One parameter family of Jordanian twists

We propose an explicit generalization of the Jordanian twist proposed in $r$-symmetrized form by Giaquinto and Zhang. We present explicit formulas for the corresponding star product and twisted coproduct. Finally, we show that our generalization coincides with the twist obtained from the simple Jordanian twist by twisting by a 1-coboundary.


Introduction
Drinfeld twists of Hopf algebras [1] provide a systematic way of producing new examples in noncommutative geometry. Given a Hopf algebra H with a coproduct ∆ 0 and a counit ǫ 0 and given an element F ∈ H ⊗ H satisfying the 2-cocycle condition and the normalization (counitality) condition [2,3,4], a new coproduct ∆ F (−) = F −1 ∆ 0 (−)F is coassociative due the 2-cocycle condition, and moreover H as an algebra, together with the new coproduct ∆ F becomes a new twisted Hopf algebra H F . Along with a Hopf algebra, many associated constructions like its representations, comodules, module algebras and so on, are twisted as well, using standard formulas involving the twist F . The systematic nature of twisting procedure makes it suitable for finding new physical models with the Hopf algebra covariance built in.
Consider the universal enveloping algebra of the 2-dimensional solvable Lie algebra with generators H, E with [H, E] = E. Define Giaquinto and Zhang in [6], Theorem 2.20 1 , proposed the Jordanian twist [5] in r-symmetrized form This twist can also be written as We shall use different notation in this paper, namely This suggests interpretation of D as relativistic dilation operator and P as momentum in some applications. We introduce a family of twists F −1 GZ,u , parametrized by parameter u, via an explicit series (6). This family interpolates between the Jordanian twists Our main interest in Jordanian twists is due to their appearance [5,7] in the study of κ-deformed Minkowski space (where also the intepretation of D and P as dilation and momentum makes sense), where κ is viewed as being linked to the scale of quantum gravity [8,9,10]. 2-cocycles can often be modified using coboundary twist T ω involving a 1cocycle ω. In an earlier paper [11], this procedure has been used to obtain certain twist F −1 R,u induced with coboundary twist [3] for every u. In that context it has been written in a form of product of three exponential factors, see also reference [12]. The fact that it is obtained via a coboundary twist from a 2-cocycle implies that it is itself a 2-cocycle.
Twists F GZ,u and F R,u generate the same Hopf algebra. It appears and it is proved in this paper that our generalized Giaquinto-Zhang twist F −1 GZ,u satisfies the same differential identity as the coboundary twist F −1 R,u which essentially shows that the two twists coincide. The importance of this result is that while F −1 GZ,u is introduced via explicit series expansion suitable for calculation, the very construction of F −1 R,u ensures that it is a 2-cocycle (which would be hard to prove directly from the definition of F −1 GZ,u ). The exposition is organized as follows. In Section 2 we define the interpolation F −1 GZ,u via an explicit expansion and show that it has claimed limits at u = 0, and then in subsection 2.1 compute the corresponding star product and the twisted coproduct ∆P µ . In 2.2 we introduce the noncommutative coordinates and their realization. Section 3 is dedicated to the Jordanian twists F R,u induced by coboundary twists, that are obtained from simple Jordanian twist F 0 (5) via twisting by a 1-cocycle. We start the section by introducing F R,u as a product of 3 exponential factors. Then we compute the corresponding deformed Hopf algebra in 3.1, introduce the corresponding noncommutative coordinates and realizations in 3.2 and compute the star products in 3.3. In Section 4 we present two different proofs both showing that F GZ,u equals F R,u . The first method in 4.1 is by showing the same ordinary differential equation and initial condition they satisfy. The second in 4.2 uses a comparison among the star products. Final Section 5 is the conclusion.

Generalization of Giaquinto-Zhang twist
We define the generalized Jordanian twist F −1 GZ,u via an explicit expansion, Twist F −1 GZ,u interpolates between F −1 0 and F −1 1 , see (5). For u −→ 0 one can easily see [14] that (6) reduces to For u = 1 For u = 1 2 this reduces to the twist introduced in

Star product and twisted coproduct ∆P
Introduce the action ⊲ of P and D on the space of formal power series in variables x µ , where µ = 0, 1, . . . , n, by formulas where v µ are such that v 2 ∈ {−1, 0, 1} and the Einstein summation rule is understood. We also denote x = (x µ ) and ∂ µ = ∂ ∂xµ . Star product is then defined for all formal power series f, g in x µ [13]. In particular, for f = e ikx and g = e iqx , where kx = k α x α and qx = q α x α are the elements of Minkowski space-time algebra and the function A is here implicitly defined. Using the actions of P and D on e ikx it follows that The following identities hold We calculate Note that It follows that Let now p µ = −i∂ µ be the momentum operator. Let us define ∆p µ by We want to show that ∆p µ is the deformed coproduct with respect to the twist where Using (19) and (17), we may rewrite (18) as and, after multiplying from the right by the denominator 1 We shall show the equality in (20) by splitting it into a sum of two equalities, (21) and (25), which are then separately proved. Descriptively, (21) involves those summands in expanding (20) where, in one of the factors, p µ is from the left side of the tensor product: To prove this equality, we first observe that by induction the equality [P, D] = P implies the commutation relation Hence i.e., We calculate the left-hand side of (21) as and the right-hand side of (21) as Comparing the terms of type P k ⊗ P l for all k and l, we find (21).
Analogously, we prove the equality of the remaining summands in (20), Now (21) and (25) add to (20). Hence this proves (18), that is ∆p µ = F GZ,u ∆ 0 (p µ ) F −1 Gz,u . Cooproduct ∆p µ satisfies the coassociativity condition Equation (18) can be rewritten as It follows that the partial derivatives of the star product compute to = iD µ (k, q)(e ikx * e iqx ) Knowing the partial derivatives (28) and the initial value (13) of the star product at x = 0, we obtain (e ikx * e iqx ) = e iDµ(k,q)x µ 1 where D µ (k, q) is given in (16). This star product is associative.

Interpolation between Jordanian twists induced with coboundary twist
Another construction for a generalized Jordanian twist is possible [11]. This twist, here denoted F R,u , has been introduced as a product of three exponential factors, (35) where u is a real parameter, u ∈ R. The symbol R in the subscript refers to the position of the dilatation generator in the formula, namely it is on the right with respect to P.
This form of the family of twists F R,u can also be easily obtained from a simple Jordanian twist using a transformation by a 1-cocycle. Indeed, according to Drinfeld ( [1,3]), if F is any normalized Drinfeld twist and ω R is any element in the Hopf algebra which is normalized, that is ǫ(ω R ) = 1, then the formula F ω := (ω −1 ⊗ ω −1 )F ∆(ω) defines a normalized Drinfeld twist again (that is, the 2-cocycle and counitality conditions are again satisfied), which is considered cohomologous to F 0 in the sense of nonabelian cohomology ( [3]). In particular, if F = 1 we get a 2-coboundary (ω −1 ⊗ ω −1 )F ∆(ω). If F = F 0 is a simple Jordanian twist, and ω = ω R = exp − u κ PD , we obtain the twist , see [11]. Thus, for any u, twist F R,u satisfies 2-cocycle and normalization condition. For u = 0, twist F R,u simplifies to F 0 and for u = 1 to F 1 .

Hopf algebra
Coalgebra sector of Hopf algebra H F R,u for deformation with F R,u has form Similar analysis as in Section 2 for ∆ F GZ,u p µ leads to the conclusion that ∆ F GZ,u D = ∆ F R,u D.

Noncommutative coordinates and realizations
In general, we consider the realizations of the form We can obtain the appropriate realization via the twist as followŝ

Star product
Using above realization ofx µ [13], we get e ikx * e iqx = e iDµ(u;k,q)xµ+iG(u;k,q) = e iDµ(u;k,q)xµ 1 where k and q belong to the n-dimensional Minkowski spacetime M 1,n−1 and where as in the equation (16), and finally Remark. Note that the quantum R-matrix computes to We point out that F −1 GZ,u and F −1 R,u lead to the same Hopf algebra, the same realizations of noncommutative coordinatesx µ and likewise for the star product e ikx * e iqx . This suggests that there must be some relation between the two twists, −k + l κ k+l (u−1) k P k D l ⊗(uP) l D k (46) Using following commutation relations and we find that the right-hand sides of (45) and of (46) agree, This shows that F −1 R,u and F −1 GZ,u as a function of parameter u satisfy the same ordinary differential equation, while the initial conditions agree. Indeed, at u = 0, Therefore F −1 R,u ≡ F −1 GZ,u .

Proof of the equality of the two twists
In the following proposition we state the conditions when two twists are equal, along with a simple proof.
Proposition. Let P be Poincaré Weyl algebra generated with momenta p µ , Lorentz generators M µν and dilatation D. Two twists F 1 ∈ U(P) ⊗ U(P) and F 2 ∈ U(P) ⊗ U(P) are identical if all star products are identical, i.e. for all f, g in Minkowski space time algebra Proof. If all star products are the same, F −1 1 and F −1 2 could differ by an element in the right ideal J 0 generated by the elements (x µ ⊗ 1 − 1 ⊗ x µ ) for all µ [18]. However, J 0 ∩ U(P) ⊗ U(P) = 0, hence F 1 = F 2 .
Since we already proved that the twists F R,u and F GZ,u give the same star products e ikx * e iqx , twists F R,u and F GZ,u must be identical. Moreover, we have proved that the noncommutative coordinatesx µ and twisted coproducts ∆p µ and ∆D from both twists are identical. Since F R,u satisfies the normalization and cocycle conditions, F GZ,u also satisfies them.

Conclusion
We have constructed a 1-parameter family F GZ,u (6) of Jordanian twists that interpolates between the simple Jordanian twists F 0 and F 1 defined in Equation (5). For u = 1 2 , F GZ,u= 1 2 coincides with F GZ [6]. We have calculated the corresponding star product e ikx * e iqx (44) and the corresponding deformed Hopf algebra. In Section 3, we have presented another interpolation between Jordanian twists cohomologous to F 0 via 1-cocycle depending on u [11]. It is pointed out that F −1 GZ,u and F −1 R,u generate the same star product and, consequently, the same deformed Hopf algebra. In Section 4, it is presented that F −1 GZ,u = F −1 R,u , that F GZ,u can be written in the form of product of three exponential factors. It automatically satisfies the cocycle condition as it is obtained from simple Jordanian twist by twisting by a 1cocycle [3]. This is a new result in the literature. We note that for twist F −1 GZ [6], the star product, an explicit form of twist F GZ and the deformed Hopf algebra structure were not known in the literature so far. Jordanian twists have been of interest in recent literature [26,27,28]. We note that our results could be useful in future applications of Jordanian twists.