Medical, Pharma, Engineering, Science, Technology and Business

**Yongsheng CHENG ^{1*} a and Yiqian SHI ^{2}**

^{1}Institute of Contemporary Mathematics & College of Mathematics and Information Science, Henan University, Kaifeng 475004, China

^{2}Department of Mathematics, University of Science and Technology of China, Hefei 230026, China

- *Corresponding
*Aut*hor: - Yongsheng CHENG

Institute of Contemporary Mathematics & College of Mathematics and Information Science,

Henan University, Kaifeng 475004, China

E-mail:

**Received date: ** April 14, 2009;** Revised date:** October 03, 2009

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

We use the general method of quantization by Drinfel'd twist element to quantize explicitly the Lie bialgebra structures on the q-analog Virasoro-like algebras studied in Comm. Algebra, 37 (2009), 1264{1274.

The study of Lie bialgebras [1,2] is now well established as an inβnitesimalization of the notion of a quantum group or Hopf algebra. A Lie bialgebra is a Lie algebra g provided with a Lie cobracket which is related to the Lie bracket by a certain compatibility condition. According to quantum groups theory, a quantum group is essentially a formal deformation of the universal enveloping algebra of a Lie algebra g, the semiclassical structure associated with such a deformation is a Lie bialgebra structure on g. Constructing quantizations of Lie bialgebras is an important method to produce new quantum groups. Using the method twisting the coproduct by a Drinfel'd twist element but keeping the product unchanged, Grunspan [3] presented the quantization of a class of inβnite dimensional Lie algebras con- taining Virasoro algebras studied in [4] (see also [5,6]). Using the same technique, Hu and Wang [7] quantized some Lie algebras presented in [8]. In a recent paper [9], the Lie bialgebra structures of q-analog Virasoro-like algebras L with the basis and brackets

for were considered, where 0 ≠ q ∈ C is a fixed non-root of unity. Here we treat L0;0 as zero. Obviously, the Lie algebra is Z^{2}-graded (however its structural constant β2 is not linearly dependent on the gradings ; β; in this case, the Lie algebra is called *non-linear*). This Lie algebra is closely related to the Virasoro and Virasoro-like algebras and the Lie algebras of Cartan type *S *and *H* (cf. [15,16]), which is probably why this type of Lie algebras has attracted some attentions in the literature (cf. [10,11,12,13,14,17,18]).

In this paper, we will use the techniques developed in [3,7] to construct the quantization of this type of bialgebra. However, since in our case the Lie algebra is *non-linear*, some of our arguments may render rather technical.

We βx a βeld F of characteristic zero. Let *A* be a unitary *R-*algebra (*R* is a ring). For z ∈ A, n ∈ Z, we set

that is

Obviously

The following lemma can be found in [3].

**Lemma 1.1.** *Let z be any element of a unitary F-algebras A. For a; d ∈ F, and m; n; r ∈ Z, one has*

where in general is the binomial coecient.

Denote by (U(),μ,τ,Δ_{0}; S_{0}; ε_{0}) the natural Hopf algebra structure on U(L) (the univer- sal enveloping algebra of the Lie algebra L), that is, the coproduct 0, the antipode S_{0} and the counit ε_{0} are respectively deβned by

The following deβnition and well-known result can be found in [2].

**Defnition 1.2.** Let (H,μ,τ,Δ_{0}; S_{0}; ε_{0}) be a Hopf algebra over a commutative ring. An element is called Drinfel'd twist element, if it is invertible such that

Lemma 1.3. Let (H,μ,τ,Δ_{0}; S_{0}; ε_{0}) be a Hopf algebra over a commutative ring, and let be a Drinfel'd twist element of then

Let (U(g),μ,τ,Δ_{0}; S_{0}; ε_{0}) be the natural Hopf algebra structure, where g is a triangular Lie bialgebra, and denote by U(g)[[t]] an associative F-algebra of formal power series with coecients in U(g). Naturally, U(g)[[t]] is equipped with an induced Hopf algebra structure arising from that on U(g).

**Defnition 1.4.** For a triangular Lie bialgebra g over F, the Hopf algebra (U(g)[[t]],μ,τ,Δ_{0}; S_{0}; ε_{0}) is called a quantization of (U(g),μ,τ,Δ_{0}; S_{0},ε_{0}) by a Drinfel'd twist element , if U(g)[[t]]=tU(g)[[t]] ≅ U(g) and is determined by its *r*-matrix *r*.

The following result is obtained in [9].

**Lemma 1.5. ***There is a triangular Lie bialgebra structure on the Lie algebras given by the r-matrix where T and E are deβned in (1:8).*

The main result of this paper is the following theorem.

**Theorem 1.6. ***Let L be the q-analog Virasoro-like algebras with [T;E] = E (cf. (1.8)), then there exists a noncommutative and noncocommutative Hopf algebra structure (U(L)[[t],μ,τ,Δ _{0}; S_{0},ε) on U(L)[[t]], such that U(L)[[t]]=tU(L)[[t]] = U(L), which preserves the product and the counit of U(L)[[t]], but the coproduct and antipode are deβned by*

where

In fact, we can introduce the operator it is easy to check that

Thus, (1.9) and (1.11) in Theorem 1.6 can be rewritten as

From above, in order to quantize the Lie bialgebra structures on q-analog Virasoro-like algebras, the key is to construct the Drinfel'd twisting, thus we have to do some necessary computation.

**Lemma 2.1. ***Let L be the q-analog Virasoro-like algebras. The following equations hold in U():*

**Proof. **Since [T;L_{β}] = cL_{β}, we have L_{β}T = (T -c)L_{β}. It is easy to see that (2.1) is true for m = 1. We can suppose that the βrst equation of (2.1) is true for m, then for m+1, we have

Thus we get (2.1) by induction on m. The second equation in (2.1), (2.2) and (2.3) can be veriβed in a similar way. Since

Similarly, we can obtain (2.5).

For a 2 F, we set

where *t* denotes a formal variable. Denote Since

**Lemma 2.2. **For a; d 2 C, one has

Therefore the elements are invertible elements with

**Proof. **Using the formula (1.5), we have

For the second equation, using (2.2) and (1.5), we have

**Lemma 2.3.*** For any positive integer m and any a 2 F, one has*

*In particular, one has*

**Proof. **In order to get the result, we want to use induction. Since it is easy to see that the result is true for m = 1; suppose that it is true for m, then it is enough to consider the condition for m + 1,

Therefore, the result is proved by induction.

**Proposition 2.4. ** *is a Drinfel'd twist element of (U(L)[[t]]μ,τ,Δ _{0}; S_{0}; ε_{0}), that is satisβes (1.6) and (1.7).*

**Proof.** The proof of (1.7) is easy, we just need to check (1.6). Since

and on the other hand,

thus, to verify (1.6), it suces to show for a βxed m that

Now, x r, s, q such that r +s = m, 0 ≤ q ≤ s, set i = q, i+k = s, then we have j = m-q, j - k = r. We see that the coecients of in both sides are equal. So the result follows.

**Lemma 2.5. **One has for any

where

**Proof. **By the second equation of (2.1) we have

this prove (2.12). For (2.10), using (2.4), we have

this proves (2.10). The following two equations give the proofs of (2.11) and (2.12):

Using (1.4) and (2.5), we have

which gives (2.12). The equations (2.14) and (2.15) follow from the following computations:

Finally,

which proves the last equation of the lemma.

Now we can prove our main theorem in this paper.

**Proof of Theorem 1.6. **For arbitrary elements, L_{β} ∈ , i = 1; 2. First, using (2.7), (2.12) and (2.10), we have

Using (2.7), (2.12) and (2.13), we have

Using (2.7) and (2.11), we have

Using (2.7), (2.14), (2.15) and (2.16), we have

This completes the proof of the theorem.

This work is supported by the National Science Foundation of China (No. 10825101), the Postdoctoral Science Foundation of China (No. 20090450810), the Natural Science Foun- dation of Henan Provincial Education Department of China (No. 2010B110003) and the Natural Science Foundation of Henan University of China (No. 2009YBZR025).

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