Interpolations between Jordanian Twists Induced by Coboundary Twists
We propose a new generalisation of the Jordanian twist (building on the previous idea from [Meljanac S., Meljanac D., Pachoł A., Pikutić D., J. Phys. A: Math. Theor. 50 (2017), 265201, 11 pages]). Obtained this way, the family of Jordanian twists allows for interpolation between two simple Jordanian...
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| Cite this: | Interpolations between Jordanian Twists Induced by Coboundary Twists / A. Borowiec, D. Meljanac. S. Meljanac, A. Pachoł // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 55 назв. — англ. |
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| citation_txt | Interpolations between Jordanian Twists Induced by Coboundary Twists / A. Borowiec, D. Meljanac. S. Meljanac, A. Pachoł // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 55 назв. — англ. |
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| description | We propose a new generalisation of the Jordanian twist (building on the previous idea from [Meljanac S., Meljanac D., Pachoł A., Pikutić D., J. Phys. A: Math. Theor. 50 (2017), 265201, 11 pages]). Obtained this way, the family of Jordanian twists allows for interpolation between two simple Jordanian twists. This new version of the twist provides an example of a new type of star product and the realization of noncommutative coordinates. Real forms of new Jordanian deformations are also discussed. Exponential formulae, used to obtain coproducts and star products, are presented with details.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 15 (2019), 054, 22 pages
Interpolations between Jordanian Twists
Induced by Coboundary Twists
Andrzej BOROWIEC †
1
, Daniel MELJANAC †
2
, Stjepan MELJANAC †
3
and Anna PACHO L †
4
†1 Institute of Theoretical Physics, University of Wroclaw,
pl. M. Borna 9, 50-204 Wroclaw, Poland
E-mail: andrzej.borowiec@ift.uni.wroc.pl
†2 Division of Materials Physics, Ruder Bošković Institute,
Bijenička c.54, HR-10002 Zagreb, Croatia
E-mail: Daniel.Meljanac@irb.hr
†3 Division of Theoretical Physic, Ruder Bošković Institute,
Bijenička c.54, HR-10002 Zagreb, Croatia
E-mail: meljanac@irb.hr
†4 Queen Mary, University of London, Mile End Rd., London E1 4NS, UK
E-mail: a.pachol@qmul.ac.uk
Received February 15, 2019, in final form July 11, 2019; Published online July 21, 2019
https://doi.org/10.3842/SIGMA.2019.054
Abstract. We propose a new generalisation of the Jordanian twist (building on the previous
idea from [Meljanac S., Meljanac D., Pacho l A., Pikutić D., J. Phys. A: Math. Theor. 50
(2017), 265201, 11 pages]). Obtained this way, the family of the Jordanian twists allows for
interpolation between two simple Jordanian twists. This new version of the twist provides an
example of a new type of star product and the realization for noncommutative coordinates.
Real forms of new Jordanian deformations are also discussed. Exponential formulae, used
to obtain coproducts and star products, are presented with details.
Key words: twist deformation; Hopf algebras; coboundary twists; star-products; real forms
2010 Mathematics Subject Classification: 81T75; 16T05; 17B37; 81R60
1 Introduction
Let H = (H,∆, S, ε) be a Hopf algebra and F ∈ H ⊗ H be a two-cocycle twist. Then new
(twisted) Hopf algebra structure on the algebra H with deformed coproduct and antipode is
denoted by HF =
(
H,∆F , SF , ε
)
, where ∆F (·) = F∆(·)F−1.
For any invertible element ω ∈ H one can define new gauge equivalent two-cocycle twist
Fω =
(
ω−1 ⊗ ω−1
)
F∆(ω) which determines the third Hopf algebra HFω =
(
H,∆Fω , SFω , ε
)
.
Notice that all three Hopf algebras share the same algebraic structure (multiplication). Its
internal automorphism, defined by the similarity transformation: α(Z) = ωZω−1, Z ∈ H
establishes, at the same time, the isomorphism between two twisted Hopf algebras HF ∼= HFω
as illustrated on the following diagram
H
∆F
−−−−−−→ H ⊗H
α ↓ ↓ α⊗ α
H
∆Fω
−−−→ H ⊗H,
mailto:andrzej.borowiec@ift.uni.wroc.pl
mailto:Daniel.Meljanac@irb.hr
mailto:meljanac@irb.hr
mailto:a.pachol@qmul.ac.uk
https://doi.org/10.3842/SIGMA.2019.054
2 A. Borowiec, D. Meljanac, S. Meljanac and A. Pacho l
i.e.,
∆F ◦ α = (α⊗ α) ◦∆Fω .
Consider now a (left) Hopf module algebra A = (A, ., ?) over the Hopf algebra H together
with a (left) Hopf action . : H ⊗ A → A, where ? denotes the multiplication in A. Changing
multiplication m? ≡ ? to m?F ≡ ?F :
a ?F b = m?F (a⊗ b) = m?
[
F−1(.⊗ .)(a⊗ b)
]
(1.1)
for a, b ∈ A, one gets new module algebra AF = (A, ., ?F ) over HF with the same action,
i.e., the module structure remains the same. It is easy to see that any invertible element
ω ∈ H provides an algebra isomorphism β : (A, ?Fω) → (A, ?F ), where β(a) ≡ ω . a, since
ω . (a ?Fω b) = (ω . a) ?F (ω . b). It turns out that the invertible map β intertwines between two
modules in the following sense: α(Z) . β(a) = β(Z . a), i.e., the diagram
H ⊗A .−−−−→ A
α⊗ β ↓ ↓ β
H ⊗A .−→ A
commutes. In particular, the coboundary twist Tω =
(
ω−1⊗ω−1
)
∆(ω) provides the Hopf algebra
isomorphism H ∼= HTω as well as module algebra isomorphism A ∼= ATω . For group-like ω, it
becomes an automorphism. It means that replacing a twist by the gauge equivalent one leads
to mathematically equivalent objects.
Let us explain this in the case of Jordanian deformations of Lie algebras.
Let’s consider a Lie algebra g. Drinfeld’s quantum groups are quantized universal enveloping
algebras Ug [14] obtained by the methods of deformation quantization of Poisson Lie groups.
More exactly, quantized objects are Hopf algebras corresponding to a given Lie bialgebra struc-
ture (g, r) [15, 16], which in turn can be determined by the classical r-matrix r ∈ g⊗g satisfying
classical Yang–Baxter equation [[r, r]] = 0 with [[ , ]] being the so-called Schouten brackets. In
the triangular case r ∈ g ∧ g the quantization is provided by an invertible, two-cocycle element
Fr ∈ Ug ⊗ Ug[[γ]] called a Drinfeld twist. Here γ is a formal parameter and Ug[[γ]] means topo-
logical completion in the topology of formal power series in γ (see, e.g., [11, 12, 32] for details).
Thus the classical r-matrix can be recovered from the quantum R-matrix as follows
Rr = F τr F
−1
r = 1⊗ 1 + γr + o
(
γ2
)
,
where τ denotes the flip map: F τ = F21. For example, in two dimensions there are only two
(non-isomorphic) Lie algebra structures: Abelian ab(2) = {x, y : [x, y] = 0} and non-Abelian
an(2) = {h, e : [h, e] = e}.1 The corresponding Lie bialgebra structures are given by the Abelian:
rAb = x ∧ y or Jordanian rJ = h ∧ e classical r-matrices. Embedding one of these algebras in
some higher-dimensional Lie algebra g as Lie subalgebra provides the twist quantization of Ug:
Abelian or Jordanian. In the latter case, deformation can be realized by the Jordanian twist of
the form
FJ = exp(ln(1 + γe)⊗ h). (1.2)
1It corresponds to the Lie group “ax+ b” of affine transformations of the real (complex) line.
Interpolations between Jordanian Twists Induced by Coboundary Twists 3
This form of the twist first appeared in [49] and then a symmetrised form (i.e., where rJ appears
in the first order in expansion of the twist) was proposed in [52, 54]2
FT = exp
(γ
2
(he⊗ 1 + 1⊗ he)
)
exp (ln(1 + γe)⊗ h) exp
(
−γ
2
∆(he)
)
. (1.3)
It was obtained by applying the coboundary twist to (1.2). There are many possible r-symmet-
risations for twists. This is due to the fact that twist deformation is defined up to the so-called
gauge transformation (in Drinfeld’s terminology), i.e., many twists can provide isomorphically
equivalent Hopf algebraic deformations, if they differ by the coboundary twist (see, e.g., [32]).
Jordanian deformations have been of interest for quite some time. For example, Jordanian
deformations of the conformal algebra were considered in [45], as well as in [2, 3, 19]. In
[2, 3, 19] the deformations of Anti de Sitter and de Sitter algebras were also investigated.
Jordanian deformations have been considered in applications in AdS/CFT correspondence [25,
26, 34, 35], as integrable deformations of sigma models in relation to deformation of AdS5 and
supergravity [20, 55]. Jordanian twists have been applied in deformation of spacetime metrics [5],
Maxwell equations and dispersion relations [1], as well as classical and quantum mechanics [36].
Gauge theories under Jordanian deformation were also investigated in [13].
The question we would like to address in the present paper is if the mathematical equivalences
of coboundary twists, in some physically relevant situations, give rise, to some extent, to physi-
cally inequivalent descriptions. For example, when one considers the star product quantization,
realizations for noncommutative coordinates, or the form of Heinsenberg algebra.
To this aim we embed our two-dimensional Lie algebra an(2) into some bigger Lie algebra
which has some potential application in physics. In the present paper we focus our attention on
the Lie algebra g = {Pµ, D}, generated by momenta Pµ (spacetime translations in n-dimensions
where µ = 0, 1, . . . , n−1) and dilatation generators D with the following commutation relations
[Pµ, D] = Pµ, [Pµ, Pν ] = 0.
This algebra can be considered as a subalgebra of some bigger Lie algebra, e.g., Poincaré–Weyl,
de Sitter, etc. This embedding is realized by choosing two elements {h, e} as h = −D and e = P
(P can be taken as any of Pµ and the formulae in the next Section 2 will hold). However, for
convenience, we choose the following notation: P = vαPα where vµ is the vector on Minkowski
spacetime M1,n−1 in n-dimensions such that v2 = vαvα ∈ {−1, 0, 1}. For the correspondence
with the κ-Minkowski spacetime [30, 31] we choose the deformation parameter as γ = − 1
κ .
Our main aim in this work is to analyse quantum deformations corresponding to gauge
equivalence of Jordanian twists (1.2), extending [41] and applying [40].
This paper is a sequel to [41], where we introduced a generalised form of r-symmetrised
twist interpolating between Jordanian twists. The main formulae are recalled here as part
of the next Section 2.1. In Section 2.2 we propose another generalised form of r-symmetrised
Jordanian twist (FR,u) providing the interpolation between Jordanian twists as well. We present
the corresponding Hopf algebra deformation, the star product form and the realization of the
noncommutative coordinates. Section 3 presents a relation between the two generalisations of
r-symmetric twists FL,u and FR,u, including the relation between the corresponding quantum
R-matrices. In Section 4 the ∗-structure and unitarity of the twists is analysed. We finish
with brief conclusions which are followed by a series of appendices complementing the results
presented in the main part of the paper. They are devoted to the explanation of the exponential
formulae which are obtained from twist realization of deformed coordinate functions. Some
applications for calculations of wave packets star products as well as coproducts of momenta are
also considered.
2These techniques have been used before in the supersymmetric case in order to unitarize super Jordanian
twist, see [6, 51, 53].
4 A. Borowiec, D. Meljanac, S. Meljanac and A. Pacho l
2 Two families of twists interpolating between Jordanian twists
In our previous work [41] we have proposed a simple generalisation of the locally r-symmetric
Jordanian twist (1.3), resulting in the one-parameter family interpolating between Jordanian
twists. All the proposed twists differed by the coboundary twists and produced the same Jor-
danian deformation of the corresponding Lie algebra. We have proposed a way, by introducing
an additional parameter u, of interpolating between the two Jordanian twists
F0 = exp
(
− ln
(
1− 1
κ
P
)
⊗D
)
(2.1)
with the logarithm on the left side of the tensor product (cf. with (1.2)) and
F1 = (F0)τ |− 1
κ
= exp
(
−D ⊗ ln
(
1 +
1
κ
P
))
with the logarithm on the right tensor factor and also with the changed sign of the deformation
parameter κ, recall τ denotes the flip map: F τ = F21. One can symmetrise these simple
Jordanian twists, into a so-called r-symmetric form, such that at the first order in the expansion
of the twist one gets the classical r-matrix corresponding to the given deformation. For the
Jordanian deformations it is always rJ . In this paper we want to present another type of such
interpolation.
First, in the below section, we recall few main formulae from [41] and then, in Section 2.2,
we shall propose another interpolation.
2.1 FL,u family of twists with dilatation on the left
The r-symmetric version of the Jordanian twist (2.1) was introduced in [54] and it was obtained
from the coboundary twist Tω by choosing ω0 = exp
(
− 1
2κDP
)
. The formula for FT follows
directly from:
(
ω−1
0 ⊗ ω
−1
0
)
F0∆(ω0).
In [41] we have introduced its generalisation in the form of one parameter family interpolating
between Jordanian twists ∀u ∈ R
FL,u = exp
(u
κ
(DP ⊗ 1 + 1⊗DP )
)
exp
(
− ln
(
1− 1
κ
P
)
⊗D
)
× exp
(
∆
(
−u
κ
DP
))
, u ∈ R. (2.2)
This generalisation simply corresponds to a modification of ω0 to ωL = exp
(
−u
κDP
)
and then
FL,u =
(
ω−1
L ⊗ ω
−1
L
)
F0∆(ωL) still differs from F0 only by the coboundary twist TωL . For this
reason the cocycle condition (see, e.g., [32]) for FL,u is automatically satisfied. For u = 1
2 (1.3)
is recovered.
Note that now FL,u contains two parameters: one real parameter u and the other κ -the
formal deformation parameter.
The reduction of FL,u, for certain values of the parameter u, to F0 (for u = 0) and to F1 (for
u = 1) were discussed in [41].
2.1.1 Hopf algebra
The deformation of the Hopf algebra Ug(µ,∆, ε, S) of the universal enveloping algebra of g
given by the twist element F ∈ Ug ⊗ Ug[[
1
κ ]] into UFg
(
µ,∆F , ε, SF
)
is provided by the de-
formation of the coproduct and antipode maps as follows: ∆F (Z) = F∆(Z)F−1, SF (Z) =[
µ((1 ⊗ S)F )]S(Z)[µ
(
(S ⊗ 1)F−1
)]
, where Z ∈ g. The coproducts, star products and realiza-
tions depend explicitly on the parameter u as well as on the parameter of deformation κ.
Interpolations between Jordanian Twists Induced by Coboundary Twists 5
We recall the coalgebra sector of the Hopf algebra UF0
g for the deformation with the twisting
element F0 from [8]
∆F0(Pµ) = Pµ ⊗ 1 +
(
1− 1
κ
P
)
⊗ Pµ, ∆F0(D) = D ⊗ 1 +
1
1− 1
κP
⊗D,
SF0(Pµ) =
−Pµ
1− 1
κP
, SF0(D) = −
(
1− 1
κ
P
)
D.
The change of the twist by the coboundary twist Tω provides a new presentation for the Hopf
algebra, and can be transformed by
(
α−1⊗α−1
)
◦∆F0 ◦α = ∆Fω where α(Z) = ωZω−1, Z ∈ H
and α−1(Z) = ω−1Zω for chosen ω. For FL,u one needs to take ωL.
Alternatively, the deformed coproducts and antipodes can be calculated directly from the
definition ∆Z = F∆(Z)F−1. The coalgebra sector of the Hopf algebra U
FL,u
g for the deformation
with FL,u, recalled from [41], is as follows
∆FL,u(Pµ) =
Pµ ⊗
(
1 + u
κP
)
+
(
1− (1−u)
κ P
)
⊗ Pµ
1⊗ 1 + u(1− u)
(
1
κ
)2
P ⊗ P
,
∆FL,u(D) =
(
D ⊗ 1
1 + u 1
κP
+
1
1− (1− u) 1
κP
⊗D
)(
1⊗ 1 + u(1− u)
(
1
κ
)2
P ⊗ P
)
.
The coproduct is coassociative. The antipodes are given by
SFL,u(Pµ) = − Pµ
1− (1− 2u) 1
κP
,
SFL,u(D) = −
(
1− (1− 2u)Pκ
1 + u
κP
)
D
(
1 +
u
κ
P
)
.
2.1.2 Coordinate realizations and star product
As a Hopf module algebra for Ug we choose the algebra of smooth (complex valued) functions on
a space time (i.e., A = C∞(Rn)⊗ C with an obvious algebraic structures determined by point-
wise multiplication and addition). This algebra includes spacetime coordinates xµ (where xµ
are considered as generators of n-dimensional Abelian Lie algebra). A natural action of g on A
(i.e., action of the momenta and the dilatation operators on coordinates) is defined, in terms of
Weyl–Heisenberg algebra generators [∂µ, x
ν ] = δνµ,3 by
Pµ . f(x) = −i
∂
∂xµ
. f(x) = −i
∂f (x)
∂xµ
and D . f(x) = xµ
∂f(x)
∂xµ
. (2.3)
The algebra of coordinates becomes noncommutative due to the twist deformation once the usual
multiplication is replaced by the star product multiplication (star product quantization) (1.1)
for any f, g ∈ C∞(Rn). This star product is associative (due to the fact that the twist FL,u
satisfies the cocycle condition).
When we choose the functions to be exponential functions eik·x and eiq·x, then we can define
new function Dµ(u; k, q) [29, 42, 43, 44, 48]:
eik·x ? eiq·x = m
[
F−1(.⊗ .)
(
eik·x ⊗ eiq·x)] = eiDµ(u;k,q)xµ , (2.4)
3This algebra has structure of a smash product and can be arranged in a Hopf algebroid structure [9, 10, 23, 24].
6 A. Borowiec, D. Meljanac, S. Meljanac and A. Pacho l
where k, q ∈ M1,n−1 (in n-dimensional Minkowski spacetime). One can calculate explicitly, see
Appendices C.1 and C.3, that in the case of twist FL,u the function Dµ(u; k, q) is given by
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)
. (2.5)
Directly from the twist we can also calculate the coordinate realizations [9, 18, 21, 22, 23, 24, 37]
and for the FL,u twist they have the following form4
x̂µ = m
[
F−1
L,u(.⊗ 1)(xµ ⊗ 1)
]
= xµ
(
1 +
u
κ
P
)
+
i
κ
vµ(1− u)D
(
1 +
u
κ
P
)
=
(
xµ +
i
κ
vµ(1− u)D
)(
1 +
u
κ
P
)
= xαφµα(P ), (2.6)
where vµ is the vector on Minkowski spacetimeM1,n−1 in n-dimensions such that v2 ∈ {−1, 0, 1}
as before. These realizations are also discussed in [27, 28, 38, 39, 46, 47].
2.2 FR,u family of twists with dilatation on the right
In this paper we want to introduce another version of the generalised Jordanian twist
FR,u = exp
(u
κ
(PD ⊗ 1 + 1⊗ PD)
)
exp
(
− ln
(
1− 1
κ
P
)
⊗D
)
exp
(
∆
(
−u
κ
PD
))
, (2.7)
where u is a real parameter u ∈ R. We point out that the sub-index R refers to the position of
the dilatation generator, it is on the right with respect to momenta generators P . Due to the
position of the dilatation operator with respect to momenta, this introduces a different formula
than the one considered in [41] and which was recalled in the previous Section 2.1, i.e., FL,u (2.2).
For the parameter u = 1
2 the twist FR, 1
2
is r-symmetric, but is not equal to FL, 1
2
. These two
twists differ at subleading order, see (3.4) in the following section. The form of the family of
twists FR,u can also be easily obtained from the simple Jordanian twist F0 by the transformation
with the coboundary twist TωR however this time with the element ωR = exp
(
−u
κPD
)
. The
twist FR,u, ∀u, satisfies the normalization and cocycle conditions.
For u = 0, twist FR,u simplifies to F0, easily seen by just plugging in u = 0 in the equa-
tion (2.7), and for u = 1, it simplifies to the twist F1. Hence FR,u provides another way of
interpolating between F0 and F1.
2.2.1 Hopf algebra
The coalgebra sector of the Hopf algebra U
FR,u
g for the deformation with FR,u can also be
calculated and has the form
∆FR,u(Pµ) =
Pµ ⊗
(
1 + u 1
κP
)
+
(
1− (1− u) 1
κP
)
⊗ Pµ
1⊗ 1 + u(1− u)
(
1
κ
)2
P ⊗ P
, (2.8)
∆FR,u(D) =
(
1⊗ 1 +
(
1
κ
)2
u(1− u)P ⊗ P
)
×
(
D ⊗ 1
1 + u 1
κP
+
1
1− (1− u) 1
κP
⊗D
)
. (2.9)
4For simplicity the matrix notation φα
µ(P ) is written as φµα(P ) throughout the paper.
Interpolations between Jordanian Twists Induced by Coboundary Twists 7
Antipodes are
SFR,u(Pµ) =
−Pµ
1− (1− 2u) 1
κP
, (2.10)
SFR,u(D) = −
(
1− (1− u)
P
κ
)
D
(
1− (1− 2u)Pκ
1− (1− u)Pκ
)
. (2.11)
2.2.2 Coordinate realizations and star product
The inverse of the family of twists F−1
R,u provides another (new) star product between the func-
tions (1.1). If we choose our functions to be exponential functions eik·x and eiq·x, then we can
define new functions Dµ(u; k, q) and G(u; k, q) in the following way [29, 40]
eik·x ? eiq·x = m
[
F−1
R,u(.⊗ .)
(
eik·x ⊗ eiq·x)] = eiDµ(u;k,q)xµ+iG(u;k,q)
= eiDµ(u;k,q)xµ 1
1 + u(1−u)
κ2
(v · k)(v · q)
, (2.12)
where k, q ∈ M1,n−1. Note the difference in the terms on the right hand side between the
formula above and the one for FL,u in (2.4). One can calculate, see Appendices C.2 and C.3,
that in the case of the twist F−1
R,u the function Dµ(u; k, q) is the same as in equation (2.5). Note
that the function Dµ(u; k, q) can be seen as rewriting the coproduct ∆(Pµ) without using the
tensor product notation (denoting left and right leg by k and q respectively). Therefore the
relation between the coproduct ∆(Pµ) and the function Dµ(u; k, q) is given by
∆(Pµ) = Dµ(u;P ⊗ 1, 1⊗ P ),
hence ∆(Pµ) uniquely determines Dµ(u; k, q), as in the case of FL,u twist.
The additional function on the right hand side of (2.12) has the following explicit form
G(u; k, q) = i ln
(
1 +
u(1− u)
κ2
(v · k)(v · q)
)
. (2.13)
We refer the reader to the Appendices C.2, C.3 and [40] for further the details of these calcu-
lations. Note that in the case of the generalisation of the twist FT (1.3) into FL,u (2.2), the
function G(u; k, q) = 0 (see also [41]).
Realizations of noncommutative coordinates x̂µ can be generally expressed in terms of Weyl–
Heisenberg algebra generated by xµ and Pµ (commutative variables).
The realization obtained from the FR,u twist has the new general form
x̂µ = xαφµα(P ) + χµ(P ).
Note that the part χµ(P ) was not present in the case of FL,u twist, see equation (2.6) in
Section 2.1.2, and also [41].
Noncommutative coordinates x̂µ, corresponding to the twist FR,u, are given by
x̂µ = m
[
F−1
R,u(.⊗ 1)(xµ ⊗ 1)
]
= xµ
(
1 +
u
κ
P
)
+
i
κ
vµ(1− u)
(
1 +
u
κ
P
)
D
=
(
xµ +
i
κ
vµ(1− u)D
)(
1 +
u
κ
P
)
+ u(1− u)
i
κ2
vµP. (2.14)
From the last line in the above formula one can read off explicitly the form of the functions
φµα(P ) and χµ(P ). In the case when u = 0, we have x̂µ = xµ + i 1
κv
µD, while in the case when
u = 1, x̂µ = xµ
(
1 + 1
κP
)
.
8 A. Borowiec, D. Meljanac, S. Meljanac and A. Pacho l
The noncommutative coordinates x̂µ satisfy
[
x̂µ, x̂ν
]
=
i
κ
(
vµx̂ν − vν x̂µ
)
,[
Pµ, x̂
ν
]
=
(
−iδνµ + i
1
κ
vν(1− u)Pµ
)(
1 + u
1
κ
P
)
. (2.15)
The above kappa deformed Weyl–Heisenberg algebra (2.15) is obtained by using the realiza-
tion (2.14). It turns out to be the same as in [41] where it was obtained from (2.6). Summari-
sing, the realizations (2.14) and (2.6) differ by the term χµ(P ) which does not change the form
of (2.15).
3 Relations between two families FL,u and FR,u
The two twists FL,u (2.2) and FR,u (2.7) are obviously related by the coboundary twists TωL
and TωR in the following sense
FR,u =
(
ω−1
R ωL ⊗ ω−1
R ωL
)
FL,u∆
(
ω−1
L ωR
)
,
where ωL = exp
(
−u
κDP
)
and ωR = exp
(
−u
κPD
)
. Hence
F−1
R,u = exp
(
∆
(u
κ
PD
))
exp
(
−∆
(u
κ
DP
))
F−1
L,u exp
(u
κ
(DP ⊗ 1 + 1⊗DP )
)
× exp
(
−u
κ
(PD ⊗ 1 + 1⊗ PD)
)
and
FL,uF
−1
R,u = ∆FL,u
(
exp
(
u
P
κ
D
)
exp
(
−uDP
κ
))
exp
(u
κ
(DP ⊗ 1 + 1⊗DP )
)
× exp
(
−u
κ
(PD ⊗ 1 + 1⊗ PD)
)
, (3.1)
where we have used the homomorphism property of the coproduct and its deformed form.
We can use the equalities5
eu
P
κ
D =
∞∑
n=0
(
uPκD
)n
n!
=
∞∑
n=0
(
u
P
κ
)n(D
n
)
and
euD
P
κ =
∞∑
n=0
(
uDP
κ
)n
n!
=
∞∑
n=0
(
u
P
κ
)n(D − 1
n
)
,
where we chose the order with P generators on the left. Now taking the difference of these two
expressions we obtain
eu
P
κ
D − euD
P
κ =
∞∑
n=0
(
u
P
κ
)n [(D
n
)
−
(
D − 1
n
)]
=
∞∑
n=1
(
u
P
κ
)n(D − 1
n− 1
)
= u
P
κ
euD
P
κ . (3.2)
5Here
(
D
n
)
denotes the generalised binomial coefficient:
(
D
n
)
= Dn
n!
= D(D−1)(D−2)...(D−(n−1))
n(n−1)(n−2)···1 .
Interpolations between Jordanian Twists Induced by Coboundary Twists 9
Therefore, we find, after multiplying both sides of (3.2) by e−uD
P
κ from the right
eu
P
κ
De−uD
P
κ = 1 + u
P
κ
. (3.3)
Inserting (3.3) into (3.1) we get
FL,uF
−1
R,u = ∆FL,u
(
1 + u
P
κ
)(
1
1⊗ 1 + uPκ ⊗ 1
)(
1
1⊗ 1 + u · 1⊗ P
κ
)
=
1
1⊗ 1 + u(1− u)Pκ ⊗
P
κ
,
which leads to
F−1
R,u = F−1
L,u
1
1⊗ 1 + u(1−u)
κ2
P ⊗ P
. (3.4)
Note that the twists FL,u and FR,u agree in the leading order of the deformation parameter, but
are different at higher orders. We point out that using star product in (2.4) and star product
in (2.12) and methods introduced in [40] one can also obtain the above relation (3.4).
Also one can write an explicit formula for the relation between RR,u and RL,u quantum
R-matrices. It has the following form
RL,u =
1
1⊗ 1 + u(1−u)
κ2
P ⊗ P
RR,u
(
1⊗ 1 +
u(1− u)
κ2
P ⊗ P
)
.
Hopf algebras
The two twists FL,u and FR,u describe two presentations of Hopf algebra (with the same corre-
sponding classical r-matrix). The coproducts and antipodes for momenta are the same
∆FL,u(Pµ) = ∆FR,u(Pµ), SFL,u(Pµ) = SFR,u(Pµ).
However this is not the case for dilatations
∆FL,u(D) 6= ∆FR,u(D), SFL,u(D) 6= SFR,u(D).
Coordinate realizations and star products
Comparing the realizations we also see the difference. Equation (2.14) has an extra term only
dependent on momenta whereas (2.6) does not.
Similarly in the formulae for the star products, the one coming from FR,u has an addition in
the form of the function G(u; k, q) which does not appear in the star product coming from FL,u.
Nevertheless, these two twists are only a change by coboundary twists from F0 and provide
equivalent Hopf algebra deformations (but with different representations).
4 Discussion on the real forms of the Jordanian deformations
For physical applications it is important to address the question if the symmetry Hopf algebras
and the deformed Weyl–Heisenberg algebra can be endowed with (compatible) ∗-structures. In
general, ∗-structure (real Lie algebra structure) can be introduced by an antilinear involutive
anti-automorphism ∗ : g→ g acting on the complex Lie algebra g. Thus the real coboundary Lie
10 A. Borowiec, D. Meljanac, S. Meljanac and A. Pacho l
bialgebra can be considered as a triple (g, ∗, r), where the skew-symmetric element r is assumed
to be anti-Hermitian, i.e.,6
r∗⊗∗ = −r = rτ .
The ∗-operation extends, by the property (XY )∗ = Y ∗X∗ (i.e., as an antilinear antiautomor-
phism), to the enveloping algebra Ug, as well as to quantized enveloping algebra, making both
of them associative ∗-algebras. Therefore, any quantized enveloping algebra admits a natural
∗-structure, inherited from the corresponding Lie bialgebra structure, which can be preserved
under twist deformation provided that the twisting element is ∗-unitary, i.e.,
F ∗⊗∗ = F−1. (4.1)
More exactly, we recall that a complex Hopf algebra H = (H,∆, S, ε) endowed with an
antilinear involutive anti-automorphism ∗ : H → H is called a real Hopf algebra or Hopf ∗-algebra
if the following compatibility conditions for coproducts, antipodes and counits are satisfied
∆(X∗) = (∆(X))∗⊗∗, S((S(X∗))∗) = X, ε(X∗) = ε(X) ∀X ∈ H,
where the standard (’unflipped’) way of ∗-operation acting on a tensor product (i.e., coproduct)
is assumed
(X ⊗ Y )∗ = X∗ ⊗ Y ∗.
In general, the twist deformation of quasitriangular Hopf algebra (H, R) give rise to quasitri-
angular Hopf algebra with new universal R-matrix RF = F τRF−1. If the real form of quasitri-
angular Hopf algebra is twisted by unitary twist then the deformed Hopf algebra is also real qua-
sitriangular Hopf algebra. More precisely, if (H,∆, S, ε, R, ∗) is a quasitriangular Hopf ∗-algebra
with R being real universal R-matrix (i.e., satisfying R∗⊗∗ = Rτ ) (resp. antireal if satisfying
R∗⊗∗ = R−1 ), then for any unitary, normalized 2-cocycle twist F = (F−1)?⊗? ∈ H⊗H the quan-
tized algebra (H,∆F , SF , ε, RF , ∗) is a quasitriangular Hopf ∗-algebra such that RF = F τRF−1
is real (resp. antireal).
In our case only two Jordanian twists F0 and F1 are unitary, i.e., satisfy the condition (4.1)
provided that we assume the following ∗-conjugations
P ∗µ = Pµ, and D∗ = −D, (4.2)
which are compatible with the commutation relations [Pµ, D] = Pµ, [Pµ, Pν ] = 0.7 Therefore, the
corresponding twisted deformations are the Hopf ∗-algebras with the same ∗-structure. Instead,
for generic value of the parameter u ∈ R two families of twists are related to each other by
F ∗⊗∗L,u = F−1
R,u,
[
∆FL,u(X)
]∗⊗∗
= ∆FR,u(X∗),
[
SFL,u(X)
]∗
= SFR,1−u(X∗)|−κ.
Therefore, the well-known purely Jordanian twists F0 and F1 remain only unitary with respect
to the conjugation (4.2). This situation can change, if we use the method proposed by S. Majid
[32, Proposition 2.3.7, p. 59] and admit deformation of the original ∗-structure. Before, one
needs to check if the twist satisfies the following condition
(S ⊗ S)
(
F ∗⊗∗
)
= F τ , (4.3)
6More detailed exposition of the background material for the present section, as well as more complete list of
references, can be found in [7].
7It can be observed that these commutation relations determine the generator D up an additive constant while
each generator Pµ can be multiplied by a constant. For this reason the conjugation D∗ = −D+c, P ∗
µ = eibPµ (c, b
are real constants) would be admissible too. In fact, our Weyl–Heisenberg ∗-algebra realisation (2.3) is compatible
with the assumption that D∗ = −D − n, P ∗
µ = Pµ, where n denotes the spacetime dimension.
Interpolations between Jordanian Twists Induced by Coboundary Twists 11
where the antipode map S and the conjugations ∗ are taken before deformation. If it does, then
one can define new quantized †-structure:
( )† = S−1(U)( )∗S−1
(
U−1
)
,
which makes the twisted Hopf algebra real. Here U is related with the twist by the formula
U =
∑
i
f
(1)
i S
(
f
(2)
i
)
and U−1 =
∑
i
S
(
f
−(1)
i
)
f
−(2)
i with the short cut notation for the twist
introduced as follows: F =
∑
i
f
(1)
i ⊗ f
(2)
i and its inverse as: F−1 =
∑
i
f
−(1)
i ⊗ f−(2)
i .
4.1 New quantized †-structure for FL,u=1
2
twist
One can explicitly check that only for one value of the parameter u, namely u = 1
2
8 the condi-
tion (4.3) for the Majid’s method is satisfied. By direct calculation one gets
l.h.s. = (S ⊗ S)
(
F ∗⊗∗
L, 1
2
)
=
(
FL, 1
2
)∣∣
−κ
and
r.h.s. = (FL, 1
2
)τ = exp
(
1
2κ
(DP ⊗ 1 + 1⊗DP )
)
(F0)τ exp
(
∆
(
− 1
2κ
DP
))
= exp
(
1
2κ
(DP ⊗ 1 + 1⊗DP )
)
(F1)|−κ exp
(
∆
(
− 1
2κ
DP
))
=
[
exp
(
− 1
2κ
(DP ⊗ 1 + 1⊗DP )
)
F1 exp
(
∆
(
1
2κ
DP
))]
|−κ
=
[
exp
(
1
2κ
(DP ⊗ 1 + 1⊗DP )
)
F0 exp
(
−∆
(
1
2κ
DP
))]
|−κ =
(
FL, 1
2
)
|−κ.
Hence
(S ⊗ S)
(
F ∗⊗∗
L, 1
2
)
=
(
FL, 1
2
)∣∣
−κ =
(
FL, 1
2
)τ
.
Therefore
X† = −SFL, 12 (X∗), ∀X ∈ H.
Analogously, one can show that FR, 1
2
also satisfies the condition (4.3)
(S ⊗ S)
(
F ∗⊗∗
R, 1
2
)
=
(
FR, 1
2
)∣∣
−κ =
(
FR, 1
2
)τ
and the new star structure
X† = −SFR, 12 (X∗), ∀X ∈ H.
It shows that for u = 1
2 we can introduce an exotic Hopf †-algebra structure which, however, in
the classical limit reduces to the standard ∗ one.
8Note that only for this value of the parameter u the twists FR,u and FL,u are r-symmetric.
12 A. Borowiec, D. Meljanac, S. Meljanac and A. Pacho l
4.2 Coboundary Hopf ∗-algebra
Actually, the best way to obtain coboundary unitary twist preserving a given ∗-Hopf algebra
structure is by using a unitary coboundary element ω∗ = ω−1. As a particular example let us
consider a one-parameter family of such elements
ωLR = exp
(
− u
2κ
(DP + PD)
)
.
Then the new generalised family of the Jordanian twists
FLR,u =
(
ω−1
LR ⊗ ω
−1
LR
)
F0∆(ωLR)
is unitary for any real parameter u and satisfies
F ∗⊗∗LR,u = F−1
LR,u
with P ∗µ = Pµ, D∗ = −D. Note that FLR,u is r-symmetric again only for u = 1
2 and differs from
both FL, 1
2
and FR, 1
2
.
FLR,u provides, via the usual twist deformation, a new Hopf ∗-algebra
∆FLR,u(Pµ) =
√
1⊗ 1 +
u(1− u)
κ2
P ⊗ P∆FL,u(Pµ)
1√
1⊗ 1 + u(1−u)
κ2
P ⊗ P
=
Pµ ⊗
(
1 + u
κP
)
+
(
1− (1−u)
κ P
)
⊗ Pµ
1⊗ 1 + u(1−u)
κ2
P ⊗ P
,
∆FLR,u(D) =
√
1⊗ 1 +
u(1− u)
κ2
P ⊗ P∆FL,u(D)
1√
1⊗ 1 + u(1−u)
κ2
P ⊗ P
=
√
1⊗ 1 +
u(1− u)
κ2
P ⊗ P
(
D ⊗ 1
1 + u
κP
+
1
1− (1−u)
κ P
⊗D
)
×
√
1⊗ 1 +
u(1− u)
κ2
P ⊗ P
and
SFLR,u(Pµ) = − Pµ
1− (1− 2u) 1
κP
,
SFLR,u(D) = −
√(
1− (1− u)Pκ
)(
1 + u
κP
) (
1− (1− 2u)
P
κ
)
D
√ (
1 + u
κP
)
1− (1− u)Pκ
(
1− (1− 2u)
P
κ
)
.
These formulae can be calculated by using the following relation between the twists FLR,u, FL,u
and FR,u:
F−1
LR,u = F−1
L,u
1√
1⊗ 1 + u(1−u)
κ2
P ⊗ P
= F−1
R,u
√
1⊗ 1 +
u(1− u)
κ2
P ⊗ P .
We can also provide the relation between quantum R-matrices
RLR,u =
√
1⊗ 1 +
u(1− u)
κ2
P ⊗ PRL,u
1√
1⊗ 1 + u(1−u)
κ2
P ⊗ P
.
Interpolations between Jordanian Twists Induced by Coboundary Twists 13
5 Conclusions
Jordanian deformations have been of interest in some recent literature [1, 2, 3, 5, 13, 19, 20, 25,
26, 34, 35, 36, 45, 50, 55]. In our previous paper [41] we have studied the simple generalisation
of the locally r-symmetric Jordanian twist. Following that idea now we have found another
possible way of interpolating between two Jordanian twists F0 and F1. In both cases, we have
introduced one real valued parameter u and obtained the family of Jordanian twists which
provides interpolation between the original unitary Jordanian twist (for u = 0) and its flipped
version (for u = 1, up to minus sign in the deformation parameter). If D∗ = −D and P ∗µ = Pµ
both twists F0 and F1 are ∗-unitary while those for u 6= 0, 1 are not. Only in the case of u = 1
2
the interpolating twists provide the real Hopf algebra structures with the deformed ∗-structures
(obtained by deformation techniques from [32]). Also only for this value of the parameter u the
twists FR,u and FL,u and FLR,u are r-symmetric. However, using unitary coboundary elements
ω∗ = ω−1 one can interpolate preserving the original real ∗-structure. The new family of
Jordanian twists FR,u (2.7), as all Jordanian twists mentioned in this work, provides the so-
called κ-Minkowski noncommutative spacetime and has the support in the Poincaré–Weyl or
conformal algebras as deformed symmetries of this noncommutative spacetime.
In this paper, starting from the twist (2.7), we have found deformed Hopf algebra symmetry
(2.8)–(2.11), star products (2.12) and corresponding realizations for noncommutative coordi-
nates x̂µ (2.14). Noncommutative coordinates induce the deformation of Weyl–Heisenberg alge-
bra (2.15) as well. Even though both of the proposed twists, the previous one FL,u from [41] and
the one introduced here FR,u provide the κ-Minkowski spacetime and have the support in the
Poincaré–Weyl or conformal algebras as deformed symmetries of this noncommutative space-
time, the realizations and star products they induce differ. The new type of star product (2.12)
obtained here contains additional term depending only on momenta G(u; k, q). The additional
term χµ(P ) also appears in the realization of noncommuting coordinates in (2.14). However,
the Weyl–Heisenberg algebra form is the same in both FL,u and FR,u deformed cases (2.15).
Therefore, mathematically equivalent deformations, may lead to differences in the physical
phenomena. Many authors [1, 13, 36] have used star products to discuss physical consequences
of deformations. In those cases the two families of Jordanian twists FL,u, FR,u would lead
to different outcomes. It would be interesting, for example, to investigate the deformation of
differential and integral calculus in the context of the new versions of Jordanian twists proposed
here and their application to second quantization [17]. Also the differences in realizations of
noncommutative coordinates could lead to different physical predictions, see, e.g., [4] where the
influence of different realizations on modified dispersion relations between energy and momenta
as well as time delay parameter was investigated.
A Exponential formula, normal ordering
and Weyl–Heisenberg algebra
We recall that in the simplest case the Weyl–Heisenberg algebra W1 (over the field C of complex
numbers) can be defined abstractly as a universal associative and unital algebra over C with
two generators x, p satisfying the relation
xp− px = i,
where i ∈ C stands for the imaginary unit. Usually, the generator p can be identified with
the derivative −i d/dx. Such realization makes it easy to remember the canonical action of W1
onto the space of polynomial functions in one variable C[x]. Therefore, any element a ∈ W1
admits a canonical presentation either in the form of differential operator a =
Np∑
s=0
as(x)ps,
14 A. Borowiec, D. Meljanac, S. Meljanac and A. Pacho l
where as(x) ∈ C[x] or in the normal ordered form a =
Nx∑
r=0
xrar(p), where ar(p) ∈ C[p], see,
e.g., [33]. This algebra can be also defined as a smash product of two polynomial algebras,
i.e., W1 = C[x] o C[p], where C[x] plays a role of module algebra over the Hopf algebra C[p].9
Using smash product definition one can naturally extend W1 to C[[x]] oC[p], i.e., replacing the
polynomial algebra C[x] by its x-adic extension C[[x]] of formal power series (see, e.g., [12]), which
bears the structure of (left) C[[x]] module. Its topological completion provides the algebra Ŵ1
with elements having a form
∞∑
r=0
xrar(p), ar(p) ∈ C[p].
For our purposes, however, one needs further extension by introducing a new commuting
formal variable k and taking Ŵ1[[k]] with elements of the form
∞∑
r=0
xrar(k, p) =
∞∑
r,s=0
xrksar,s(p),
where ar,s(p) ∈ C[p] as before and ar(k, p) =
∞∑
s=0
ar,s(p)k
s ∈ C[p][[k]]. Now we are in position to
formulate the following
Proposition A.1. In this framework:
i) For any element φ ≡ φ(p) ∈ C[p] there exists a unique element Φ ≡ Φ(k, p) ∈ C[p][[k]]
such that
exp(ikxφ(p)) =
∞∑
r=0
(ix)r
r!
[Φ(k, p)]r
holds true in Ŵ1[[k]]. One should notice that the last expression can be denoted as
: exp(ixΦ(k, p)):, where : . . . : denotes normal ordering of the generators x, p (i.e., x’s
left from p’s).
ii) Moreover
Φ(k, p) = J(k, p)− p,
where
J(k, p) = e−ikxφ(p)peikxφ(p). (A.1)
iii) J(k, p) turns out to be a unique (formal) solution of the (formal) partial differential equa-
tion
∂
∂k
J(k, p) = φ(J(k, p)) (A.2)
with the boundary condition: J(0, p) = p.
Proof. Since exp(ikxφ(p)) ∈ Ŵ1[[k]], one can employ the adjoint action adpa = [p, a] to calcu-
late
exp(ikxφ(p)) =
∞∑
n=0
(ix)n
n!
Φn(k, p),
(adp)
m exp(ikxφ(p)) =
∞∑
n=m
(ix)n−m
(n−m)!
Φn(k, p),
(adp)
m exp(ikxφ(p))|x=0 = Φm(k, p).
9For the definition see, e.g., [10] and references therein.
Interpolations between Jordanian Twists Induced by Coboundary Twists 15
Hence
exp(ikxφ(p)) =
∞∑
n=0
(ix)n
n!
(
(adp)
n(exp(ikxφ(p))
)∣∣
x=0
. (A.3)
On the other hand, introducing J(k, p) by formula (A.1), one gets (cf. (A.3))
adp
(
eikxφ(p)
)
=
[
p, eikxφ(p)
]
= eikxφ(p)
(
e−ikxφ(p)peikxφ(p) − p
)
= eikxφ(p)(J(k, p)− p) = eikxφ(p)Φ(k, p),
i.e., Φ(k, p) = J(k, p)− p. Similarly, by induction
(adp)
n
(
eikxφ(p)
)
= eikxφ(p)(Φ(k, p))n.
It follows that
Φn(k, p) = (adp)
n
(
eikxφ(p)
)∣∣
x=0
= (Φ(k, p))n = (J(k, p)− p)n,
hence
exp(ikxφ(p)) =
∞∑
n=0
(ix)n
n!
(Φ(k, p))n = : exp(ikxΦ(k, p)):.
Finally, differentiation of (A.1) gives
∂
∂k
J(k, p) = e−ikxφ(p)[−ix, p]φ(p)eikxφ(p) = e−ikxφ(p)φ(p)eikxφ(p) = φ(J(k, p)).
This provides the equation (A.2) together with the boundary condition: J(0, p) = p. �
B The multidimensional case
Replacing W1 by the Weyl–Heisenberg algebra WN with 2N generators xα, pβ:
xαxβ − xβxα = pαpβ − pβpα = 0, xαpβ − pβxα = iδαβ ,
where α, β = 0, 1, . . . , N − 1, allows for the following generalisation [48]
Proposition B.1.10 For arbitrary realization φ(p) for the noncommutative coordinates
x̂µ = xαφµα(p) it holds:
i)
exp
(
ikαx̂
α
)
= : exp
(
ixαΦα(k, p)
)
:,
where : . . . : denotes normal ordering of the generators x, p (i.e., x’s left from p’s).
ii)
Φµ(k, p) = e−ikαx̂αpµeikαx̂α − pµ = Jµ(k, p)− pµ,
i.e.,
Jµ(k, p) = e−ikαx̂αpµeikαx̂α . (B.1)
10Proof of Proposition B.1 and it’s generalisations will be given elsewhere.
16 A. Borowiec, D. Meljanac, S. Meljanac and A. Pacho l
iii) Jµ(k, p) satisfies
d
dλ
Jµ(λk, p) = kαφ
α
µ(J(λk, p)) (B.2)
with the boundary condition: Jµ(0, p) = pµ.
The last relation (B.2) can be shown in the following way. Applying kα
∂
∂kα
on both sides
to (B.1) we have
e−ikx̂[−ikx̂, pµ]eikx̂ = kα
∂
∂kα
(Jµ(k, p)).
Using the form of the realization for x̂µ = xαφµα(p), we obtain
kα
∂
∂kα
(Jµ(k, p)) = kαφ
α
µ(J(k, p)).
In order to simplify this equation we introduce the change of variables: kα → λkα and use the
identity
λ
d
dλ
= λkα
∂
∂(λkα)
= kα
∂
∂kα
,
which leads to (B.2).
Further generalisation admits more general class of realizations. If
x̂µ = xαφµα(p) + χµ(p)
for φµα(p) and χµ(p) as in (2.14) then we have
exp
(
ikαx̂
α
)
= : exp
(
ixα(Jα(k, p)− pα)
)
: exp(iQ(k, p)).
Therefore we get
e−ikαxβφαβ (p)pµeikx̂ = Jµ(k, p)eiQ(k,p).
Applying kα
∂
∂kα
on both sides we finally obtain
kα
∂
∂kα
Q(k, p) = kαχ
α(J(k, p)),
i.e., after changing variables kα → λkα we find
d
dλ
Q(λk, p) = kαχ
α(J(λk, p)).
Note that if χα(p) 6= 0 then
eikαx̂α B eiqβx
β
= eiJα(k,q)xα+iQ(k,q),
where the action is defined in (C.1) below. For kα = 0 it holds: Jµ(0, q) = qµ and Q(0, q) = 0.
If qµ = 0 then Jµ(k, 0) = Kµ(k) and Q(k, 0) = g(k).
Interpolations between Jordanian Twists Induced by Coboundary Twists 17
C Star products
C.1 Star products for FL,u: general formulas
If we start with realization
x̂µ = xαφµα(P ),
then, for action ., defined by
xµ . f(x) = xµf(x), Pµ . f(x) = −i
∂f(x)
∂xµ
, (C.1)
it holds
eik·x̂ . 1 = eikµxαφ
µ
α(P ) . 1 = eiK(k)·x,
eik·x̂ . eiq·x = eikµxαφ
µ
α(P ) . eiq·x = eiJ(k,q)·x,
where functions Kµ(k) and Jµ(k, q) can be calculated from the following differential equations
dKµ(λk)
dλ
= φµα(K(λk))kα, (C.2)
dJµ(λk, q)
dλ
= φµα(J(λk, q))kα, (C.3)
with boundary conditions Kµ(0) = 0 and Jµ(k, 0) = Kµ(k), Jµ(0, q) = qµ.
The star product is given by
eik·x ? eiq·x = eiK−1(k)·x̂ . eiq·x = eiJ(K−1(k),q)·x = eiD(k,q)·x,
where
D(k, q) = J
(
K−1(k), q
)
with the inverse function of Kµ(k) defined as Kµ
(
K−1(k)
)
= K−1
µ (K(k)) = kµ.
C.2 Star products for FR,u: general formulas
For a more general realization given by
x̂µ = xαφµα(P ) + χµ(P ),
it holds
eik·x̂ . 1 = eiK(k)·x+ig(k),
eik·x̂ . eiq·x = eiJ(k,q)·x+iQ(k,q).
K(k), J(k, q) satisfy the same differential equation as in Appendix C.1. Similarly we can
determine g(k) and Q(k, q) by differentiating with respect to λ [29, 40]
dg(λk)
dλ
= k · χ(K(λk)), (C.4)
dQ(λk, q)
dλ
= k · χ(J(λk, q)). (C.5)
18 A. Borowiec, D. Meljanac, S. Meljanac and A. Pacho l
The boundary condition is Q(k, 0) = g(k), g(0) = Q(0, q) = 0. This gives [29, 40]
g(k) = Q(k, 0) =
∫ 1
0
dλ[k · χ(K(λk))],
Q(k, q) =
∫ 1
0
dλ[k · χ(J(λk, q))].
The star product is
eik·x ? eiq·x = eiK−1(k)·x̂−g(K−1(k)) . eiq·x = eiJ(K−1(k),q)·x+iQ(K−1(k),q)−iQ(K−1(k),0).
Now we take
D(k, q) = J
(
K−1(k), q
)
,
G(k, q) = Q
(
K−1(k), q
)
−Q
(
K−1(k), 0
)
. (C.6)
Therefore, it follows that
eik·x ? eiq·x = eiD(k,q)·x+iG(k,q).
C.3 FL,u and FR,u: explicit calculations
Realizations of noncommutative coordinates for FL,u is (2.6)
x̂µ =
[
xµ +
i(1− u)
κ
vµD
](
1 +
u
κ
P
)
, u ∈ [0, 1],
and for FR,u (2.14)
x̂µ =
(
xµ +
i(1− u)
κ
vµD
)(
1 +
u
κ
P
)
+
iu(1− u)
κ2
vµP, u ∈ [0, 1].
From these realizations the form of the function φµα(P ) can be read as
φµα(P ) =
(
δµα −
(1− u)
κ
vµPα
)(
1 +
u
κ
P
)
,
where P = vαPα is used as a shortcut.
Note that it is the same for both realizations (2.6), (2.14). Therefore, both of these realizations
have the same form of the functions K(k), K−1(k), J(k, q) and D(k, q), see below for the explicit
calculations.
To obtain Kµ(k) and Jµ(λk, q), the following differential equations (C.2), (C.3) need to be
solved
dKµ(λk)
dλ
= φµα(K(λk))kα =
(
δµα −
(1− u)
κ
vµKα(λk)
)(
1 +
u
κ
K(λk)
)
kα, (C.7)
dJµ(λk, q)
dλ
= φµα(J(λk, q))kα =
(
δµα −
(1− u)
κ
vµJα(λk, q)
)(
1 +
u
κ
J(λk, q)
)
kα, (C.8)
after using the explicit form of φ. Here the shortcut notation P = vαPα has been extended to
K(λk) = vαKα(λk) and J(λk, q) = vαJα(λk, q). The solution of the first equation (C.7) is
Kµ(k) = kµ
e
1
κ
v·k − 1
1
κv · k
1
(1− u)e
1
κ
v·k + u
. (C.9)
Interpolations between Jordanian Twists Induced by Coboundary Twists 19
The inverse function
(
K−1
)µ
(k) is
(
K−1
)µ
(k) = kµ
1
1
κv · k
ln
(
1 + u
κ(v · k)
1− (1−u)
κ (v · k)
)
.
The function Jµ(k, q) is calculated similarly. The solution of the second equation (C.8) is
Jµ(k, q) =
Kµ(k)
(
1 + u
κ(v · q)
)
+
(
1− (1−u)
κ (v ·K(k))
)
qµ
1 + u(1−u)
κ2
(v ·K(k))(v · q)
(C.10)
and Kµ(k) = Jµ(k, 0).
For FR,u case, we have χµ(P ) = iu(1−u)
κ2
vµP , so we can determine g(k) and Q(k, q) by differ-
entiating with respect to λ, i.e., using (C.4), (C.5),
dg(λk)
dλ
= k · χ(K(λk)) =
iu(1− u)
κ2
(k · v)(v ·K(λk)),
dQ(λk, q)
dλ
= k · χ(J(λk, q)) =
iu(1− u)
κ2
(k · v)(v · J(λk, q)).
To find the solution to the first equation we use (C.9)
v ·K(k) =
κ
(
e
1
κ
v·k − 1
)
(1− u)e
1
κ
v·k + u
,
and for the second equation we take (C.10)
v · J(k, q) =
(v ·K(k))
(
1 + u
κ(v · q)
)
+
(
1− (1−u)
κ (v ·K(k))
)
(v · q)
1 + u(1−u)
κ2
(v ·K(k))(v · q)
.
So g(k) and Q(k, q) are calculated as
g(k) = Q(k, 0) =
iu(1− u)
κ2
(k · v)
∫ 1
0
dλ[v ·K(λk)],
Q(k, q) =
iu(1− u)
κ2
(k · v)
∫ 1
0
dλ[v · J(λk, q)].
The solutions are
g(k) = Q(k, 0) = i ln
(
ue−
(1−u)
κ
v·k + (1− u)e
u
κ
v·k
)
,
Q(k, q) = i ln
(
u
(
1− (1− u)
1
κ
v · q
)
e−
(1−u)
κ
v·k + (1− u)
(
1 + u
1
κ
v · q
)
e
u
κ
v·k
)
.
Using equations (C.6), it follows that
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)
and
G(u; k, q) = i ln
(
1 +
u(1− u)
κ2
(v · k)(v · q)
)
,
i.e.,
eiG(u;k,q) =
1
1 + u(1−u)
κ2
(v · k)(v · q)
.
Compare the above with (2.5) and (2.13).
20 A. Borowiec, D. Meljanac, S. Meljanac and A. Pacho l
Acknowledgments
This work has been supported by COST (European Cooperation in Science and Technology)
Action MP1405 QSPACE. AB is supported by Polish National Science Center (NCN), project
UMO-2017/27/B/ST2/01902. We are grateful to Zoran Škoda for his comments. We would like
to thank the referees for their constructive input.
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22 A. Borowiec, D. Meljanac, S. Meljanac and A. Pacho l
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1 Introduction
2 Two families of twists interpolating between Jordanian twists
2.1 FL,u family of twists with dilatation on the left
2.1.1 Hopf algebra
2.1.2 Coordinate realizations and star product
2.2 FR,u family of twists with dilatation on the right
2.2.1 Hopf algebra
2.2.2 Coordinate realizations and star product
3 Relations between two families FL,u and FR,u
4 Discussion on the real forms of the Jordanian deformations
4.1 New quantized †-structure for FL,u=12 twist
4.2 Coboundary Hopf *-algebra
5 Conclusions
A Exponential formula, normal ordering and Weyl–Heisenberg algebra
B The multidimensional case
C Star products
C.1 Star products for FL,u: general formulas
C.2 Star products for FR,u: general formulas
C.3 FL,u and FR,u: explicit calculations
References
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| id | nasplib_isofts_kiev_ua-123456789-210241 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T21:24:52Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Borowiec, A. Meljanac, D. Meljanac, S. Pachoł, A. 2025-12-04T13:09:22Z 2019 Interpolations between Jordanian Twists Induced by Coboundary Twists / A. Borowiec, D. Meljanac. S. Meljanac, A. Pachoł // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 55 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 81T75; 16T05; 17B37; 81R60 arXiv: 1812.05535 https://nasplib.isofts.kiev.ua/handle/123456789/210241 https://doi.org/10.3842/SIGMA.2019.054 We propose a new generalisation of the Jordanian twist (building on the previous idea from [Meljanac S., Meljanac D., Pachoł A., Pikutić D., J. Phys. A: Math. Theor. 50 (2017), 265201, 11 pages]). Obtained this way, the family of Jordanian twists allows for interpolation between two simple Jordanian twists. This new version of the twist provides an example of a new type of star product and the realization of noncommutative coordinates. Real forms of new Jordanian deformations are also discussed. Exponential formulae, used to obtain coproducts and star products, are presented with details. This work has been supported by COST (European Cooperation in Science and Technology) Action MP1405 QSPACE. AB is supported by the Polish National Science Center (NCN), project UMO-2017/27/B/ST2/01902. We are grateful to Zoran Škoda for his comments. We would like to thank the referees for their constructive input. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Interpolations between Jordanian Twists Induced by Coboundary Twists Article published earlier |
| spellingShingle | Interpolations between Jordanian Twists Induced by Coboundary Twists Borowiec, A. Meljanac, D. Meljanac, S. Pachoł, A. |
| title | Interpolations between Jordanian Twists Induced by Coboundary Twists |
| title_full | Interpolations between Jordanian Twists Induced by Coboundary Twists |
| title_fullStr | Interpolations between Jordanian Twists Induced by Coboundary Twists |
| title_full_unstemmed | Interpolations between Jordanian Twists Induced by Coboundary Twists |
| title_short | Interpolations between Jordanian Twists Induced by Coboundary Twists |
| title_sort | interpolations between jordanian twists induced by coboundary twists |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210241 |
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