra W x,r A (2) 2N using generators and relations similarly to the definition of the twisted affine Lie algebra of type A (2) 2N given in Section 2.
We also justified our definition of the deformed W -algebra of type A 2N . We compare Definition 3.3 with other definitions. In [8], the deformed W -algebras of types A (1) N , and A (2) 2N were proposed as the intersection of the kernels of the screening operators. We recall the definition based on the screening operators for A (2) 2N . Let H x,r be the vector space spanned by the formal power series currents of the form We propose another definition of the deformed W -algebra. From (3.5), the W -currents In this study, our definitions W x,r A 2N were based on generators and relations. We have introduced three definitions of the deformed W -algebra for the twisted algebra of the type A 2N are isomorphic as associative algebras 2N . (5.1) The author believes that this conjecture can be extended to arbitrary affine Lie algebras. Some necessary conditions of isomorphism (5.1) in Conjecture 5.1 can be indicated immediately. From (3.5), we obtain the following inclusion: 2N .
We establish a homomorphism of associative algebras φ ∈ Hom C W x,r A 2N .
If we assume that φ is injective, the isomorphism on the left side in (5.1) is obtained. In other words, no independent relations other than (3.3) and (3.4) exist in W x,r A 2N . We propose two results to support this claim. In the classical limit the second Hamiltonian structure {·, ·} of the q-Poisson algebra [7,8,9,15] was obtained from the quadratic relations (see (3.6) and (3.7)). In the conformal limit all defining relations of the W -algebra W β A (1) N , N = 1, 2, are obtained from the quadratic relations of W x,r A (1) N upon the assumption that the currents T i (z) have the form of expansion for small parameter ℏ (see [1,Appendix]).
The definition of the deformed W -algebra W x,r (g) for non-twisted affine Lie algebra g was formulated in terms of the quantum Drinfeld-Sokolov reduction in [16]. Formulating the definition of the deformed W -algebras W x,r (g) in terms of the quantum Drinfeld-Sokolov reduction for twisted affine Lie algebra or affine Lie superalgebra [4,6,10,12,13] is still a problem that needs to be solved.
It remains an open challenge to identify quadratic relations of the deformed W -algebras W x,r (g) for the affine Lie algebras g except for types A (1) N and A (2) 2N . We believe that this paper presents a key step towards extending our construction for general affine Lie algebras g. In [8] and [6] the free field construction of the basic W -current T 1 (z) of W x,r g was suggested in the case when the underlying simple finite-dimensional Lie algebra • g is of classical type, for g of type A (1) N , Λ 1 (z) + · · · + Λ N (z) + Λ 0 (z) + Λ N (z) + · · · + Λ 1 (z) for g of types B (1) 2N , D N +1 , Λ 1 (z) + · · · + Λ N (z) + Λ N (z) + · · · + Λ 1 (z) for g of types C Here we omit details of free field constructions of Λ i (z). The free field construction of T 1 (z) has similar form to that for g of type A 2N −1 , and D (2) N +1 . We would like to draw your attention to the following analogy. Let g be an affine Lie algebras of one of the types B be the integrable highest weight representation of U q ( • g ) with the highest weight Λ 1 . Let V be the evaluation representation corresponding to V Λ 1 of the quantum affine algebra U q (g) with a spectral parameter z ∈ C × . Let n be the dimension of V Λ 1 . We have The evaluation representation V of U q g is self-dual except for g of type A (1) N . Hence, we obtain the duality of the representations of U q g , which is similar as that in (3.3). As an analogy, we expect the duality of the W -currents, N , for the deformed W algebras W x,r (g). Here c i , 0 ≤ i ≤ n, are constants.