On q-deformed infinite-dimensional n-algebra

The $q$-deformation of the infinite-dimensional $n$-algebra is investigated. Based on the structure of the $q$-deformed Virasoro-Witt algebra, we derive a nontrivial $q$-deformed Virasoro-Witt $n$-algebra which is nothing but a sh-$n$-Lie algebra. Furthermore in terms of the pseud-differential operators on the quantum plane, we construct the (co)sine $n$-algebra and the $q$-deformed $SDiff(T^2)$ $n$-algebra. We prove that they are the sh-$n$-Lie algebras for the case of even $n$. An explicit physical realization of the (co)sine $n$-algebra is given.

by carrying out a Sugawara construction on a q-analogue of an infinite dimensional Heisenberg algebra. It is well-known that there is a remarkable connection between the Virasoro algebra and the Korteweg-de Vries (KdV) equation [10,11]. For the q-deformed Virasoro algebra, Chaichian et al. [12] showed that it generates the sympletic structure which can be used for a description of the discretization of the KdV equation.
The Nambu 3-algebra was introduced in [13,14] as a natural generalization of a Lie algebra for higher-order algebraic operations. Recently Bagger and Lambert [15], and Gustavsson [16] (BLG) found that 3-algebras play an important role in world-volume description of multiple M2-branes. Due to BLG theory, there has been considerable interest in the 3-algebra and its application. More recently there has been the progress in constructing the infinite-dimensional 3-algebras, such as V-W [17,18], Kac-Moody [19] and w ∞ 3-algebras [20,21]. Moreover the relation between the infinite-dimensional 3-algebras and the integrable systems has also been paid attention [22,23].
The structure and property of q-deformed algebra are now very well understood. But for the q-deformed 3-algebra, it has not been dealt with in such detail. Much less is known about its structure and property. Recently Curtright et al. [17], constructed a V-W algebra through the use of su(1,1) enveloping algebra techniques. It is worthwhile to mention that this ternary algebra depends on a parameter z and is only a Nambu-Lie algebra when z = ±2i. Ammar et al. [24] presented a q-deformation of this 3-algebra and noted it carrying the structure of ternary Hom-Nambu-Lie algebra.
In order to achieve a better understanding of the q-deformed n-algebra, in this Letter we focus on the q-deformation of the null V-W n-algebra. Based on the well-known structure of the q-deformed V-W algebra, we construct the nontrivial q-deformed V-W n-algebra and explore its intriguing features.

n-Lie algebra and sh-n-Lie algebra
To avoid too many technicalities, we will give here only the definitions of n-Lie algebra [25] and sh-n-Lie algebra [26].
The notion of n-Lie algebra or Filippov n-algebra was introduced by Filippov [25]. It is a natural generalization of Lie algebra. For a linear space V , an n-Lie algebra structure is defined by a multilinear map called Nambu bracket [., · · · , .]: V ⊗n → V satisfying the following properties: (1). Skew-symmetry (1) (2). Fundamental identity (FI) or Filippov condition For the case of 3-algebra, the corresponding fundamental identity is We have already seen that an n-Lie algebra A is a vector space A endowed with an n-ary skew-symmetric multiplication satisfying the FI condition. We now turn to the notion of sh-n-Lie algebra.
In terms of the Lévi-Cività symbol, i.e., ǫ i 1 ···ip the sh-Jacobi's identity (4) can also be expressed as When n = 3, the sh-Jacobi's identity (4) becomes We have briefly introduced the n-Lie algebra and sh-n-Lie algebra. It should be noted that any n-Lie algebra is a sh-n-Lie algebra, but a sh-n-Lie algebra is a n-Lie algebra if and only if any adjoint operator is a derivation.
3 q-deformed V-W 3-algebra 3.1 q-deformed V-W algebra As a start before investigating the q-deformed 3-algebra, let us recall the case of q-deformed algebra. The deformation of the commutator is defined by It possesses the following properties [4,9]: and the q-Jacobi identity The Virasoro algebra is an infinite dimensional Lie algebra and plays important roles in physics. The V-W algebra is indeed the centerless Virasoro algebra. It is given by For the generators L 0 , L 1 and L −1 , it can be easily seen that they satisfy the SU (1, 1) algebra: To construct the deformed V-W algebra, let us take the q-deformed generators where the q-deformed oscillator is deformed by the following relations [27]- [29]: Substituting the q-generators (13) into the commutator (8) and using the q-deformed oscillator (14), it leads to the so-called q-deformed V-W algebra [1] where [k] = q k −q −k q−q −1 . In the limit q → 1, (15) reduces to the V-W algebra (11) From q-deformed V-W algebra (15), we note that the generators L 0 , L 1 and L −1 of (13) comprise the SU q (1, 1) algebra, This q-deformed su(1, 1) algebra has been well investigated in the literature [1,30].
Let us define the star product by Then we have By means of (18), one can confirm the following q-Jacobi identity [4] satisfied by the q-deformed V-W algebra (15): 3.2 q-deformed V-W 3-algebra and su q (1, 1) 3-algebra We have introduced the q-deformed algebra in the previous subsection. Let us now turn our attention to the case of 3-algebra. The operator Nambu 3-bracket is defined to be a sum of single operators multiplying commutators of the remaining two [13], i.e., For the q-deformed V-W algebra (15), we have already seen that the q-Jacobi identity (19) is guaranteed to hold. It is worth to emphasize that the star product (17) plays a pivotal role in the q-Jacobi identity. In terms of the star product (17), let us define the q-3-bracket as follows: By means of (15) and (17), we may derive the following q-deformed 3-algebra from (21): Performing lengthy but straightforward calculations, we find that (22) satisfies the sh-Jacobi's identity (7), but the FI condition (3) does not hold. It is easy to verify that the skew-symmetry holds for this ternary algebra Therefore the q-deformed V-W 3-algebra (22) is indeed a sh-3-Lie algebra. In the limit q → 1, (22) reduces to the null 3-algebra derived in [18], The FI condition (3) is trivially satisfied for this null 3-algebra.
It is known that the su(1, 1) algebra is a subalgebra of V-W algebra. From (24), we have the null su(1, 1) 3-algebra, Let us turn to discuss the q-deformation of (25). Taking the generators to be L 0 , L 1 and L −1 in (22), it leads to the su q (1, 1) 3-algebra An intriguing feature is that for the null su(1, 1) 3-algebra, its q-deformed 3-algebra is nontrivial.
Moreover it is worth to emphasize that this su q (1, 1) 3-algebra satisfies the FI condition (3).

q-deformed V-W n-algebra
Now encouraged by the possibility of constructing the nontrivial sh-3-Lie algebra (22), it would be interesting to study further and see whether one could construct the q-deformed V-W nalgebra with a genuine sh-n-Lie algebra structure. In this section we give affirmative answer to this question.
The n-bracket with n ≥ 3 is defined by Here we denote a notational convention used frequently in the rest of this paper. Namely for any arbitrary symbol Z, the hat symbolẐ stands for the term that is omitted.
Let us define a q-n-bracket as follows: where the general star product is given by in which (x = 2, y = −1) for n = 3, (x = n − 1, y = −2) for odd n ≥ 4 and (x = n, y = 0) for even n ≥ 5. As done in the case of q-3-bracket (21), we introduce the general star product (29) into the q-n-bracket here. It should be noted that the general star product (29) will play an important role in deriving the desired q-deformed V-W n-algebra.
Let us confirm this by the mathematical induction for n. Equation (22) indicates that (30) is satisfied for n = 3. We suppose (30) is satisfied for n. By means of (27), we have q xi 1 +(2+y)Σ n j=2 i j +2 · · · q xis+(2+y)Σ n j=1,j =s i j +2 · · · q xi l +(2+y)Σ n Substituting (x = n − 1, y = −2) for odd n and (x = n, y = 0) for even n into (31), respectively, we find that the determinate A is zero. After a straightforward calculation for the second determinate in (31), we obtain the explicit form of (n + 1)-bracket (31) which shows that (30) is satisfied for n + 1. Now the proof is completed.
Let us turn to the case of the sh-Jacobi's identity with respect to the q-n-bracket (30). We first focus on (30) with odd n. In terms of the Lévi-Cività symbol (5), we can rewrite (2n + 1)-bracket (30) as Then let us use the expression (32) to calculate L i 1 , · · · , L i 2n+1 , L i 2n+2 , · · · , L i 4n+1 . It leads Substituting (33) into the left-hand side of (6), we obtain where the power of q is given by and the following formula is useful in simplifying expression: From the expression of α (35), we observe that the coefficients of two different j µ should be equal. Since ǫ is completely antisymmetric, it is easy to see that (34) equals zero.
It indicates that the sh-Jacobi's identity is satisfied by (30) with odd n. For the case of (30) with even n, by the similar way, we can confirm the corresponding sh-Jacobi's identity. Taking the above results, we may conclude that the sh-Jacobi's identity (4) does hold for (30). Since the structure constants are determined by the the determinate, n-bracket (30) is anticommutative.
Based on the above analysis, it is clear that the q-deformed V-W n-algebra is indeed a sh-n-Lie algebra.
As an example, let us list first few q-deformed V-W n-algebras as follows: For the q-deformed V-W 3-algebra, we note that there exists a nontrivial sub-3-algebra, i.e., su q (1, 1) 3-algebra. As to the case of the q-deformed V-W n-algebra (30), we can easily see that this sh-n-Lie algebra only admits the null su q (1, 1) n-algebra for n ≥ 4.

Summary
The V-W algebra is the centerless Virasoro algebra. Its q-deformation has been well investigated in the literature. One has already known that in the usual way, the V-W n-algebra is null. In this paper, we investigated the q-deformation of the null V-W n-algebra and constructed the nontrivial q-deformed V-W n-algebra. It is of interest to note that it satisfies the sh-Jacobi's identity, but the FI condition fails. Thus this q-deformed V-W n-algebra is indeed a sh-n-Lie algebra. Furthermore we pointed out that a special case is that of n = 3. For the q-deformed Virasoro-Witt 3-algebra, we found that there exists a nontrivial finite-dimensional sub-3-algebra, i.e., su q (1, 1) 3-algebra.
Our investigation revealed a deep connection between the q-deformed infinite-dimensional n-algebra and the sh-n-Lie algebra. It sheds new light on the sh-n-Lie algebra. It would be interesting to study further and see whether there exist the central extension terms for the sh-n-Lie algebra derived in this paper. Furthermore the application of this sh-n-Lie algebra in physics might be of interest.