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Journal of Generalized Lie Theory and Applications
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Quantization of the q-analog Virasoro-like algebras

Yongsheng CHENG1* a and Yiqian SHI 2

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

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

*Corresponding Author:
Yongsheng CHENG
Institute of Contemporary Mathematics & College of Mathematics and Information Science,
Henan University, Kaifeng 475004, China
E-mail:
[email protected]

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

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Abstract

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.

Introduction

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 basisequation and brackets

equation

for equation were considered, where 0 ≠ q ∈ C is a fixed non-root of unity. Here we treat L0;0 as zero. Obviously, the Lie algebra equation is Z2-graded (however its structural constant equationβ2 is not linearly dependent on the gradings ; β; in this case, the Lie algebra equation 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

equation

equation that is

equation

equation

Obviously equation

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

equation

equation

where in general equation is the binomial coecient.

Denote by (U(equation),μ,τ,Δ0; S0; ε0) 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 S0 and the counit ε0 are respectively deβned by

equation

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

Defnition 1.2. Let (H,μ,τ,Δ0; S0; ε0) be a Hopf algebra over a commutative ring. An element equation is called Drinfel'd twist element, if it is invertible such that

equation

Lemma 1.3. Let (H,μ,τ,Δ0; S0; ε0) be a Hopf algebra over a commutative ring, and let equation be a Drinfel'd twist element of equation then

equation

equation

equation

Let (U(g),μ,τ,Δ0; S0; ε0) 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,Δ0; S0; ε0) is called a quantization of (U(g),μ,τ,Δ0; S00) by a Drinfel'd twist element equation, if U(g)[[t]]=tU(g)[[t]] ≅ U(g) and equationis determined by its r-matrix r.

equation

The following result is obtained in [9].

Lemma 1.5. There is a triangular Lie bialgebra structure on the Lie algebras equation given by the r-matrix equation 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,Δ0; S0,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

equation

where

equation

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

equation

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

equation

equation

Proof of the main results

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(equation):

equation

equation

Proof. Since [T;Lβ] = cLβ, we have LβT = (T -c)Lβ. 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

equation

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

equation

equation

equation

Similarly, we can obtain (2.5).

For a 2 F, we set

equation

where t denotes a formal variable. Denote equation Since equation

equation

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

equation

Therefore the elements equation are invertible elements with equation

Proof. Using the formula (1.5), we have

equation

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

equation

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

equation

In particular, one has

equation

Proof. In order to get the result, we want to use induction. Since equation 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,

equation

Therefore, the result is proved by induction.

Proposition 2.4. equation is a Drinfel'd twist element of (U(L)[[t]]μ,τ,Δ0; S0; ε0), that is equation satisβes (1.6) and (1.7).

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

equation

and on the other hand,

equation

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

equation

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 ofequation in both sides are equal. So the result follows.

Lemma 2.5. One has for any equation

equation

where

equation

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

equation

equation

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

equation

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

equation

equation

equation

Using (1.4) and (2.5), we have

equation

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

equation

Finally,

equation

equation

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, Lβequation, i = 1; 2. First, using (2.7), (2.12) and (2.10), we have

equation

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

equation

Using (2.7) and (2.11), we have

equation

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

equation

equation

This completes the proof of the theorem.

Acknowledgments

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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