Quantum Deformations of Relativistic Symmetries
Quantum Deformations of Relativistic Symmetries∗
V.N. Tolstoy†
Institute of Nuclear Physics, Moscow State University,
119 992 Moscow, Russia; e-mail: tolstoy@nucl-th.sinp.msu.ru
Abstract
We discussed quantum deformations of D = 4 Lorentz and Poincaré algebras. In
the case of Poincaré algebra it is shown that almost all classical r-matrices of S. Za-
krzewski classification correspond to twisted deformations of Abelian and Jordanian
types. A part of twists corresponding to the r-matrices of Zakrzewski classification
are given in explicit form.
1 Introduction
The quantum deformations of relativistic symmetries are described by Hopf-algebraic
deformations of Lorentz and Poincaré algebras. Such quantum deformations are classified
by Lorentz and Poincaré Poisson structures. These Poisson structures given by classical
r-matrices were classified already some time ago by S. Zakrzewski in [1] for the Lorentz
algebra and in [2] for the Poincaré algebra. In the case of the Lorentz algebra a complete
list of classical r-matrices involves the four independent formulas and the corresponding
quantum deformations in different forms were already discussed in literature (see [3, 4, 5,
6, 7]). In the case of Poincaré algebra the total list of the classical r-matrices, which satisfy
the homogeneous classical Yang-Baxter equation, consists of 20 cases which have various
numbers of free parameters. Analysis of these twenty solutions shows that each of them
can be presented as a sum of subordinated r-matrices which almost all are of Abelian
and Jordanian types. A part of twists corresponding to the r-matrices of Zakrzewski
classification are given in explicit form.
2 Preliminaries
Let r be a classical r-matrix of a Lie algebra g, i.e. r ∈
∧ g and r satisfies to the classical
Yang–Baxter equation (CYBE)
[r12, r13 + r23] + [r13, r23] = Ω , (2.1)
∗Invited talk at the XXII Max Born Symposium ”Quantum, Super and Twistors”, September 27-29,
2006 Wroclaw (Poland), in honour of Jerzy Lukierski.
†Supported by the grants RFBR-05-01-01086 and FNRA NT05-241455GIPM.
http://arxiv.org/abs/0704.0081v1
where Ω is g-invariant element, Ω ∈ (
∧ g)g. We consider two types of the classical
r-matrices and corresponding twists.
Let the classical r-matrix r = rA has the form
xi ∧ yi , (2.2)
where all elements xi, yi (i = 1, . . . , n) commute among themselves. Such an r-matrix is
called of Abelian type. The corresponding twist is given as follows
= exp
= exp
xi ∧ yi
. (2.3)
This twisting two-tensor F := Fr
satisfies the cocycle equation
F 12(∆⊗ id)(F ) = F 23(id⊗∆)(F ) , (2.4)
and the ”unital” normalization condition
(ǫ⊗ id)(F ) = (id⊗ ǫ)(F ) = 1 . (2.5)
The twisting element F defines a deformation of the universal enveloping algebra U(g)
considered as a Hopf algebra. The new deformed coproduct and antipode are given as
follows
∆(F )(a) = F∆(a)F−1 , S(F )(a) = uS(a)u−1 (2.6)
for any a ∈ U(g), where ∆(a) is a co-product before twisting, and u =
i S(f
i ) if
i ⊗ f
Let the classical r-matrix r = rJ(ξ) has the form
rJ(ξ) = ξ
xν ∧ yν
, (2.7)
where the elements xν , yν (ν = 0, 1, . . . , n) satisfy the relations
[x0, y0] = y0 , [x0, xi] = (1− ti)xi , [x0, yi] = tiyi ,
[xi, yj] = δijy0 , [xi, xj ] = [yi, yj] = 0 , [y0, xj ] = [y0, yj] = 0 ,
(2.8)
(i, j = 1, . . . , n), (ti ∈ C). Such an r-matrix is called of Jordanian type. The corresponding
twist is given as follows [8, 9]
= exp
xi ⊗ yi e
−2tiσ
exp(2x0 ⊗ σ) , (2.9)
1Here entering the parameter deformation ξ is a matter of convenience.
2It is easy to verify that the two-tensor (2.7) indeed satisfies the homogenous classical Yang-Baxter
equation (2.1) (with Ω = 0), if the elements xν , yν (ν = 0, 1, . . . , n) are subject to the relations (2.8).
where σ := 1
ln(1 + ξy0).
Let r be an arbitrary r-matrix of g. We denote a support of r by Sup(r)4. The
following definition is useful.
Definition 2.1 Let r1 and r2 be two arbitrary classical r-matrices. We say that r2 is
subordinated to r1, r1 ≻ r2, if δr1(Sup(r2)) = 0, i.e.
(x) := [x⊗ 1 + 1⊗ x, r1] = 0 , ∀x ∈ Sup(r2) . (2.10)
If r1 ≻ r2 then r = r1 + r2 is also a classical r-matrix (see [15]). The subordination
enables us to construct a correct sequence of quantizations. For instance, if the r-matrix
of Jordanian type (2.7) is subordinated to the r-matrix of Abelian type (2.2), rA ≻ rJ ,
then the total twist corresponding to the resulting r-matrix r = rA+ rJ is given as follows
Fr = Fr
. (2.11)
The further definition is also useful.
Definition 2.2 A twisting two-tensor Fr(ξ) of a Hopf algebra, satisfying the conditions
(2.4) and (2.5), is called locally r-symmetric if the expansion of Fr(ξ) in powers of the
parameter deformation ξ has the form
Fr(ξ) = 1 + c r +O(ξ
2) . . . (2.12)
where r is a classical r-matrix, and c is a numerical coefficient, c 6= 0.
It is evident that the Abelian twist (2.3) is globally r-symmetric and the twist of Jordanian
type (2.9) does not satisfy the relation (2.12), i.e. it is not locally r-symmetric.
3 Quantum deformations of Lorentz algebra
The results of this section in different forms were already discussed in literature (see
[3, 4, 5, 6, 7]).
The classical canonical basis of the D = 4 Lorentz algebra, o(3, 1), can be described
by anti-Hermitian six generators (h, e±, h
′, e′±) satisfying the following non-vanishing
commutation relations5:
[h, e±] = ±e± , [e+, e−] = 2h , (3.1)
[h, e′±] = ±e
± , [h
′, e±] = ±e
± , [e±, e
∓] = ±2h
′ , (3.2)
[h′, e′±] = ∓e± , [e
−] = −2h , (3.3)
and moreover
x∗ = −x (∀ x ∈ o(3, 1)) . (3.4)
3The corresponding twists for Lie algebras sl(n), so(n) and sp(2n) were firstly constructed in the
papers [10, 11, 12, 13].
4The support Sup(r) is a subalgebra of g generated by the elements {xi, yi} if r =
xi ∧ yi.
5Since the real Lie algebra o(3, 1) is standard realification of the complex Lie sl(2,C) these relations
are easy obtained from the defining relations for sl(2,C), i.e. from (3.1).
A complete list of classical r-matrices which describe all Poisson structures and generate
quantum deformations for o(3, 1) involve the four independent formulas [1]:
r1 = α e+ ∧ h , (3.5)
r2 = α (e+ ∧ h− e
+ ∧ h
′) + 2β e′+ ∧ e+ , (3.6)
r3 = α (e
+ ∧ e− + e+ ∧ e
−) + β (e+ ∧ e− − e
+ ∧ e
−)− 2γ h ∧ h
′ , (3.7)
r4 = α
e′+ ∧ e− + e+ ∧ e
− − 2h ∧ h
± e+ ∧ e
+ . (3.8)
If the universal R-matrices of the quantum deformations corresponding to the classical
r-matrices (3.5)–(3.8) are unitary then these r-matrices are anti-Hermitian, i.e.
r∗j = −rj (j = 1, 2, 3, 4) . (3.9)
Therefore the ∗-operation (3.4) should be lifted to the tensor product o(3, 1) ⊗ o(3, 1).
There are two variants of this lifting: direct and flipped, namely,
(x⊗ y)∗ = x∗ ⊗ y∗ (∗ − direct) , (3.10)
(x⊗ y)∗ = y∗ ⊗ x∗ (∗ − flipped) . (3.11)
We see that if the ”direct” lifting of the ∗-operation (3.4) is used then all parameters in
(3.5)–(3.8) are pure imaginary. In the case of the ”flipped” lifting (3.11) all parameters
in (3.5)–(3.8) are real.
The first two r-matrices (3.5) and (3.6) satisfy the homogeneous CYBE and they are
of Jordanian type. If we assume (3.10), the corresponding quantum deformations were
described detailed in the paper [6] and they are entire defined by the twist of Jordanian
type:
= exp (h⊗ σ) , σ =
ln(1 + αe+) (3.12)
for the r-matrix (3.5), and
= exp
σ ∧ ϕ
exp (h⊗ σ − h′ ⊗ ϕ) , (3.13)
(1 + αe+)
2+ (αe′+)
, ϕ = arctan
1 + αe+
(3.14)
for the r-matrix (3.6). It should be recalled that the twists (3.12) and (3.13) are not
locally r-symmetric. A locally r-symmetric twist for the r-matrix (3.5) was obtained in
[14] and it has the following complicated formula:
= exp
∆(h)−
sinhαe+
⊗ e−αe++ eαe+⊗ h
sinhαe+
α∆(e+)
sinhα∆(e+)
, (3.15)
where ∆ is a primitive coproduct.
The last two r-matrices (3.7) and (3.8) satisfy the non-homogeneous (modified) CYBE
and they can be easily obtained from the solutions of the complex algebra o(4,C) ≃
sl(2,C)⊕ sl(2,C) which describes the complexification of o(3, 1). Indeed, let us introduce
the complex basis of Lorentz algebra (o(3, 1) ≃ sl(2;C) ⊕ sl(2,C)) described by two
commuting sets of complex generators:
(h + ıh′) , E1± =
(e± + ıe
±) , (3.16)
(h− ıh′) , E2± =
(e± − ıe
±) , (3.17)
which satisfy the relations (compare with (3.1))
[Hk, Ek±] = ±Ek± , [Ek+, Ek−] = 2Hk (k = 1, 2) . (3.18)
The ∗-operation describing the real structure acts on the generatorsHk, and Ek± (k = 1, 2)
as follows
H∗1 = −H2 , E
1± = −E2± , H
2 = −H1 , E
2± = −E1± . (3.19)
The classical r-matrix r3, (3.7), and r4, (3.8), in terms of the complex basis (3.16), (3.17)
take the form
r3 = r
1 + r
r′3 := 2(β + ıα)E1+ ∧ E1− + 2(β − ıα)E2+ ∧ E2− ,
r′′3 := 4ıγ H2 ∧H1 ,
(3.20)
r4 = r
4 + r
r′4 := 2ıα(E1+ ∧ E1− −E2+ ∧ E2− − 2H1 ∧H2) ,
r′′4 := 4ıλE1+ ∧ E2+
(3.21)
For the sake of convenience we introduce parameter6λ in r′′4 . It should be noted that
r′3, r
3 and r
4 are themselves classical r-matrices. We see that the r-matrix r
3 is simply
a sum of two standard r-matrices of sl(2;C), satisfying the anti-Hermitian condition
r∗ = −r. Analogously, it is not hard to see that the r-matrix r4 corresponds to a Belavin-
Drinfeld triple [15] for the Lie algebra sl(2;C) ⊕ sl(2,C)). Indeed, applying the Cartan
automorphism E2± → E2∓, H2 → −H2 we see that this is really correct (see also [16]).
We firstly describe quantum deformation corresponding to the classical r-matrix r3
(3.20). Since the r-matrix r′′3 is Abelian and it is subordinated to r
3 therefore the algebra
o(3, 1) is firstly quantized in the direction r′3 and then an Abelian twist corresponding
to the r-matrix r′′3 is applied. We introduce the complex notations z± := β ± ıα. It
should be noted that z− = z
+ if the parameters α and β are real, and z− = −z
the parameters α and β are pure imaginary. From structure of the classical r-matrix r′3
it follows that a quantum deformation Ur′
(o(3, 1)) is a combination of two q-analogs of
U(sl(2;C)) with the parameter qz
and qz
, where qz
:= exp z±. Thus Ur′3(o(3, 1))
(sl(2;C))⊗Uq
(sl(2;C)) and the standard generators q±H1z
, E1± and q
, E2± satisfy
6We can reduce this parameter λ to ± 1
by automorphism of o(4,C).
the following non-vanishing defining relations
qH1z+ E1± = q
E1± q
, [E1+, E1−] =
q2H1z+ − q
qz+ − q
, (3.22)
qH2z− E2± = q
E2± q
, [E2+, E2−] =
q2H2z− − q
qz− − q
. (3.23)
In this case the co-product ∆r′
and antipode Sr′
for the generators q±H1z
, E1± and q
E2± can be given by the formulas:
(q±H1z+ ) = q
⊗ q±H1z+ , ∆r′1
(E1±) = E1± ⊗ q
+ q−H1z+ ⊗ E1± , (3.24)
(q±H2z− ) = q
⊗ q±H2z− , ∆r′1
(E2±) = E2± ⊗ q
+ q−H2z− ⊗ E2± , (3.25)
(q±H1z+ ) = q
, Sr′
(E1±) = −q
E1± , (3.26)
(q±H2z
) = q∓H2z− , Sr′1
(E2±) = −q
E2± . (3.27)
The ∗-involution describing the real structure on the generators (3.8) can be adapted to
the quantum generators as follows
(q±H1z+ )
∗ = q∓H2
, E∗1± = −E2± , (q
)∗ = q∓H1
, E∗2± = −E1± , (3.28)
and there exit two ∗-liftings: direct and flipped, namely,
(a⊗ b)∗ = a∗ ⊗ b∗ (∗ − direct) , (3.29)
(a⊗ b)∗ = b∗ ⊗ a∗ (∗ − flipped) (3.30)
for any a ⊗ b ∈ Ur′
(o(3, 1)) ⊗ Ur′
(o(3, 1)), where the ∗-direct involution corresponds to
the case of the pure imaginary parameters α, β and the ∗-flipped involution corresponds
to the case of the real deformation parameters α, β. It should be stressed that the Hopf
structure on Ur′
(o(3, 1)) satisfy the consistency conditions under the ∗-involution
(a∗) = (∆r′
(a))∗, Sr′
((Sr′
(a∗))∗) = a (∀x ∈ Ur′
(o(3, 1)) . (3.31)
Now we consider deformation of the quantum algebra Ur′
(o(3, 1)) (secondary quan-
tization of U(o(3, 1))) corresponding to the additional r-matrix r′′3 , (3.20). Since the
generators H1 and H2 have the trivial coproduct
(Hk) = Hk ⊗ 1 + 1⊗Hk (k = 1, 2) , (3.32)
therefore the unitary two-tensor
:= qH1∧H2ıγ (F
= F−1
) (3.33)
satisfies the cocycle condition (2.4) and the ”unital” normalization condition (2.5). Thus
the complete deformation corresponding to the r-matrix r3 is the twisted deformation of
(o(3, 1)), i.e. the resulting coproduct ∆r
is given as follows
(x) = Fr′′
(x)F−1
(∀x ∈ Ur′
(o(3, 1)) . (3.34)
and in this case the resulting antipode Sr
does not change, Sr
. Applying the
twisting two-tensor (3.33) to the formulas (3.24) and (3.25) we obtain
(q±H1z+ ) = q
⊗ q±H1z+ , ∆r′1(q
) = q±H2z− ⊗ q
, (3.35)
(E1+) = E1+ ⊗ q
qH2ıγ + q
q−H2ıγ ⊗ E1+ , (3.36)
(E1−) = E1− ⊗ q
q−H2ıγ + q
qH2ıγ ⊗ E1− , (3.37)
∆r3(E2+) = E2+ ⊗ q
q−H1ıγ + q
qH1ıγ ⊗ E2+ , (3.38)
∆r3(E2−) = E2− ⊗ q
qH1ıγ + q
q−H1ıγ ⊗ E2− . (3.39)
Next, we describe quantum deformation corresponding to the classical r-matrix r4
(3.21). Since the r-matrix r′4(α) := r
4 is a particular case of r3(α, β, γ) := r3, namely
r′4(α) = r3(α, β = 0, γ = α), therefore a quantum deformation corresponding to the r-
matrix r′4 is obtained from the previous case by setting β = 0, γ = α, and we have the
following formulas for the coproducts ∆r′
) = q
(k = 1, 2) , (3.40)
(E1+) = E1+ ⊗ q
H1+H2
+ q−H1−H2
⊗ E1+ , (3.41)
(E1−) = E1− ⊗ q
H1−H2
+ q−H1+H2
⊗ E1− , (3.42)
(E2+) = E2+ ⊗ q
−H1−H2
ξ + q
H1+H2
ξ ⊗ E2+ , (3.43)
(E2−) = E2− ⊗ q
H1−H2
ξ + q
−H1+H2
ξ ⊗ E2− , (3.44)
where we set ξ := ıα.
Consider the two-tensor
:= exp
λE1+q
H1+H2
ξ ⊗ E2+q
H1+H2
. (3.45)
Using properties of q-exponentials (see [17]) is not hard to verify that Fr′′
satisfies the co-
cycle equation (2.4). Thus the quantization corresponding to the r-matrix r4 is the twisted
q-deformation Ur′
(o(3, 1)). Explicit formulas of the co-products ∆r
(·) = F
(·)F−1
and antipodes Sr4(·) in the complex and real Cartan-Weyl bases of Ur′4(o(3, 1)) will be
presented in the outgoing paper [7].
4 Quantum deformations of Poincare algebra
The Poincaré algebra P(3, 1) of the 4-dimensional space-time is generated by 10 elements:
the six-dimensinal Lorentz algebra o(3, 1) with the generators Mi, Ni (i = 1, 2, 3):
[Mi, Mj ] = ıǫijk Mk, [Mi, Nj ] = ıǫijk Nk, [Ni, Nj ] = −ıǫijk Mk, (4.1)
and the four-momenta Pj, P0 (j = 1, 2, 3) with the standard commutation relations:
[Mj , Pk] = ıǫjkl Pl , [Mj , P0] = 0 , (4.2)
[Nj, Pk] = −ıδjk P0 , [Nj , P0] = −ıPj . (4.3)
The physical generators of the Lorentz algebra, Mi, Ni (i = 1, 2, 3), are related with the
canonical basis h, h′, e±, e
± as follows
h = ıN3 , e± = ı(N1 ± M2), (4.4)
h′ = −ıM3 , e
± = ı(±N2 −M1). (4.5)
The subalgebra generated by the four-momenta Pj, P0 (j = 1, 2, 3) will be denoted by P
and we also set P± := P0 ± P3.
S. Zakrzewski has shown in [2] that each classical r-matrix, r ∈ P(3, 1) ∧ P(3, 1), has
a decomposition
r = a+ b+ c , (4.6)
where a ∈ P ∧P, b ∈ o(3, 1) ∧P, c ∈ o(3, 1) ∧ o(3, 1) satisfy the following relations
[[c, c]] = 0 , (4.7)
[[b, c]] = 0 , (4.8)
2[[a, c]] + [[b, b]] = tΩ (t ∈ R) , (4.9)
[[a, b]] = 0 . (4.10)
Here [[·, ·]] means the Schouten bracket. Moreover a total list of the classical r-matrices
for the case c 6= 0 and also for the case c = 0, t = 0 was found.7 It was shown that
there are fifteen solutions for the case c = 0, t = 0, and six solutions for the case c 6= 0
where there is only one solution for t 6= 0. Thus Zakrzewski found twenty r-matrices which
satisfy the homogeneous classical Yang-Baxter equation (t = 0 in (4.9)). Analysis of these
twenty solutions shows that each of them can be presented as a sum of subordinated r-
matrices which almost all are of Abelian and Jordanian types. Therefore these r-matrices
correspond to twisted deformations of the Poincaré algebra P(3, 1). We present here
r-matrices only for the case c 6= 0, t = 0:
r1 = γh
′ ∧ h+ α(P+ ∧ P− − P1 ∧ P2) , (4.11)
r2 = γe
+ ∧ e+ + β1(e+ ∧ P1 − e
+ ∧ P2 + h ∧ P+) + β2h
′ ∧ P+ , (4.12)
r3 = γe
+ ∧ e+ + β(e+ ∧ P1 − e
+ ∧ P2 + h ∧ P+) + αP1 ∧ P+ , (4.13)
r4 = γ(e
+ ∧ e+ + e+ ∧ P1+ e
+ ∧ P2− P1 ∧ P2) + P+ ∧ (α1P1+ α2P2) , (4.14)
r5 = γ1(h ∧ e+ − h
′ ∧ e′+) + γ2e+ ∧ e
+ . (4.15)
The first r-matrix r1 is a sum of two subordinated Abelian r-matrices
r1 := r
1 + r
1 , r
1 ≻ r
r′1 = α(P+ ∧ P− − P1 ∧ P2) , r
1 := γh
′ ∧ h .
(4.16)
Therefore the total twist defining quantization in the direction to this r-matrix is the
ordered product of two the Abelian twits
= Fr′′
= exp
γh′ ∧ h
α(P+ ∧ P− − P1 ∧ P2)
. (4.17)
7Classification of the r-matrices for the case c = 0, t 6= 0 is an open problem up to now.
The second r-matrix r2 is a sum of three subordinated r-matrices where two of them
are of Abelian type and one is of Jordanian type
r2 := r
3 + r
2 + r
2 , r
2 ≻ r
2 ≻ r
r′2 := β1(e+ ∧ P1 − e
+ ∧ P2 + h ∧ P+) ,
r′′2 := γe
+ ∧ e+ , r
2 := β2h
′ ∧ P+ .
(4.18)
Corresponding twist is given by the following formulas
= Fr′′′
, (4.19)
where
= exp
β1(e+ ⊗ P1 − e
+ ⊗ P2)
exp(2h⊗ σ+) ,
= exp(γe′+ ∧ e+) , Fr′′′2 = exp(β2h
′ ∧ σ+) .
(4.20)
Here and below we set σ+ :=
ln(1 + β1P+).
The third r-matrix r3 is a sum of two subordinated r-matrices where one is of Abelian
type and another is a more complicated r-matrix which we call mixed Jordanian-Abelian
r3 := r
3 + r
3 , r
3 ≻ r
r′3 := β1(e+ ∧ P1 − e
+ ∧ P2 + h ∧ P+) + αP1 ∧ P+ ,
r′′3 := γe
+ ∧ e+ .
(4.21)
Corresponding twist is given by the following formulas
= Fr′′
, (4.22)
where
= exp
β1(e+ ⊗ P1 − e
+ ⊗ P2)
exp(αP1 ∧ σ+) exp(2h⊗ σ+) ,
= exp(γe′+ ∧ e+) .
(4.23)
The fourth r-matrix r4 is a sum of two subordinated r-matrices of Abelian type
r4 := r
4 + r
4 , r
4 ≻ r
r′4 := P+ ∧ (α1P1 + α+P2) ,
r′′4 := γ(e
+ − P1) ∧ (e+ + P2) .
(4.24)
Corresponding twist is given by the following formulas
= Fr′′
, (4.25)
where
= exp
(P+ ⊗ (α1P1 + α2P2)
= exp
γ(e′+ − P1) ∧ (e+ + P2)
(4.26)
The fifth r-matrix r5 is the r-matrix of the Lorentz algebra, (3.6), and the correspond-
ing twist is given by the formula (3.13).
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http://arxiv.org/abs/q-al/9602001
http://xxx.lanl.gov/abs/hep-th/0301033
arXiv:hep-th/0604146
http://xxx.lanl.gov/abs/math.QA/0402433
http://xxx.lanl.gov/abs/math.QA/0701079
	Introduction
	Preliminaries
	Quantum deformations of Lorentz algebra
	Quantum deformations of Poincare algebra
