One Parameter Family of Jordanian Twists

We propose an explicit generalization of the Jordanian twist proposed in r-symmetrized form by Giaquinto and Zhang. It is proven that this generalization satisfies the 2-cocycle condition. We present explicit formulas for the corresponding star product and twisted coproduct. Finally, we show that ou...

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Опубліковано в: :	Symmetry, Integrability and Geometry: Methods and Applications
Дата:	2019
Автори:	Meljanac, D., Meljanac, S., Škoda, Z., Štrajn, R.
Формат:	Стаття
Мова:	Англійська
Опубліковано:	Інститут математики НАН України 2019
Онлайн доступ:	https://nasplib.isofts.kiev.ua/handle/123456789/210306
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	One Parameter Family of Jordanian Twists / D. Meljanac, S. Meljanac, Z. Škoda, R. Štrajn // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 39 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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author	Meljanac, D. Meljanac, S. Škoda, Z. Štrajn, R.
author_facet	Meljanac, D. Meljanac, S. Škoda, Z. Štrajn, R.
citation_txt	One Parameter Family of Jordanian Twists / D. Meljanac, S. Meljanac, Z. Škoda, R. Štrajn // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 39 назв. — англ.
collection	DSpace DC
container_title	Symmetry, Integrability and Geometry: Methods and Applications
description	We propose an explicit generalization of the Jordanian twist proposed in r-symmetrized form by Giaquinto and Zhang. It is proven that this generalization satisfies the 2-cocycle condition. 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-cochain.
first_indexed	2025-12-07T21:25:05Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 15 (2019), 082, 16 pages One Parameter Family of Jordanian Twists Daniel MELJANAC †1, Stjepan MELJANAC †2, Zoran ŠKODA †3 and Rina ŠTRAJN †4 †1 Division of Materials Physics, Institute Rudjer Bošković, Bijenička cesta 54, P.O. Box 180, HR-10002 Zagreb, Croatia E-mail: Daniel.Meljanac@irb.hr †2 Theoretical Physics Division, Institute Rudjer Bošković, Bijenička cesta 54, P.O. Box 180, HR-10002 Zagreb, Croatia E-mail: meljanac@irb.hr †3 Department of Teachers’ Education, University of Zadar, Franje Tudjmana 24, 23000 Zadar, Croatia E-mail: zskoda@unizd.hr †4 Department of Electrical Engineering and Computing, University of Dubrovnik, Ćira Carića 4, 20000 Dubrovnik, Croatia E-mail: rina.strajn@unidu.hr Received April 16, 2019, in final form October 19, 2019; Published online October 25, 2019 https://doi.org/10.3842/SIGMA.2019.082 Abstract. We propose an explicit generalization of the Jordanian twist proposed in r- symmetrized form by Giaquinto and Zhang. It is proved that this generalization satisfies the 2-cocycle condition. 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-cochain. Key words: noncommutative geometry; Jordanian twist 2010 Mathematics Subject Classification: 53D55; 16T05 1 Introduction Drinfeld twists of Hopf algebras [7] provide a systematic way of producing new examples in noncommutative geometry. Given a Hopf algebra H with a coproduct ∆0, a counit ε0, and an element F ∈ H ⊗ H satisfying a 2-cocycle condition and a normalization (couni- tality) condition [3, 8, 9, 27], often called a Drinfeld twist, one defines a new coproduct, ∆F (−) = F−1∆0(−)F , which is coassociative due to the 2-cocycle condition. Moreover, H as an algebra, together with the new coproduct ∆F becomes a new twisted Hopf algebra HF . 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 the twisting procedure makes it suitable for finding new physical models with the Hopf algebra covariance built in. In 1989, a new construction of a deformation 2-cocycle is proposed by Coll, Gerstenhaber and Giaquinto in [4]. Their construction starts with a k-algebra A with multiplication µA and two derivations φ, ψ : A→ A satisfying [φ, ψ] = λψ for some λ ∈ k. The action of the 2-dimensional Lie algebra L generated by φ and ψ on A by derivations extends to a unique action . of the Hopf algebra U(L) on A making it a U(L)-module algebra. They prove [4, 12] that µA ◦ (φ⊗ψ) is a Hochschild 2-cocycle which may be integrated to yield a formal deformation of A. The deformed multiplication on A[[t]] is given by µA ◦ (1 ⊗ 1 + tφ ⊗ 1)1⊗ψ◦ (. ⊗ .) for λ = 1. This mailto:Daniel.Meljanac@irb.hr mailto:meljanac@irb.hr mailto:zskoda@unizd.hr mailto:rina.strajn@unidu.hr https://doi.org/10.3842/SIGMA.2019.082 2 D. Meljanac, S. Meljanac, Z. Škoda and R. Štrajn formula involves the element (1⊗ 1 + tφ⊗ 1)1⊗ψ = ∞∑ n=0 tnφn ⊗ ( ψ n ) = ∞∑ n=0 tn n! φn ⊗ ψ(ψ − 1) · · · (ψ − n+ 1) ∈ (U(L)⊗ U(L))[[t]], which is a (Drinfeld) 2-cocycle twist for the Hopf algebra U(L)[[t]]. They provide several exam- ples. Their construction is reanalyzed in detail in [12] and the 2-cocycle twist has been redis- covered in [35]. Algebras obtained by variants of their method are now often called Jordanian deformations. Most studied examples are a Jordanian deformation of the universal enveloping algebra U(sl(2)) (and its dual) and more general Jordanian quantum groups, leading to the cor- responding Jordanian classical r-matrices and quantum R-matrices (some of which were known before, e.g., [26, Example 1, due D. Gurevich] and [5, Section 2.2]). Regarding that U(sl(2)) can be embedded into Yangian Y (sl(2)), it is not surprising that more elaborate versions of Jordanian twists are used to obtain new deformations of Yangians [20, 23, 37], with applications to integrable models, chain models in particular [21]. A comprehensive study of a related class of classical r-matrices can be found in [19]. Here we present another approach to new Jordanian deformations. Closer to the setup of our paper, consider the universal enveloping algebra of the 2-dimensional solvable Lie algebra with generators H and E with [H,E] = E. Define H〈m〉 = H(H + 1) · · · (H +m− 1), H〈0〉 = 1. Giaquinto and Zhang in [10, Theorem 2.20]1 proposed the Jordanian twist [35] in r-symmetrized form F−1GZ = ∞∑ m=0 tm m! m∑ r=0 (−1)r ( m r ) Em−rH〈r〉 ⊗ ErH〈m−r〉. This twist can also be written as F−1GZ = ∞∑ m=0 tm ∞∑ r=0 (−E)m−r ( −H r ) ⊗ Er ( −H m− r ) = ∞∑ k,l=0 tk+l(−E)k ( −H l ) ⊗ El ( −H k ) . We shall use a different notation in this paper, namely E = P, H = −D, [D,P ] = −P. This suggests an interpretation of D as the relativistic dilation operator and P as the momentum in some applications. We introduce a family of twists F−1GZ,u, parametrized by parameter u, via an explicit series (2.1). This family interpolates between the Jordanian twists F−10 and F−11 , where F0 = exp ( − ln ( 1− 1 κ P ) ⊗D ) and F1 = exp ( −D ⊗ ln ( 1 + 1 κ P )) . (1.1) Our main interest in Jordanian twists is due to their appearance [2, 39] in the study of κ-deformed Minkowski space (where the intepretation of D and P as the dilation and momentum operators also makes sense), where κ is viewed as being linked to the scale of quantum gravity [24, 25, 38]. 1The twist F in [10] is renamed here as F−1. One Parameter Family of Jordanian Twists 3 Any Drinfeld twist F can be modified by any 1-cochain ω ∈ H, producing a new twist( ω−1 ⊗ ω−1 ) F∆(ω), see [27]. In an earlier paper [1], this procedure has been used to obtain a certain twist F−1R,u for every u. In that context, it has been written in the form of a product of three exponential factors, see also reference [31]. Regarding that it is obtained from a 2-cocycle by modification by a 1-cochain implies that it is itself a 2-cocycle. Twists FGZ,u and FR,u generate the same Hopf algebra. It is proved in this paper that our generalized Giaquinto–Zhang twist F−1GZ,u satisfies the same differential identity as F−1R,u, including the initial condition; consequently the two twists coincide. The importance of this result is that while F−1GZ,u is introduced via an explicit series expansion more suited for other calculations, the very construction of F−1R,u ensures that it is a 2-cocycle; we however also exhibit an elaborate proof of the 2-cocycle condition, directly from the definition of F−1GZ,u. The exposition is organized as follows. In Section 2, we define an interpolation F−1GZ,u via an explicit expansion and show that it has the claimed limits at u = 0 and u = 1. In Section 2.1, we prove directly from the definition that F−1GZ,u satisfies the 2-cocycle condition. In Section 2.2, we compute the corresponding star product and in Section 2.3 the twisted coproduct ∆pµ. In Section 2.4 we introduce noncommutative coordinates and their realizations. Section 3 is dedicated to the family FR,u of Jordanian twists obtained from a simple Jordanian twist F0 (1.1) via twisting by a 1-cochain. We start the section by introducing FR,u as a product of three exponential factors. Then we compute the corresponding deformed Hopf algebra in Section 3.1, introduce the corresponding noncommutative coordinates and realizations in Section 3.2 and compute the star products in Section 3.3. In Section 4, we present two different proofs both showing that FGZ,u equals FR,u. The first proof in Section 4.1 is by showing that they solve the same Cauchy problem (an ordinary differential equation with initial condition). The second proof in Section 4.2 uses a comparison among the star products. The final Section 5 is the conclusion. Appendix A is added presenting a proof of an identity used in the proof in Section 2.1 of the 2-cocycle condition for F−1GZ,u. 2 Generalization of the Giaquinto–Zhang twist We define the generalized Jordanian twist F−1GZ,u via an explicit expansion, F−1GZ,u = ∞∑ k,l=0 ( 1 κ )k+l ((u− 1)P )k ( D l ) ⊗ (uP )l ( D k ) . (2.1) The twist F−1GZ,u interpolates between F−10 and F−11 . For u→ 0 one can easily see [2] that (2.1) reduces to F−10 = ∞∑ m=0 ( −1 κ )m Pm ⊗ ( D m ) = eln(1− 1 κ P )⊗D. For u = 1 F−11 = ∞∑ m=0 ( 1 κ )m(D m ) ⊗ Pm = eD⊗ln(1+ 1 κ P ). For u = 1 2 this reduces to the twist introduced in [10], where t = 1 2κ , E = P, and H = −D. 4 D. Meljanac, S. Meljanac, Z. Škoda and R. Štrajn 2.1 2-cocycle condition Theorem 1. For arbitrary u, twists F−1GZ,u satisfy the 2-cocycle condition given by( (∆0 ⊗ 1)F−1GZ,u )( F−1GZ,u ⊗ 1 ) = ( (1⊗∆0)F−1GZ,u )( 1⊗F−1GZ,u ) . (2.2) Proof. If we write fn := ∑ k+l=n ((u− 1)P )k ( D l ) ⊗ (uP )l ( D k ) then F−1GZ,u = ∞∑ n=0 ( 1 κ )n fn with fn not depending on κ and f0 = 1 ⊗ 1. In terms of fi, the 2-cocycle condition becomes a sequence of equations for all n, n∑ i=0 ((∆0 ⊗ 1)fi) (fn−i ⊗ 1) = n∑ i=0 ((1⊗∆0)fi) (1⊗ fn−i). In the first order in 1/κ, f1 = (u− 1)P ⊗D + uD ⊗ P, (∆0 ⊗ 1)f1 + f1 ⊗ 1 = (1⊗∆0)f1 + 1⊗ f1, and in the second order, f2 = (u− 1)2P 2 ⊗ ( D 2 ) + (u− 1)uPD ⊗ PD + u2 ( D 2 ) ⊗ P 2, (∆0 ⊗ 1)f2 + ((∆0 ⊗ 1)f1)(f1 ⊗ 1) + f2 ⊗ 1 = (1⊗∆0)f2 + ((1⊗∆0)f1)(1⊗ f1) + 1⊗ f2. For general order n, it should hold that ∞∑ k1,k2,l1,l2=0 k1+k2=k, l1+l2=l, k+l=n [ ∆0 ( P k1 ( D l1 )) ⊗ P l1 ( D k1 )][ P k2 ( D l2 ) ⊗ P l2 ( D k2 ) ⊗ 1 ] = ∞∑ k1,k2,l1,l2=0 k1+k2=k, l1+l2=l, k+l=n [ P k1 ( D l1 ) ⊗∆0 ( P l1 ( D k1 ))][ 1⊗ P k2 ( D l2 ) ⊗ P l2 ( D k2 )] . This can be rewritten as k∑ k1=0 l∑ l1=0 [ ∆0 ( P k1 ( D l1 )) ⊗ P l1 ( D k1 )][ P k−k1 ( D l − l1 ) ⊗ P l−l1 ( D k − k1 ) ⊗ 1 ] = k∑ k1=0 l∑ l1=0 [ P k1 ( D l1 ) ⊗∆0 ( P l1 ( D k1 ))][ 1⊗ P k−k1 ( D l − l1 ) ⊗ P l−l1 ( D k − k1 )] , k∑ k1=0 l∑ l1=0 [ ∆0 ( P k1 )( P k−k1 ⊗ P l2−l1 )((D − k + k1)⊗ 1 + 1⊗ (D − l + l1) l1 ) ⊗ P l1 ( D k1 )] One Parameter Family of Jordanian Twists 5 × [( D l − l1 ) ⊗ ( D k − k1 ) ⊗ 1 ] = k∑ k1=0 l∑ l1=0 [ P k1 ⊗ ( ∆0 ( P l1 )( P k−k1 ⊗ P l−l1 ))] × [( D l1 ) ⊗ ( (D − k + k1)⊗ 1 + 1⊗ (D − l + l1) k1 )(( D l − l1 ) ⊗ ( D k − k1 ))] . Let us compare the terms of type PA⊗PB ⊗PC with A+B +C = k+ l = n on both sides. We see only the terms with C = l1 on the left-hand side and only the terms with A = k1 on the right-hand side. We also need to take into account ∆0 ( P k1 ) = k1∑ a=0 ( k1 a ) P k1−a ⊗ P a on the left-hand side and ∆0 ( P l1 ) = l1∑ b=0 ( l1 b ) P b ⊗ P l1−b on the right-hand side to obtain k∑ k1=0 k−a=A k1∑ a=0 ( k1 a )( (D − k + k1)⊗ 1 + 1⊗ (D − l + C) C )(( D l − C ) ⊗ ( D k − k1 )) ⊗ ( D k1 ) = l∑ l1=0 l−b=C l1∑ b=0 ( l1 b )( D l1 ) ⊗ ( (D − k +A)⊗ 1 + 1⊗ (D − l + l1) A )(( D l − l1 ) ⊗ ( D k −A )) , k∑ k−a=A k1=0 ( k1 k −A )( (D − k + k1)⊗ 1 + 1⊗ (D − l + C) C )(( D l − C ) ⊗ ( D k − k1 )) ⊗ ( D k1 ) = l∑ l−b=C l1=0 ( l1 l − C )( D l1 ) ⊗ ( (D − k +A)⊗ 1 + 1⊗ (D − l + l1) A )(( D l − l1 ) ⊗ ( D k −A )) for every k, l ∈ N0, and all A ≤ k, C ≤ l. In terms of the new variables x = D ⊗ 1⊗ 1, y = 1⊗D ⊗ 1, z = 1⊗ 1⊗D, and taking into account that k + l = n, we reduce the 2-cocycle condition to the identity ( x l − C ) k∑ k1=k−A ( k1 k −A )( x+ y − k − l + k1 + C C )( y k − k1 )( z k1 ) = ( z k −A ) l∑ l1=l−C ( l1 l − C )( y + z − k − l + l1 +A A )( x l1 )( y l − l1 ) (2.3) for all A ≤ k and C ≤ l. This is restated as (A.1), and then proved, in Appendix A. � For C = 0 the identity (2.3) reduces to k∑ k1=k−A ( k1 k −A )( y k − k1 )( z k1 ) = ( z k −A )( y + z − k +A A ) . 6 D. Meljanac, S. Meljanac, Z. Škoda and R. Štrajn 2.2 Star product We now introduce an action . of P and D on the space of formal power series in variables xµ, where µ = 0, 1, . . . , n, by formulas (P . f)(x) = −ivµ ∂f(x) ∂xµ , (D . f)(x) = xµ ∂f(x) ∂xµ , where the constants vµ are such that v2 ∈ {−1, 0, 1} and the Einstein summation rule is under- stood. We also denote x = (xµ) and ∂µ = ∂ ∂xµ . A star product ∗ is then defined as f ∗ g = m ( F−1GZ,u(.⊗ .)(f ⊗ g) ) for all formal power series f , g in xµ [30]. In particular, for f = eikx and g = eiqx, eikx ∗ eiqx = m ( F−1GZ,u(.⊗ .) ( eikx ⊗ eiqx )) =: eA(u;k,q,x), (2.4) where kx = kαxα and qx = qαxα are elements of the Minkowski space-time algebra, the function A is implicitly defined by (2.4) and m denotes the multiplication map on usual functions. Using the actions of P and D on eikx, it follows that P . eikx = (v · k)eikx, P . eiqx = (v · q)eiqx, where (v · k) = vαkα, (v · q) = vαqα. For j < l,( P j ( D l ) . eikx )∣∣∣∣ x=0 = 0,( Pn ( D n ) . eikx )∣∣∣∣ x=0 = (v · k)n,( Pn+1 ( D n ) . eikx )∣∣∣∣ x=0 = (n+ 1)(v · k)n+1. The following identities hold( D n ) . eikx = (ikx)n n! eikx, n ∈ N0,( P j ( D l ) . eikx )∣∣∣∣ x=0 = ( P j . (ikx)l l! eikx )∣∣∣∣ x=0 = ( j l ) (vαkα)j . Then we have( eikx ∗ eiqx )∣∣ x=0 = ∞∑ n=0 ( u(u− 1) κ2 )n (v · k)n(v · q)n = 1 1− u(u−1) κ2 (v · k)(v · q) = eA(u;k,q,x) \|x=0 . (2.5) We now calculate the partial derivatives of the star product, ∂µ ( eikx ∗ eiqx )∣∣ x=0 = ∞∑ j,l=0 ( 1 κ )j+l{ ikµ((u− 1)P )j . [ (v · k)l l! + (v · k)l−1 (l − 1)! ] eikx } One Parameter Family of Jordanian Twists 7 × { (uP )l . (v · q)j j! eiqx } ∣∣∣∣ x=0 + ∞∑ j,l=0 ( 1 κ )j+l { ((u− 1)P )j . (v · k)l l! eikx }{ iqµ(vP )l . [ (v · q)j j! + (v · q)j−1 (j − 1)! eiqx ]}∣∣∣∣ x=0 = i(kµ + qµ) ∞∑ n=0 ( u(u− 1) κ2 )n (v · k)n(v · q)n + i(kµ + qµ) ∞∑ n=0 ( u(u− 1) κ2 )n n(v · k)n +(v · q)n + i [ kµ u κ (v · q) + qµ (u− 1) κ (v · k) ] ∞∑ n=0 ( u(u− 1) κ2 )n (n+ 1)(v · k)n(v · q)n = iDµ(k, q) 1 1− u(u−1) κ2 (v · k)(v · q) and ∂µ ( eikx ∗ eiqx )∣∣ x=0 = i ( kµ ( 1 + u κ (v · q) ) + qµ ( 1 + u− 1 κ (v · k) )) × ∞∑ n=0 ( u(u− 1) κ2 )n (n+ 1)(v · k)n(v · q)n = iDµ(k, q) 1 1− u(u−1) κ2 (v · k)(v · q) = iDµ(k, q) ( eikx ∗ eiqx )∣∣ x=0 . Note that iDµ(k, q) = ( ∂A(u; k, q, x) ∂xµ )∣∣∣∣ x=0 . It follows that Dµ(k, q) = kµ ( 1 + u κ(v · q) ) + qµ ( 1 + u−1 κ (v · k) ) 1− u(u−1) κ2 (v · k)(v · q) . (2.6) 2.3 Twisted coproduct ∆(pµ) Let now pµ = −i∂µ be the momentum operator. Let us define ∆pµ by ∆pµ = Dµ(p⊗ 1, 1⊗ p) = pµ ⊗ ( 1 + u κP ) + ( 1 + u−1 κ P ) ⊗ pµ 1⊗ 1− u(u−1) κ2 P ⊗ P , P = vαpα. (2.7) We want to show that ∆pµ is the deformed coproduct with respect to the twist FGZ,u, ∆pµ = FGZ,u∆0pµF−1GZ,u, (2.8) where ∆0pµ = pµ ⊗ 1 + 1⊗ pµ. (2.9) Using (2.9) and (2.7), we may rewrite (2.8) as F−1GZ,u pµ ⊗ ( 1 + u κP ) + ( 1 + u−1 κ P ) ⊗ pµ 1⊗ 1− u(u−1) κ2 P ⊗ P = (pµ ⊗ 1 + 1⊗ pµ)F−1GZ,u 8 D. Meljanac, S. Meljanac, Z. Škoda and R. Štrajn and, after multiplying from the right by the denominator 1⊗ 1− u(u−1) κ2 P ⊗ P , as F−1GZ,u ( pµ ⊗ ( 1 + u κ P ) + ( 1 + u− 1 κ P ) ⊗ pµ ) = (pµ ⊗ 1 + 1⊗ pµ)F−1GZ,u ( 1⊗ 1− u(u− 1) κ2 P ⊗ P ) . (2.10) We shall show the equality in (2.10) by splitting it into a sum of two equalities, (2.11) and (2.12), which are then separately proved. Descriptively, (2.11) involves all those summands in ex- panded (2.10) where, in one of the factors, pµ is at the left side from the tensor product, F−1GZ,u ( pµ ⊗ ( 1 + u κ P )) = (pµ ⊗ 1)F−1GZ,u ( 1⊗ 1− u(u− 1) κ2 P ⊗ P ) . (2.11) To prove this equality, we first observe that by induction the equality [P,D] = P implies the commutation relation pµ ( D k ) = ( D + 1 k ) pµ. Hence P ( D k ) = ( D + 1 k ) P, i.e., ( D k ) P = P ( D − 1 k ) . We calculate the left-hand side of (2.11) as F−1GZ,u ( pµ ⊗ ( 1 + u κ P )) = ∞∑ k,l=0 (( u− 1 κ P )k (D l ) ⊗ ( uP κ )l (D k ))( pµ ⊗ 1 + u κ pµ ⊗ P ) = ∞∑ k,l=0 (( (u− 1)P κ )k (D l ) ⊗ ( uP κ )l (D k ) + ( (u− 1)P κ )k (D l ) ⊗ ( uP κ )l+1(D − 1 k )) (pµ ⊗ 1) and the right-hand side of (2.11) as (pµ ⊗ 1)F−1GZ,u ( 1⊗ 1− u(u− 1) κ2 P ⊗ P ) = ∞∑ k,l=0 (( (u− 1)P κ )k (D + 1 l ) ⊗ ( uP κ )l (D k ) − ( (u− 1)P κ )k+1(D l ) ⊗ ( uP κ )l+1(D − 1 k )) (pµ ⊗ 1). One Parameter Family of Jordanian Twists 9 Comparing the terms of type P k ⊗ P l for all k and l, we find( (u− 1)P κ )k (D l ) ⊗ ( uP κ )l (D k ) + ( (u− 1)P κ )k l ( D l − 1 ) ⊗ ( uP κ )l (D k ) D − k D = ( (u− 1)P κ )k (D + 1) ( D l − 1 ) ⊗ ( uP κ )l (D k ) − ( (u− 1)P κ )k l ( D l − 1 ) ⊗ ( uP κ )l k D ( D k ) and ( (u− 1)P κ )k (D l ) ⊗ ( uP κ )l (D k ) = ( (u− 1)P κ )k l ( D l − 1 ) ⊗ ( uP κ )l (D k ) + ( (u− 1)P κ )k (D + 1) ( D l − 1 ) ⊗ ( uP κ )l (D k ) = ( u− 1)P κ )k ( D l − 1 ) (D − l + 1)⊗ ( uP κ )l (D k ) = ( (u− 1)P κ )k (D l ) ⊗ ( uP κ )l (D k ) , which proves (2.11). Analogously, we prove the equality of the remaining summands in (2.10), F−1GZ,u (( 1 + (u− 1) 1 κ P ) ⊗ pµ ) = (1⊗ pµ)F−1GZ,u ( 1⊗ 1− u(u− 1) κ2 P ⊗ P ) . (2.12) Now (2.11) and (2.12) add to (2.10). Hence this proves (2.8), that is ∆pµ = FGZ,u∆0(pµ)F−1GZ,u. The coproduct ∆pµ satisfies the coassociativity condition (∆⊗ 1)∆pµ = (1⊗∆)∆pµ. Equation (2.8) can be rewritten as ∆0pµF−1GZ,u = F−1GZ,u∆pµ. (2.13) This enables us to obtain explicit formulas for the derivatives of the star product and for the star product from Section 2.2. Namely, for the partial derivatives of the star product, we compute ∂µ ( eikx ∗ eiqx ) (2.4) = ∂µmF−1GZ,u ( eikx ⊗ eiqx ) = m(∂µ ⊗ 1 + 1⊗ ∂µ)F−1GZ,u ( eikx ⊗ eiqx ) = mi∆0(pµ)F−1GZ,u ( eikx ⊗ eiqx ) (2.13) = imF−1GZ,u∆(pµ) ( eikx ⊗ eiqx ) (2.7) = miF−1GZ,uDµ(p⊗ 1, 1⊗ p) ( eikx ⊗ eiqx ) = imF−1GZ,uDµ(k, q) ( eikx ⊗ eiqx ) = iDµ(k, q)mF−1GZ,u ( eikx ⊗ eiqx ) (2.4) = iDµ(k, q) ( eikx ∗ eiqx ) , (2.14) 10 D. Meljanac, S. Meljanac, Z. Škoda and R. Štrajn where m denotes the multiplication map for usual functions. Knowing the partial deriva- tives (2.14) and the initial value (2.5) of the star product at x = 0, we finally obtain eikx ∗ eiqx = eiDµ(k,q)x µ 1 1− u(u−1) κ2 (v · k)(v · q) , (2.15) where Dµ(k, q) is given in (2.6). This star product is associative in agreement with the fact that twists F−1GZ,u satisfy the 2-cocycle condition (2.2). 2.4 Noncommutative coordinates and realizations Here we introduce noncommutative coordinates x̂µ, the commutation relations among them and their realizations. We use realizations of elements of noncommutative algebras via a Heisenberg algebra with generators xµ, pν , [xµ, xν ] = 0, [pµ, pν ] = 0, [xµ, pν ] = −iδµ,ν . The following expression defines noncommutative coordinates x̂µ [30], x̂µ = m ( F−1GZ,u(.⊗ 1)(xµ ⊗ 1) ) = xµ ( 1 + u κ P ) + i κ vµ(1− u) ( 1 + u κ P ) D = ( xµ + (1− u) i κ vµD )( 1 + u κ P ) + u(1− u) κ2 ivµP. Noncommutative coordinates x̂µ satisfy a κ-deformed Heisenberg algebra that corresponds to the κ-Minkowski space [11, 15, 16, 24, 25, 29, 32, 34] [x̂µ, x̂ν ] = i κ (vµx̂ν − vν x̂µ), [pµ, x̂ν ] = ( −iδµ,ν + i κ vν(1− u)pµ )( 1 + u κ P ) . In the case u = 0, x̂µ = xµ + i κ vµD. In the case u = 1, x̂µ = xµ ( 1 + u κ P ) . Using this realization of x̂µ and the method from [30] we obtain the same star product (2.15). 3 Interpolation between Jordanian twists induced by a 1-cochain Another construction for a generalized Jordanian twist is possible [1]. This twist, here de- noted FR,u, has been introduced as a product of three exponential factors, FR,u = exp (u κ (PD ⊗ 1 + 1⊗ PD) ) exp ( − ln ( 1− 1 κ P ) ⊗D ) exp ( ∆0 ( −u κ PD )) , (3.1) 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 . The classical r-matrix corresponding to twists FGZ,u (2.1) and FR,u (3.1) does not depend on the parameter u, namely r = 1 κ (D ⊗ P − P ⊗D). One Parameter Family of Jordanian Twists 11 The above form (3.1) of the family of twists FR,u is obtained from a simple Jordanian twist F0, using a transformation by a 1-cochain. Namely, according to Drinfeld [7, 27], if F is any nor- malized Drinfeld twist and ωR is any element in the Hopf algebra satisfying the normalization ε(ωR) = 1, then the formula Fω := ( ω−1⊗ω−1 ) F∆(ω) defines a normalized Drinfeld twist again (that is, the 2-cocycle and counitality conditions are satisfied again). In particular, if F = 1⊗ 1 we get a 2-coboundary twist ( ω−1⊗ω−1 ) ∆(ω). If the two twists, F and Fω, transform one into another by a 1-cochain, we say that they are cohomologous in the sense of nonabelian coho- mology [27]. In this case, twisted Hopf algebras HF and HFω are isomorphic [27] and, for each H-module algebra M , the corresponding twistings MF and MFω are also mutually isomorphic as algebras. If ω is group like, Fω is evidently obtained from F by an inner automorphism. Re- garding that cohomologous twists give isomorphic mathematical objects, one sometimes thinks of these twists as gauge equivalent. If F = F0 is a simple Jordanian twist, and ω = ωR = exp ( −u κPD ) , we obtain the twist FR,u = FωR = ( ω−1R ⊗ ω−1R ) F0∆(ωR), see [1]. This also shows that, for any u, twist FR,u satisfies the 2-cocycle and normalization conditions. Regarding that u appeared by gauge trans- forming F0, we can view u as a gauge parameter (the reader should not confuse u with a spectral parameter involved in some other Jordanian deformations). For u = 0, twist FR,u simplifies to F0 and for u = 1 to F1. 3.1 Hopf algebra The coalgebra sector of the Hopf algebra HFR,u for the deformation with FR,u is given by the formulas ∆FR,upµ = pµ ⊗ ( 1 + u 1 κP ) + ( 1− (1− u) 1 κP ) ⊗ pµ 1⊗ 1 + u(1− u) ( 1 κ )2 P ⊗ P , ∆FR,uD = ( 1⊗ 1 + u(1− u) κ2 P ⊗ P )( D ⊗ 1 1 + u κP + 1 1− 1−u κ P ⊗D ) , SFR,u(pµ) = − pµ 1− (1− 2u) 1 κP , SFR,u(D) = − ( 1− 1− u κ P ) D ( 1− 1−2u κ P 1− 1−u κ P ) . A similar analysis as in Section 2 for ∆FGZ,upµ leads to the conclusion that ∆FGZ,uD = ∆FR,uD. 3.2 Noncommutative coordinates and realizations In general, we consider realizations of the form x̂µ = xαϕαµ(p) + χ(p). We can obtain the appropriate realization via the twist as follows x̂µ = m ( F−1R,u(.⊗ 1)(xµ ⊗ 1) ) = ( xµ + i κ vµ(1− u)D )( 1 + u κ P ) + u(1− u) i κ2 vµP. 3.3 Star product Using the above realization of x̂µ [30], we get eikx ∗ eiqx = eiDµ(u;k,q)xµ+iG(u;k,q) = eiDµ(u;k,q)xµ 1 1 + u(1−u) κ2 (v · k)(v · q) , 12 D. Meljanac, S. Meljanac, Z. Škoda and R. Štrajn where k and q belong to the n-dimensional Minkowski spacetime M1,n−1 and where Dµ(u; k, q) = kµ ( 1 + u κ(v · q) ) + ( 1− 1−u κ (v · k) ) qµ 1 + u(1−u) κ2 (v · k)(v · q) as in equation (2.6), and finally G(u; k, q) = i ln ( 1 + u(1− u) κ2 (v · k)(v · q) ) . Remark. Note that the corresponding quantum R-matrix is given by RR,u = F21 R,uF−1R,u = exp(u(PD ⊗ 1 + 1⊗ PD)R0 exp(−u(PD ⊗ 1 + 1⊗ PD)), where R0 = exp ( −D ⊗ ln ( 1− P κ )) exp ( ln ( 1− P κ ) ⊗D ) = ∞∑ k,l=0 ( −D l )( −P κ )k ⊗ ( −P κ )l (D k ) . Both twists, F−1GZ,u and F−1R,u, lead to the same Hopf algebra, the same realizations of non- commutative coordinates x̂µ and likewise for the star product eikx ∗ eiqx. This suggests that there must be a close relation between the two twists, F−1GZ,u and F−1R,u. In the next section, we present a proof that indeed F−1GZ,u = F−1R,u. 4 Proofs of the equality of the two twists 4.1 Differentiation with respect to parameter u Differentiating F−1R,u from equation (3.1) with respect to the parameter u gives κ dF−1R,u du = (P ⊗D +D ⊗ P )F−1R,u + [ PD ⊗ 1 + 1⊗ PD,F−1R,u ] . (4.1) Differentiating F−1GZ,u from equation (2.1) with respect to u gives κ dF−1GZ,u du = (P ⊗D +D ⊗ P )F−1GZ,u + (P ⊗ 1− 1⊗ P ) ∞∑ k,l=0 −k + l κk+l (u− 1)kP k ( D l ) ⊗ (uP )l ( D k ) . (4.2) Using the commutation relations[ PD,P k ( D l )] = (l − k)P k+1 ( D l ) and [ PD,P l ( D k )] = (k − l)P l+1 ( D k ) , One Parameter Family of Jordanian Twists 13 we find that the right-hand sides of (4.1) and of (4.2) agree, r.h.s. = (P ⊗D +D ⊗ P )F−1GZ,u + [ PD ⊗ 1 + 1⊗ PD,F−1GZ,u ] . This shows that F−1R,u and F−1GZ,u as functions of the parameter u satisfy the same ordinary differential equation, while the initial conditions agree. Indeed, at u = 0, F−1R,u=0 = F−10 = F−1GZ,u=0. Therefore F−1R,u ≡ F −1 GZ,u. 4.2 Proof of the equality of the two twists In the following proposition we state the conditions under which the two twists are equal, along with a simple proof. Proposition 1. Let P be the Poincaré Weyl algebra generated with momenta pµ, Lorentz gene- rators Mµν and dilatation D. Two twists F1 ∈ U(P)⊗U(P) and F2 ∈ U(P)⊗U(P) are identical if all the star products are identical, i.e., for all f and g in the Minkowski space time algebra, f ∗ g = m ( F−11 (.⊗ .)(f ⊗ g) ) = m ( F−12 (.⊗ .)(f ⊗ g) ) . Proof. If all star products are the same, F−11 and F−12 could differ by an element in the right ideal J0 generated by the elements (xµ ⊗ 1 − 1 ⊗ xµ) for all µ [14, 17, 18]. However, J0 ∩ U(P)⊗ U(P) = 0, hence F1 = F2. � Since we already proved that the twists FR,u and FGZ,u give the same star products eikx∗eiqx, the twists FR,u and FGZ,u must be identical. Moreover, we have proved that the noncommutative coordinates x̂µ and twisted coproducts ∆pµ and ∆D from both twists are identical. Since FR,u satisfies the normalization and 2-cocycle conditions, FGZ,u also satisfies them. 5 Conclusion We have constructed a 1-parameter family FGZ,u (2.1) of Jordanian twists that interpolates between the simple Jordanian twists F0 and F1 defined in equation (1.1). We explicitly proved that F−1GZ,u satisfies the 2-cocycle condition (2.2). For u = 1 2 , FGZ,u= 1 2 coincides with FGZ [10]. We have calculated the corresponding star product eikx ∗ eiqx (2.15) and the corresponding deformed Hopf algebra structure. In Section 3, we have presented another interpolation between Jordanian twists cohomologous to F0 via a 1-cochain depending on u [1]. It is pointed out that F−1GZ,u and F−1R,u generate the same star product and the same deformed Hopf algebra. In Section 4, a new result is presented that F−1GZ,u = F−1R,u, implying that FGZ,u can be written in the form of a product of three exponential factors. Twist FR,u automatically satisfies the 2-cocycle condition as it is obtained from a simple Jordanian twist by twisting by a 1-cochain [27]. We note that for the twist F−1GZ [10], the star product, an explicit form of the twist FGZ and the deformed Hopf algebra structure, were not known in the literature so far. Jordanian twists have been of interest in the recent literature [6, 13, 22, 28, 33, 36]. We note that our results could be useful in future applications of Jordanian twists. 14 D. Meljanac, S. Meljanac, Z. Škoda and R. Štrajn A Appendix Lemma 1. If x, y, z are mutually commuting variables, and k, l, A, C with A ≤ k, C ≤ l nonnegative integers, then( x l − C ) k∑ k1=k−A ( k1 k −A )( z k1 )( x+ y − k − l + k1 + C C )( y k − k1 ) = ( z k −A ) l∑ l1=l−C ( l1 l − C )( x l1 )( y + z − k − l + l1 +A A )( y l − l1 ) . (A.1) Proof. To make the proof more transparent, we make a change of summation indices i = k1 − k +A, j = l1 − l + C, hence k1 = i+ k −A and l1 = j + l − C, to restate equation (A.1) as( x l − C ) A∑ i=0 ( i+ k −A k −A )( z i+ k −A )( x+ y − l + i−A+ C C )( y A− i ) = ( z k −A ) C∑ j=0 ( j + l − C l − C )( x j + l − C )( y + z − l + j − C +A A )( y C − j ) . (A.2) We remind the reader of the simple identity( r s )( w r ) = ( w s )( w − s r − s ) , which we apply in (A.2) for w = z on the left and for w = x on the right, to obtain an equivalent statement,( x l − C )( z k −A ) A∑ i=0 ( z − k +A i )( x+ y − l + i−A+ C C )( y A− i ) = ( z k −A )( x l − C ) C∑ j=0 ( x j + l − C )( y + z − k + j − C +A A )( y C − j ) . (A.3) We expand( x+ y − l + i−A+ C C ) = C∑ j=0 ( x− l + C j )( y + i−A C − j ) on the left-hand side, and( y + z − k + j − C +A A ) = A∑ i=0 ( z − k +A i )( y + j − C A− i ) on the right-hand side of (A.3). Now both sides involve double summation over i and j. For each fixed pair (i, j), compare the corresponding summands on the two sides. The factors involving x and z are identical on both sides. It remains to check that the factors involving y agree. 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[39] Tolstoy V.N., Twisted quantum deformations of Lorentz and Poincaré algebras, arXiv:0712.3962. https://doi.org/10.1023/A:1022869414679 https://arxiv.org/abs/q-alg/9708024 https://doi.org/10.1016/0370-2693(92)90894-A https://doi.org/10.1016/0370-2693(92)90894-A https://doi.org/10.1016/0370-2693(91)90358-W https://doi.org/10.1070/RM1986v041n05ABEH003441 https://doi.org/10.1017/CBO9780511613104 https://doi.org//10.1103/PhysRevD.99.126012 https://arxiv.org/abs/1903.08679 https://doi.org/10.1142/S0217751X11053948 https://doi.org/10.1142/S0217751X11053948 https://arxiv.org/abs/1004.4647 https://doi.org/10.1016/j.physletb.2017.01.006 https://doi.org/10.1016/j.physletb.2017.01.006 https://arxiv.org/abs/1610.06716 https://doi.org/10.1088/1751-8121/aa72d7 https://arxiv.org/abs/1612.07984 https://doi.org/10.1142/S0217732310032652 https://arxiv.org/abs/0912.5087 https://doi.org/10.1103/PhysRevD.92.105015 https://arxiv.org/abs/1509.02115 https://doi.org/10.1140/epjc/s2006-02584-8 https://doi.org/10.1140/epjc/s2006-02584-8 https://arxiv.org/abs/hep-th/0605133 https://doi.org/10.1088/1751-8113/48/44/445202 https://doi.org/10.1088/1751-8113/48/44/445202 https://arxiv.org/abs/1507.03523 https://doi.org/10.1023/A:1021460515598 https://doi.org/10.1023/A:1021460515598 https://arxiv.org/abs/q-alg/9608011 https://arxiv.org/abs/0704.0081 https://arxiv.org/abs/0712.3962 1 Introduction 2 Generalization of the Giaquinto–Zhang twist 2.1 2-cocycle condition 2.2 Star product 2.3 Twisted coproduct (p) 2.4 Noncommutative coordinates and realizations 3 Interpolation between Jordanian twists induced by a 1-cochain 3.1 Hopf algebra 3.2 Noncommutative coordinates and realizations 3.3 Star product 4 Proofs of the equality of the two twists 4.1 Differentiation with respect to parameter u 4.2 Proof of the equality of the two twists 5 Conclusion A Appendix References
id	nasplib_isofts_kiev_ua-123456789-210306
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn	1815-0659
language	English
last_indexed	2025-12-07T21:25:05Z
publishDate	2019
publisher	Інститут математики НАН України
record_format	dspace
spelling	Meljanac, D. Meljanac, S. Škoda, Z. Štrajn, R. 2025-12-05T09:30:29Z 2019 One Parameter Family of Jordanian Twists / D. Meljanac, S. Meljanac, Z. Škoda, R. Štrajn // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 39 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53D55; 16T05 arXiv: 1904.03993 https://nasplib.isofts.kiev.ua/handle/123456789/210306 https://doi.org/10.3842/SIGMA.2019.082 We propose an explicit generalization of the Jordanian twist proposed in r-symmetrized form by Giaquinto and Zhang. It is proven that this generalization satisfies the 2-cocycle condition. 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-cochain. We thank Anna Pachol for useful discussions. Z.Š. has been partly supported by the Croatian Science Foundation under the Project "New Geometries for Gravity and Spacetime" (IP-2018-01-7615) and by the grant 18-00496S of the Czech Science Foundation. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications One Parameter Family of Jordanian Twists Article published earlier
spellingShingle	One Parameter Family of Jordanian Twists Meljanac, D. Meljanac, S. Škoda, Z. Štrajn, R.
title	One Parameter Family of Jordanian Twists
title_full	One Parameter Family of Jordanian Twists
title_fullStr	One Parameter Family of Jordanian Twists
title_full_unstemmed	One Parameter Family of Jordanian Twists
title_short	One Parameter Family of Jordanian Twists
title_sort	one parameter family of jordanian twists
url	https://nasplib.isofts.kiev.ua/handle/123456789/210306
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One Parameter Family of Jordanian Twists

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