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q-Deformation of W(2,2) Lie algebra associated with quantum groups

An explicit realization of the W(2,2) Lie algebra is presented using the famous bosonic and fermionic oscillators in physics, which is then used to construct the q-deformation of this Lie algebra. Furthermore, the quantum group structures on the q-deformation of this Lie algebra are completely determined.

different and thus the quantum group (associated with the deformation) constructed in this paper seems to be new.
The W (2, 2) Lie algebra (denoted by W) considered in the present paper is an infinite dimensional Lie algebra with basis L n , W n (for n ∈ Z) and the Lie brackets One sees that it is different from that defined in [36], since we drop here the central element. But our next aim is to develop the central extensions of the q-deformation W q .
In this paper, an explicit realization (Lemma 2.1) of the W (2, 2) Lie algebra defined in (1.1) is given using the famous bosonic and fermionic oscillators in physics ((2.1) and (2.3)). As a result, the q-deformation W q (Proposition 2.2) of the W (2, 2) Lie algebra is obtained by exact calculations.
Based on this, Hopf algebraic structures of W q are proposed and thus the quantum group of W q are completely determined. It seems to us that the results in our paper may be of some potential use in mathematical physics. Throughout this paper, F denotes a field of characteristic zero, F * the multiplicative group of nonzero element of F. All vector spaces and algebras are assumed to be over F. Denote by Z, Z + , Z * the sets of integers, nonnegative, nonzero integers, respectively. §2. Realization of W and its q-Deformation W q In this section, we propose a version of realizing the W(2, 2) Lie algebra defined in (1.1). Based on this, a quantum deformation of this Lie algebra is constructed. One sees that the q-deformed W(2, 2) Lie algebra W q here is rather different from that given in [11].
(2.6) From(2.5) it follows by induction on n that [a, (a + ) n ] (q −n ,q n ) = [n] q (a + ) n−1 (2.7) for arbitrary n, where the general notation Furthermore, one can also deduce that Now we have the following result: The generators L n and W n (for n ∈ Z) defined in (2.4) satisfy the following relations: Hence equation (2.9) holds for all m, n ∈ Z. Similarly, one can get equations (2.10) and (2.11) using the facts that b + commutes with both a and a + and that (b + ) 2 = 0.
Proposition 2.2 says that the algebra with generators L n , W n (for n ∈ Z) and the relations (2.9)-(2.11) realizes the W (2, 2) Lie algebra W in the q → 1 limit. We call it the q-deformation of the W (2, 2) Lie algebra W, which will be denoted by W q in the sequel.
Furthermore, we can generalize this deformation by setting where c ∈ F * and c = 1. Then by induction, one has Now we can show that the expressions (2.4) of L n and W n satisfy the following relations: for all m, n ∈ Z. From these we see the algebra generated by L n and W n in (2.4) with relations (2.12)-(2.14) is also a realization of the W (2, 2) Lie algebra in the q → 1 limit, which is called the generalized q-deformation of W (2, 2) Lie algebra and is denoted by W c q . Note that one gets W q (see Proposition 2.2) when c = −1. §3. Quantum Group Structures of W q In this section, we give a direct construction of the Hopf algebraic structures of the q-deformed W (2, 2) Lie algebra W q defined in previous section.
Fix a q ∈ F * such that q is not a root of unity. Then U q is defined as the associative algebra (with 1 and over F) with generators T , T −1 , L n ,W n for n ∈ Z and relations: Before giving the construction of the Hopf algebraic structures on U q , we have to check whether or not these six relations (R1)-(R6) above ensure a nontrivial associative algebra U q . The following proposition gives a positive answer.
Let I be the two-sided ideal of T (M ) generated by for all m, n ∈ Z and where T −n = (T −1 ) n . Set S(M ) := T (M )/I. It is obvious that S(M ) is also a Z-graded associative algebra with a basis where d, d ′ , k i , l j , i p , j q ∈ Z + , (i, p = 1, 2, · · · m; j, q = 1, 2, · · · n); i 1 < i 2 < · · · < i m , j 1 < j 2 < · · · < j n . LetJ be another two-sided ideal of T (M ) generated by the elements of form together with that in (3.1) and (3.2). Then set Our aim is to show thatB defined in (3.4) is also a basis of U q . Let be a subset of T (M ) and let U ′ be the subspace of T (M ) spanned byB ′ . We now claim that However, v (m) is a nonzero linear combination of the elements inB ′ , it is impossible to appear in ker π. This contradiction impliesB is a basis of U q . Since it is clear that U q ∼ = U q /J, where J is the two-sided ideal of U q generated by T T −1 − 1, we obtain that U q is a nontrivial associative algebra with basisB With the above proposition in hand, we can safely proceed with the construction of the Hopf algebraic structures on U q now. This will be done by several lemmas below.

Lemma 3.4 There is a unique homomorphism of F-algebras
We have to show that ∆(T ), ∆(T −1 ), ∆(L n ) and ∆(W n ) satisfy the relations (R1)-(R6). This is trivial for (R1). For (R2) and (R3) it follows directly from (3.9)- (3.12). Now look at (R4): we have Similarly, we get Then it follows Hence, (R4) is preserved by ∆ and so is it for (R5) and (R6), which can be checked by the similar method. That means ∆ is an algebraic homomorphism. Consequently, U q is a bialgebra.
Proof. We simply have to check that all the generators of U q are mapped both ways by (1 ⊗ ∆)∆ and (∆ ⊗ 1)∆ to the same image, which simply involves straightforward calculations. We shall take L n as an example (others can be done similarly).
for all n ∈ Z. Moreover, the following diagrams are commutative . So we have the homomorphism ε. For the commutativity of the diagrams, it can be easily checked on the generators.
The homomorphism ε from Lemma 3.6 is called the counit of U q .
In addition, one has S 2 = id.
Proof. We need to show that S(T ), S(T −1 ), S(L m ), S(W m ) satisfy the relations (R1)-(R6) in U opp q . Let us denote the multiplication in U opp q by a " · " in order to distinguish it from that in U q .

Now (R1) is obvious and it is easy to see that
One can check similarly that (R3) is preserved by S. Now consider (R4): it follows from (R2) that Similarly, one has q m−n S(L m ) · S(L n ) = q n−m T −m−n L n L m T −m−n . Then we have namely, the map S preserves (R4). One can similarly check that (R5) and (R6) are also preserved by S. So there is indeed a homomorphism S : U q → U opp q or an antihomomorphism S : U q → U q satisfying (3.14)-(3.16). Now S 2 is an ordinary homomorphism from U q to U q . One can check easily on the generators that S 2 = id, which implies that S is bijective.
The map S in Lemma 3.7 is called the antipode of U q . It is clear that the inverse S −1 of S is an antiautomorphism, which is given by Lemma 3.8 The following diagrams are commutative Proof. Let us restrict ourselves to the left diagram. The map f = m • (1 ⊗ S) • ∆ acts on generators as follows: as predicted by the diagram.
To conclude the proof we have to check: then also f (uv) = ι • ε(uv). That is not obvious, since S and m are not ring homomorphisms. We We assume that f (v) = ι • ε(v), so this element is a scalar multiple of 1 and thus central in U q . Therefore In general, an F-algebra A together with algebra homomorphisms ∆ : A → A⊗A and ε : A → F and a linear map S : A → A is called a Hopf algebra, if ∆ is coassociative and if the diagrams in Lemmas 3.6 and 3.8 (with U q is replaced by A) commute. One calls ∆ the comultiplication, ε the counit and S the antipode of the Hopf algebra. A Hopf algebra A is called cocommutative, if P • ∆ = ∆ with P (u ⊗ v) = v ⊗ u for all u and v in A. So the Lemmas 3.4-3.8 say: Theorem 3.9 (U q , ∆, ε, S) defined by (R1)-(R6) and (3.9)-(3.16) is a Hopf algebra, which is neither cocommutative nor commutative.
Corollary 3.10 As vector spaces, one has is the enveloping algebra of W q generated by L n and W n (n ∈ Z) with relations (2.9)-(2.11).
for any n ∈ Z.
and S(L r+1 n ) = (−1) r T −rn L r n T −rn − T −n L n T −n = (−1) r+1 T −rn L r n T −n T −rn L n T −n Since 1990s, there have been intensive explorations of quantized universal enveloping algebras, namely, quantum groups, which were first introduced independently by Drinfeld [7,8] and Jimbo [16,17] around 1985 in order for them to construct solutions to the quantum Yang-Baxter equations. Since then quantum groups are found to have numerous applications in various areas ranging from statistical physics via symplectic geometry and knot theory to modular representations of reductive algebraic groups. For this reason, the interests in quantum groups, quantum deformations of Lie algebras as well as Lie bialgebras have been growing in the physical and mathematical literatures, especially those of Cartan type and Block type, which are closely related to the Virasoro algebra and the W -infinity algebra W 1+∞ (e.g., [20,26,29,30,31,32,33,34,35,36]). In particular, the q-deformed Virasoro algebra, q-deformed oscillator and q-deformed Heisenberg algebra have been investigated in a number of papers (see e.g. [1,3,4,14,27,25]). Among these kinds of algebras, the q-deformation of the Virasoro algebra has been most intensively considered [1,10,13,19,23,24], which can be viewed as a typical example of the physical application of the quantum group. In addition, two-parameter deformation of Lie algebras has also been considered by some authors (see e.g. [2,5]), while the more general quantum Lie algebras have been investigated in [6,9,15,28]. Roughly speaking, quantum Lie algebras in the context of these deformations are universal enveloping algebras deformed by one or more parameter(s) (q-deformation) and possess structures of Hopf algebras. However, the essential reason for the name "quantum" algebra is that it becomes the conventional Lie algebra in the q → 1 limit (classical limit).
The W (2, 2) Lie algebra was introduced by Zhang-Dong in [37] for the study of classification of vertex operator algebras generated by vectors of weight 2. Later the Harish-Chandra modules of this Lie algebra were investigated in [21], while the classification of irreducible weight modules was discussed in [22]. The derivations, central extensions and automorphism groups of this Lie algebra were determined in [12]. Recently, the Verma modules over the W (2, 2) Lie algebra was investigated in [18]. A quantum group structure of the q-deformed W (2, 2) Lie algebra was also given in [11]. Nonetheless, there are still plenty rooms for our reconsideration on this matter, since our definition of the q-deformation W q (Proposition 2.2) of this Lie algebra with its origin from physics, is rather 1 Supported by NSF grants 10825101 of China * Corresponding author: lmyuan@mail.ustc.edu.cn different and thus the quantum group (associated with the deformation) constructed in this paper seems to be new.
The W (2, 2) Lie algebra (denoted by W) considered in the present paper is an infinite dimensional Lie algebra with basis L n , W n (for n ∈ Z) and the Lie brackets One sees that it is different from that defined in [37], since we drop here the central element. But our next aim is to develop the central extensions of the q-deformation W q .
In this paper, an explicit realization (Lemma 2.1) of the W (2, 2) Lie algebra defined in (1.1) is given using the famous bosonic and fermionic oscillators in physics ((2.1) and (2.3)). As a result, the q-deformation W q (Proposition 2.2) of the W (2, 2) Lie algebra is obtained by exact calculations.
Based on this, Hopf algebraic structures of W q are proposed and thus the quantum group of W q are completely determined. It seems to us that the results in our paper may be of some potential use in mathematical physics. Throughout this paper, F denotes a field of characteristic zero, F * the multiplicative group of nonzero element of F. All vector spaces and algebras are assumed to be over F. Denote by Z, Z + , Z * the sets of integers, nonnegative, nonzero integers, respectively. §2. Realization of W and its q-Deformation W q In this section, we propose a version of realizing the W(2, 2) Lie algebra defined in (1.1). Based on this, a quantum deformation of this Lie algebra is constructed. One sees that the q-deformed W(2, 2) Lie algebra W q here is rather different from that given in [11].
In the oscillator, the bosonic oscillator a and its hermitian conjugate a + obey the commutation relations: More details can be consulted in [23]. for arbitrary n, where the general notation Furthermore, one can also deduce that for all m, n ∈ Z.
Proof. Obviously, equation (2.9) holds for m = n since both sides are equal to 0. Assume that n = m. Then it follows from (2.4), (2.6), (2.7) and (2.8) that Hence equation (2.9) holds for all m, n ∈ Z. Similarly, one can get equations (2.10) and (2.11) using the facts that b + commutes with both a and a + and that (b + ) 2 = 0.
Proposition 2.2 says that the algebra with generators L n , W n (for n ∈ Z) and the relations (2.9)-(2.11) realizes the W (2, 2) Lie algebra W in the q → 1 limit. We call it the q-deformation of the W (2, 2) Lie algebra W, which will be denoted by W q in the sequel. for all m, n ∈ Z. From these we see that the algebra generated by L n and W n in (2.4) with relations (2.12)-(2.14) is also a realization of the W (2, 2) Lie algebra when q → 1, which is called the generalized q-deformation of W (2, 2) Lie algebra and is denoted by W c q . Note that one gets W q (see Proposition 2.2) when c = −1. §3. Quantum Group Structures of W q In this section, we give a direct construction of the Hopf algebraic structures of the q-deformed W (2, 2) Lie algebra W q defined in previous section.
Fix a q ∈ F * such that q is not a root of unity. Then U q is defined as the associative algebra (with 1 and over F) with generators T , T −1 , L n ,W n for n ∈ Z and relations: Before giving the construction of the Hopf algebraic structures on U q , we have to check whether or not these six relations (R1)-(R6) above ensure a nontrivial associative algebra U q . The following proposition gives a positive answer.
Let I be the two-sided ideal of T (M ) generated by for all m, n ∈ Z and where T −n = (T −1 ) n . Set S(M ) := T (M )/I. It is obvious that S(M ) is also a Z-graded associative algebra with a basis where d, d ′ , k i , l j , i p , j q ∈ Z + , (i, p = 1, 2, · · · m; j, q = 1, 2, · · · n); i 1 < i 2 < · · · < i m , j 1 < j 2 < · · · < j n . LetJ be another two-sided ideal of T (M ) generated by the elements of form together with that in (3.1) and (3.2). Then set Our aim is to show thatB defined in (3.4) is also a basis of U q . Let be a subset of T (M ) and let U ′ be the subspace of T (M ) spanned byB ′ . We now claim that However, v (m) is a nonzero linear combination of the elements inB ′ , it is impossible to appear in ker π. This contradiction impliesB is a basis of U q . Since it is clear that U q ∼ = U q /J, where J is the two-sided ideal of U q generated by T T −1 − 1, we obtain that U q is a nontrivial associative algebra with basisB where d ∈ Z, k i , l j , i p , j q ∈ Z + , (i, p = 1, 2, · · · m; j, q = 1, 2, · · · n); i 1 < i 2 < · · · < i m , j 1 < j 2 < · · · < j n .
With the above proposition in hand, we can safely proceed with the construction of the Hopf algebraic structures on U q now. This will be done by several lemmas below.

Lemma 3.4 There is a unique homomorphism of F-algebras
Proof. It is clear that ∆(T m ) = T m ⊗ T m for arbitrary m ∈ Z. We have to show that ∆(T ), ∆(T −1 ), ∆(L n ) and ∆(W n ) satisfy the relations (R1)-(R6). This is trivial for (R1). For (R2) and (R3) it follows directly from (3.9)- (3.12). Now look at (R4): we have Similarly, we get Then it follows Hence, (R4) is preserved by ∆ and so is it for (R5) and (R6), which can be checked by the similar method. That means ∆ is an algebraic homomorphism. Consequently, U q is a bialgebra.

Lemma 3.5
The comultiplication ∆ on U q is coassociative.
Proof. We simply have to check that all the generators of U q are mapped both ways by (1 ⊗ ∆)∆ and (∆ ⊗ 1)∆ to the same image, which simply involves straightforward calculations. We shall take L n as an example (others can be done similarly).
The homomorphism ε from Lemma 3.6 is called the counit of U q . namely, the map S preserves (R4). One can similarly check that (R5) and (R6) are also preserved by S. So there is indeed a homomorphism S : U q → U opp q or an antihomomorphism S : U q → U q satisfying (3.14)-(3.16). Now S 2 is an ordinary homomorphism from U q to U q . One can check easily on the generators that S 2 = id, which implies that S is bijective.

Lemma 3.7 There is a unique antiautomorphism
The map S in Lemma 3.7 is called the antipode of U q . It is clear that the inverse S −1 of S is an antiautomorphism, which is given by Lemma 3.8 The following diagrams are commutative L n → L n ⊗ T n + T n ⊗ L n → L n ⊗ T −n + T n ⊗ (−T −n L n T −n ) → L n T −n − L n T −n = 0, W n → W n ⊗ T n + T n ⊗ W n → W n ⊗ T −n + T n ⊗ (−T −n W n T −n ) → W n T −n − W n T −n = 0, as predicted by the diagram.
To conclude the proof we have to check: If f (u) = ι • ε(u) and f (v) = ι • ε(v) for u, v ∈ U q , then also f (uv) = ι • ε(uv). That is not obvious, since S and m are not ring homomorphisms. We We assume that f (v) = ι • ε(v), so this element is a scalar multiple of 1 and thus central in U q . Therefore In general, an F-algebra A together with algebra homomorphisms ∆ : A → A⊗A and ε : A → F and a linear map S : A → A is called a Hopf algebra, if ∆ is coassociative and if the diagrams in Lemmas 3.6 and 3.8 (with U q is replaced by A) commute. One calls ∆ the comultiplication, ε the counit and S the antipode of the Hopf algebra. A Hopf algebra A is called cocommutative, if P • ∆ = ∆ with P (u ⊗ v) = v ⊗ u for all u and v in A. So the Lemmas 3.4-3.8 say: Theorem 3.9 (U q , ∆, ε, S) defined by (R1)-(R6) and (3.9)-(3.16) is a Hopf algebra, which is neither cocommutative nor commutative. Corollary 3.10 As vector spaces, one has where U q = U (W q ) is the enveloping algebra of W q generated by L n and W n (n ∈ Z) with relations (2.9)-(2.11).