Quantizations of \(D=3\) Lorentz symmetry
Eur. Phys. J. C
Quantizations of D = 3 Lorentz symmetry
J. Lukierski 1
V. N. Tolstoy 0 1
0 Lomonosov Moscow State University, Skobeltsyn Institute of Nuclear Physics , Moscow 119991, Russian Federation
1 Institute for Theoretical Physics, University of Wroc?aw , pl. Maxa Borna 9, 50205 Wroc?aw , Poland
Using the isomorphism o(3; C) sl(2; C) we develop a new simple algebraic technique for complete classification of quantum deformations (the classical r matrices) for real forms o(3) and o(2, 1) of the complex Lie algebra o(3; C) in terms of real forms of sl(2; C): su(2), su(1, 1) and sl(2; R). We prove that the D = 3 Lorentz symmetry o(2, 1) su(1, 1) sl(2; R) has three different Hopfalgebraic quantum deformations, which are expressed in the simplest way by two standard su(1, 1) and sl(2; R) qanalogs and by simple Jordanian sl(2; R) twist deformation. These quantizations are presented in terms of the quantum CartanWeyl generators for the quantized algebras su(1, 1) and sl(2; R) as well as in terms of quantum Cartesian generators for the quantized algebra o(2, 1). Finally, some applications of the deformed D = 3 Lorentz symmetry are mentioned.

The search for quantum gravity is linked with studies of
noncommutative spacetimes and quantum deformations
of spacetime symmetries. The considerations of simple
dynamical models in quantized gravitational background
(see e.g. [1?3]) indicate that the presence of quantum
gravity effects generates noncommutativity of spacetime
coordinates, and as well the Liealgebraic spacetime symmetries
(e.g. Lorentz, Poincar?) are modified into quantum
symmetries, described by noncocommutative Hopf algebras, named
by Drinfeld quantum deformations or quantum group [4]. We
recall that in relativistic theories the basic role is played by
Lorentz symmetries and the Lorentz algebra, i.e. all aspects
of their quantum deformations should be studied in a very
detailed and careful way.
For classifications, constructions and applications of
quantum Hopf deformations of an universal enveloping
algebra U (g) of a Lie algebra g, Lie bialgebras (g, ?) play an
essential role (see e.g. [4?7]). Here the cobracket ? is a linear
skewsymmetric map g ? g ? g with the relations consistent
with the Lie bracket in g:
?([x , y]) = [x ? 1 + 1 ? x , ?(y)] ? [y ? 1 + 1 ? y, ?(x )],
(? ? id)?(x ) + cycle = 0 (1.1)
for any x , y ? g. The first relation in (1.1) is a condition
of the 1cocycle and the second one is the coJacobi identity
(see [4,7]). The Lie bialgebra (g, ?) is a correct
infinitesimalization of the quantum Hopf deformation of U (g) and the
operation ? is an infinitesimal part of the difference between
a coproduct and an opposite coproduct ? in the Hopf
algebra, ?(x ) = h?1( ? ? ) mod h where h is a
deformation parameter. Any two Lie bialgebras (g, ?) and (g, ? )
are isomorphic (equivalent) if they are connected by a
gautomorphism ? satisfying the condition
for any x ? g. Of special interest here are the quasitriangle
Lie bialgebras (g, ?(r)) := (g, ?, r ), where the cobracket
?(r) is given by the classical r matrix r ? g ? g as follows:
It is easy to see from (1.2) and (1.3) that two
quasitriangular Lie bialgebras (g, ?(r)) and (g, ?(r )) are isomorphic iff the
classical r matrices r and r are isomorphic, i.e. (? ? ?)r =
r . Therefore for a classification of all nonequivalent
quasitriangular Lie bialgebras (g, ?(r)) of the given Lie algebra
g we need find all nonequivalent (nonisomorphic)
classical r matrices. Because nonequivalent quasitriangular Lie
bialgebras uniquely determine nonequivalent
quasitriangular quantum deformations (Hopf algebras) of U (g) (see [4,5])
therefore the classification of all nonequivalent
quasitriangular Hopf algebras is reduced to the classification of the
nonequivalent classical r matrices.
In this paper we investigate the quantum deformations
of D = 3 Lorentz symmetry. Firstly, following [8], we
obtain the complete classifications of the nonequivalent
(nonisomorphic) classical r matrices for complex Lie
algebra sl(2; C) and its real forms su(2), su(1, 1) and sl(2; R)
with the help of explicit formulas for the automorphisms of
these Lie algebras in terms of the Cartan?Weyl bases. In
the case of sl(2; C) there are two nonequivalent classical
r matrices ? standard and Jordanian ones. For su(2)
algebra there is only the standard nonequivalent r matrix. These
results are well known. For the su(1, 1) case we obtained
three nonequivalent r matrices ? standard, quasistandard
and quasiJordanian ones. In the case of sl(2; R) we find also
three nonequivalent r matrices ? standard, quasistandard
and Jordanian ones. Then using isomorphisms o(2, 1)
su(1, 1) sl(2; R) we express these r matrices in terms of
the Cartesian basis of the D = 3 Lorentz algebra o(2, 1) and
we see that two systems with three r matrices for su(1, 1) and
sl(2; R) algebras coincides. Thus we obtain that the
isomorphic Lie algebras su(1, 1) and sl(2; R) have the isomorphic
systems of their quasitriangle Lie bialgebras. In the case of
o(2, 1) we see that the D = 3 Lorentz algebra has two
standard qdeformations and one Jordanian. These Hopf
deformations are presented in explicit form in terms of the quantum
Cartan?Weyl generators for the quantized universal
enveloping algebras of su(1, 1) and sl(2; R) and also in the terms of
the quantum Cartesian generators.
It should be noted that the full list of the nonequivalent
classical r matrices for sl(2; R) and o(2, 1) Lie algebras has
been obtained early by different methods [9,10] (see also
[11?13]), however, the complete list of the nonequivalent
Hopf quantizations for these Lie algebras has not been
presented in the literature. Furthermore, there was put forward
the incorrect hypothesis that the isomorphic Lie algebra
su(1, 1) and sl(2; R) do not have any isomorphic
quasitriangular Lie bialgebras (see [14]).
The isomorphic Lie algebras o(2, 1), sl(2; R), su(1, 1)
and their quantum deformations play very important role
in physics as well as in mathematical considerations, so
the structure of these deformations should be understood
with full clarity. The o(2, 1) Lie algebra has been used as
D = 1 conformal algebra describing basic symmetries in
conformal classical and quantum mechanics [15]; in such
a case the o(2, 1) algebra is realized as a nonlinear
realization on the onedimensional time axis [16,17] and can
be extended to osp(12) describing D = 1 N = 2
supersymmetric conformal algebra [18]. In fieldtheoretic
framework the o(2, 1) Lie algebra describes Lorentz symmetries
of threedimensional relativistic systems with planar d = 2
space sector, which are often discussed as simplified
version of the fourdimensional relativistic case. Due to the
isomorphism o(2, 2) o(2, 1) ? o(2, 1) our results can be
also applied to the description of D = 3 AdS symmetries
[19]. We recall that o(2, 2) symmetry has been employed
in Chern?Simons formulation of D = 3 gravity [20?22],
with Lorentzian signature and nonvanishing negative
cosmological constant. Subsequently, the quantum deformations of
D = 3 Chern?Simons theory have been used for the
description of D = 3 quantum gravity as deformed D = 3
topological QFT [23,24]. Threedimensional deformed spacetime
geometry is also a basis of historical Ponzano?Regge
formulation of D = 3 quantum gravity [25], which was further
developed into spin foam [26] and causal triangulation [27]
approaches.
In mathematics and mathematical physics the importance
of o(2, 1) and its deformations follows also from the unique
role of the o(2, 1) algebra as the lowestdimensional rank one
noncompact simple Lie algebra, endowed only with unitary
infinitedimensional representations. One can point out that
the program of deformations of infinitedimensional
modules of quantumdeformed U (su(1, 1)) algebra has been
initiated already more than 20 years ago (see e.g. [28,29]). The
(2 + 1)dimensional models are also important in the theory
of classical and quantum integrable systems [30,31] with
their symmetries described by Poisson?Lie groups in
classical case and after quantization by quantum groups. In
particular recently, using a sigma model formulation of (super)string
actions (see e.g. [32]), the integrable deformations of string
target (super)spaces were obtained by Yang?Baxter
deformations [33?37] of the principal as well as coset sigma models
with symmetries, which may contain Ad S2 o(2; 1) and
Ad S3 o(2, 2) factors [38?40].
The plan of our paper is the following. In Sect. 2 we
consider the complex Lie algebra o(3; C) and its all real
forms: o(3) su(2) and o(2, 1) su(1, 1) sl(2; R).
In Sect. 3 we classify all classical r matrices for these real
forms and in Sect. 4 we provide the explicit isomorphisms
between the real su(1, 1), sl(2; R) and o(2, 1) bialgebras. In
Sect. 5 all three Hopfalgebraic quantizations (explicit
quantum deformations) of the real D = 3 Lorentz symmetry are
presented in detail: quantized bases, coproducts and
universal Rmatrices are given. In Sect. 6 we present short summary
and outlook.
2 Complex D = 3 Euclidean Lie algebra o(3; C) and its
real forms
We first recall different most popular bases of the complex
D = 3 Euclidean Lie algebra o(3; C): metric, Cartesian and
Cartan?Weyl bases (see [8]).
The metric basis contains in its commutation relations an
explicit metric, namely, the complex D = 3 Euclidean Lie
algebra o(3; C) is generated by three Euclidean basis
elements Li j = ?L ji ? o(3; C) (i, j = 1, 2, 3) satisfying the
relations
[Li j , Lkl ] = g jk Lil ? g jl Lik + gil L jk ? gik L jl ,
where gi j is the Euclidean metric: gi j = diag (1, 1, 1). The
Euclidean algebra o(3; C), as a linear space, is a linear
envelope of the basis {Li j } over C.
The Cartesian (or physical) basis of o(3; C) is related with
the generators Li j as follows:
From (2.1) and (2.2) we get
If we consider a Lie algebra over R with the commutation
relations (2.3) then we get the compact real form o(3) :=
o(3; R) with the antiHermitian basis
Ii? = ?Ii (i = 1, 2, 3) for o(3).
The real form o(2, 1) is given by the formulas
Ii? = (?1)i?1 Ii (i = 1, 2, 3) for o(2, 1).
For the description of quantum deformations and in particular
for the classification of classical r matrices of the complex
Euclidean algebra o(3; C) and its real forms o(3) and o(2, 1)
it is convenient to use the Cartan?Weyl (CW) basis of the
isomorphic complex Lie algebra sl(2; C) and its real forms
su(2), sl(1, 1) and sl(2, R). In the case of o(3) the su(2)
Cartan?Weyl basis can be chosen as follows:
H := ? I3, E? := ? I1 ? I2,
E , E?] = 2H,
[H, E?] = ?E?, [ +
H ? = H, E ?? = E?,
where the conjugation (?) is the same as in (2.4).1
For the real form o(2, 1) we will use two CW bases of
sl(2; C) real forms: sl(1, 1) and sl(2, R). Such bases are
given by
H := ? I2, E? := ? I1 ? I3,
[H, E?] = ?E?, [E+, E?] = 2H
H := ? I3, E? := ? I1 ? I2,
E , E?] = 2H
[H , E?] = ?E?, [ +
for su(1, 1), (2.7)
for sl(2, R). (2.8)
The two bases {E?, H } and {E?, H } have the same
commutation relations but they have different reality properties,
namely
H ? = H, E ?? = ?E?
H ? = ?H , E?? = ?E?
for su(1, 1),
1 The basis elements E?, H over C with the defining relations in the
second line of (2.6) generate the complex Lie algebra sl(2; C). The
relations in the first line of (2.6) reproduce the isomorphism between
o(3; C) and sl(2; C).
where the conjugation (?) is the same as in (2.5).2 The
relations between the su(1, 1) and su(2, R) bases look as follows:
?
H = ? 2 (E+ ? E?),
3 Classical rmatrices of sl(2; C) and its real forms: su(2), su(1, 1) and sl(2; R)
By definition any classical r matrix of arbitrary complex or
real Lie algebra g, r ? g?g, satisfy the classical Yang?Baxter
equation (CYBE):
[[r, r ]] = ? .
Here [[?, ?]] is the Schouten bracket which for any monomial
skewsymmetric twotensors r1 = x ? y and r2 = u ? v
(x , y, u, v ? g) is given by3
[[x ? y, u ? v]] := x ? ([y, u] ? v + u ? [y, v])
? y ? ([x , u] ? v + u ? [x , v])
= [[u ? v, x ? y]],
and ? is the ginvariant element which in the case of g :=
sl(2; C) looks as follows:
(sl(2; C)) = ? (4E? ? H ? E+)
where ? ? C, and E?, H is the CW basis of sl(2; C) with
the defining relations on the second line of (2.6).
Firstly we show that any twotensor of sl(2; C) ? sl(2; C)
is a classical sl(2; C) r matrix. Indeed, let
be an arbitrary element of sl(2; C) ? sl(2; C), where
r+ := E+ ? H, r0 := E+ ? E?, r? := H ? E?, (3.5)
are the basis elements of sl(2; C)?sl(2; C). Because all terms
(3.5) are classical r matrices, moreover, [[r?, r?]] = 0, as
well as the Schouten brackets of the elements r? with r0 are
also equal to zero, [[r?, r0]] = 0, we have
[[r, r ]] = 2?+??[[r+, r?]] + ?02[[r0, r0]]
= (?02 + ?+??) (4E? ? H ? E+) ? ? .
Thus an arbitrary element (3.4) is a classical r matrix, and
if its coefficients ??, ?0 satisfy the condition ? := ?02 +
+ ? = 0 then it satisfies the homogeneous CYBE; if ? :=
? ?
2
?0 + ?+?? = 0 it satisfies the nonhomogeneous CYBE.
2 It should be noted that in the case of su(1, 1) the Cartan generator H
is compact, while for the case su(2, R) the generator H is noncompact.
3 For general polynomial (a sum of monomials) twotensors r1 and r2
one can use the bilinearity of the Schouten bracket.
We shall call the parameter ? = ?02 +?+?? in (3.6) the ?
characteristic of the classical r matrix (3.4). It is evident that
the ? characteristic of the classical r matrix r is invariant
under the sl(2; C)automorphisms, i.e. any two r matrices
r and r , which are connected by a sl(2; C)automorphism,
have the same ? characteristic, ? = ? . We can show also
that any two sl(2; C) r matrices r and r with the same ?
characteristic can be connected by a sl(2; C)automorphism.
There are two types of explicit sl(2; C)automorphisms
which were presented in [8]. First type connecting the
classical r matrices with zero ? characteristic is given by the
formulas (see (3.15) in [8])4:
?0(E?) = ? ?1(??? E+ ? 2???0 H + ??+ E?),
?0(H ) = ??0 E+ + (???+ + ???) H + ???0 E?,
where ? is a nonzero rescaling parameter (including ? = 1),
? takes two values +1 or ?1, and the parameters ??i (i =
+, 0, ?) satisfy the conditions
r := ?+ E+ ? H + ?0 E+ ? E? + ?? H ? E?,
r := ?+ E+ ? H + ?0 E+ ? E? + ?? H ? E?,
where ?02 + ?+?? = 0 and ?02 + ?+?? = 0. Moreover, we
suppose that the parameters ?? and ?? satisfy the additional
relations:
??+ ? ?? = ??+ ? ? ?1??? = 0,
where the parameters ? and ? are the same as in (3.7). One
can check that the following formula is valid:
?+ E+ ? H + ?0 E+ ? E? + ?? H ? E?
= ?+?0(E+) ? ?0(H ) + ?0?0(E+) ? ?0(E?)
where ?0 is the sl(2; C)automorphism (3.7) with the
following parameters:
+ =
? =
?0(??+ + ? ?1???) ? ?0(??+ + ??) ,
(??+ ? ??)(??+ ? ? ?1???)
?(??+ + ??)(??+ + ? ?1???) + 4?0?0
2(??+ ? ??)(??+ ? ? ?1???)
(??+ + ??)(??+ + ? ?1???) + 4??0?0
2(??+ ? ??)(??+ ? ? ?1???)
It is easy to check that as expected Eqs. (3.12) satisfy the
conditions (3.8).
Let us assume in (3.9), (3.11) and (3.12) that the
parameters ?0 and ?? are equal to zero. Then the general classical
r matrix r in (3.9), satisfying the homogeneous CYBE, is
reduced to usual Jordanian form by the automorphism (3.7)
with the parameters:
??0 = ??+??0 ?? , ??? = ??+??? ?? . (3.13)
Second type of sl(2; C)automorphism connecting the
classical r matrices with nonzero ? characteristic is given
as follows:5
?1(E?) =
(??0 + 1) E+ + 2??? H ? ??0??+?2 1 E? ,
Let us consider two general r matrices with nonzero ?
characteristics:
r := ?+ E+ ? H + ?0 E+ ? E? + ?? H ? E?,
r := ?+ E+ ? H + ?0 E+ ? E? + ?? H ? E?,
where the parameters ??, ?0 and ??, ?0 can be equal to zero
provided that ? = ?02 + ?+?? = ? = (?0)2 + ?+?? = 0,
i.e. the two r matrices r and r have the same nonzero ?
characteristic ? = ? = 0. One can check the following
relation:
?+ E+ ? H + ?0 E+ ? E? + ?? H ? E?
= ?+?1(E+) ? ?1(H ) + ?0?1(E+) ? ?0(E?)
(?0 + ?0)2 ? (?+ ? ??+)(?? ? ? ?1??) ,
??0 = (?0 + ?0)2 + (?+ ? ??+)(?? ? ? ?1??)
?? 2(?0 + ?0)(?? ? ? ?1??) .
? = (?0 + ?0)2 + (?+ ? ??+)(?? ? ? ?1??)
It is easy to check that Eqs. (3.17) satisfy the condition ??02 +
??+??? = 1.
If we assume in (3.15)?(3.17) that the parameters ?? are
equal to zero then the general classical r matrix r in (3.15),
4 Equations (3.7) are obtained from (3.15) in [8] by the substitution:
?0/(k?+ ? ??) = ?2??0, ??/(k?+ ? ??) = ???.
5 Equations (3.14) are obtained from (3.14) in [8] by the substitution
?0 = 2??0, ?? = ????, D = 4.
satisfying the nonhomogeneous CYBE, is reduced to the
usual standard form by the automorphism (3.14) with the
following parameters:
where ?0 and ?0 are real numbers and we use the
conditions (3.22). The r matrices ri (i = 1, 2, 3) satisfy the
nonhomogeneous CYBE
Finally for sl(2, C) we obtain the following wellknown
result:
For the complex Lie algebra sl(2, C) there exist up to sl(2, C)
automorphisms two solutions of CYBE, namely Jordanian r J
and standard rst :
r J = ? E+ ? H, [[r J , r J ]] = 0,
rst = ? E+ ? E?, [[rst , rst ]] = ?
where the complex parameter ? in (3.19) can be removed by
the rescaling automorphism: ?(E+) = ??1 E+, ?(E?) =
? E?, ?(H ) = H ; in (3.20) the parameter ? = e?? ?  for
? ? ?2 is effective.
The general nonreduced expression (3.4) is convenient
for the application of reality conditions
? E?
where is the conjugation associated with corresponding
real form ( = ?, ?), and ?i? (i = +, 0, ?) means the
complex conjugation of the number ?i . It should be noted that
for any classical r matrix r , r is again a classical r matrix.
Moreover, if r matrix is antireal (antiHermitian),6 i.e. it
satisfies the condition (3.21), then its ? characteristic is real.
Indeed, applying the conjugation to the relation (3.6) we
have for the leftside [[r, r ]] = ?[[r , r ]] = ?[[r, r ]]
and for the rightside: (? ) = ?? ? for all real forms
su(2), su(1, 1), su(2; R). It follows that the parameter ? is
real, ? ? = ? .
I. The compact real form su(2) (H ? = H , E ?? = E?).
In this case it follows from (3.21) that
2
If in (3.4) ? = ?0 +?+?? = 0 then ?0?0? +?+?+? = 0 and it
follows that ?0 = ?+ = ?? = 0, i.e. any classical r matrix,
which satisfies the homogeneous CYBE and the su(2) reality
condition, is equal zero.
If in (3.4) ? = ?02 + ?+?? = 0 we have three possible
su(2) real classical r matrices:
6 The antireal property r = ?r is a direct consequence of the reality
condition for the cobracket ?(x) := [x ?1+1?x, r ], namely ?(x) =
?(x ) for ?x ? g (= {su(2), su(1, 1), sl(2, R)}).
where all ?i (i = 1, 2, 3) are positive: ?1 = ?02 > 0, ?2 =
?+?+? > 0, ?3 = ?02 + ?+?+? > 0.
Let the classical r matrices (3.15) be su(2)antireal, i.e.
their parameters satisfy the reality conditions (3.22). It
follows that the functions (3.17) for ? = e?? have the same
conjugation properties, i.e. ??0? = ??0, ???? = ???, and we see
that the automorphism (3.14) with such parameters is
su(2)real, i.e.:
?1(E?)? = ?1(E ??) = ?1(E?),
?1(H )? = ?1(H ?) = ?1(H ).
We see that the r matrices r2 and r3 in (3.23) can be reduced
to the standard r matrix rst := r1 using Eq. (3.16).
It is easy to see that the standard r matrix rst = r1 in (3.23)
effectively depends only on positive values of the parameter
? := ?0. Indeed, we see that
where ? is the simple su(2) automorphism: ?(E?) = E?,
?(H ) = ?H , i.e. any negative value of parameter ? in rst
can be replaced by the positive one.
We obtain the following result:
For the compact real form su(2) there exists up to the su(2)
automorphisms only one solution of CYBE and this solution
is the usual standard classical r matrix rst :
rst := ? E+ ? E?, [[rst , rst ]] = ? ,
II. The noncompact real form su(1, 1) (H ? = H , E ?? =
I?f??02=+??e+??????=0, 0anidn w(3e.4h)atvheenth?e0f?o0?llo?w?in+g??+?f=am0il,yi.oef.
su(1, 1) homogeneous CYBE solutions:
r? := ?0 ei? ??00 E+ ? H + E+ ? E? ? e?i? ??00 H ? E? ,
where ?0 is real. By using the su(1, 1)real rescaling
automorphism ?(E?) = (?? ei? ?0 )?1 E?, ?(H ) = H we can
?0
reduce the ?family (3.29) to rq J := ?0(? E+ ? H + E+ ?
E? + ? H ? E?):
r? = ?0 e?? ??00 E+ ? H + E+ ? E? ? e??? ??00 H ? E?
= ?0(? (?(E+) ? ?(E?)) ? ?(H ) + ?(E+) ? ?(E?)).
(3.30)
We shall call a su(1, 1)real r matrix ?quasiJordanian? if
it cannot be reduced to Jordanian form by a su(1, 1)real
automorphism, but after complexification of su(1, 1) it can
be reduced to Jordanian form by an appropriate complex
sl(2, C)automorphism. Thus all r matrices in the ?family
(3.29) are quasiJordanian and they are connected with each
other by the su(1, 1)real rescaling automorphism. We take
rq J as an representative of the ?family. It is easy to see that
the quasiJordanian r matrix rq J effectively depends only on
positive values of the parameter ?0, indeed,
rq J = ?0(i E+ ? H + E+ ? E? + i H ? E?)
= ??0(i ?(E+) ? ?(H ) + i ?(E+) ? ?(E?)
+ ?(H ) ? ?(E?)),
where ? is the simple su(1, 1) automorphism ?(E?) =
?E?, ?(H ) = ?H , i.e. any negative value of parameter
?0 in rq J can be changed into a positive one.
In the case ?02 + ?+?? = 0 in (3.4) we have four
versions of su(1, 1)real classical r matrices. Two of them are
characterized by positive value of ?i , (i = 1, 2):
where ?0 and ?0 are real (see (3.28)), and ?1 = ?02 > 0,
?2 = ?0?0? ? ?+?+? > 0. The remaining two are with
negative values of ?i , (i = 3, 4):
where ?0 is real (see (3.28)), and ?3 = ??+?+? < 0,
?4 = ?0 ?0 ? ? ?+ ?+ ? < 0.
Let the classical r matrices (3.15) be su(1, 1)antireal,
i.e. their parameters satisfy the reality conditions (3.28). In
such case the functions (3.17) for ? = e?? have the same
conjugation properties, i.e. ??0? = ??0, ???? = ????, and we
see that the automorphism (3.14) with these parameters is
su(1, 1)real, i.e.:
?1(E?)? = ?1(E ??) = ??1(E?),
?1(H )? = ?1(H ?) = ?1(H ).
It allows to reduce the r matrix r2 to the standard r matrix
rst := r1 for ?1 = ?2 > 0 and the r matrix r4 to the r
matrix r3 for ?3 = ?4 < 0 by use of Eq. (3.16). By analogy
to the notation of quasiJordanian r matrix we shall call the
r matrices r3 and r4 as quasistandard ones and take rqst :=
?(E+ + E?) ? H as their representative.7
Finally for su(1, 1) we obtain the following results:
For the noncompact real form su(1, 1) there exist up to
su(1.1) automorphisms three solutions of CYBE, namely
quasiJordanian rq J , standard rst and quasistandard rqst :
?
rq J = 2 (? (E+ ? E?) ? H + E+ ? E?), [[rq J , rq J ]] = 0,
rst = ? E+ ? E?, [[rst , rst ]] = ?2 ,
rqst = ?(E+ + E?) ? H, [[rqst , rqst ]] = ??2 ,
where ? effectively is a positive number.
III. The noncompact real form sl(2; R) (H ? = ?H , E?? =
?E?).
In this case from (3.21) we obtain
i.e. all parameters ?i (i = +, 0, ?) are purely imaginary.
Consider the case ?02 + ?+?? = 0 in (3.4). We have three
su(2; R) solutions of the homogeneous CYBE:
r1 = ?+ E+ ? H , r2 = ?? H ? E?,
r3 = ?+, E+ ? H + ?0 E+ ? E? + ?? H ? E?, (3.39)
where all parameters ?i (i = +, ?), ?i (i = +, 0, ?) are
purely imaginary, and ?02 + ?+?? = 0.
If the classical r matrices (3.9), where all generators H ,
E? are replaced by H , E?, are sl(2; R)antireal, i.e. their
parameters satisfy the reality conditions (3.38), then for the
real parameter ? all functions (3.12) are real, i.e. ??0? = ??0,
???? = ???. We see that the automorphism of the type (3.7)
with such parameters is sl(2; R)real, i.e.:
It allows to reduce the r matrices r2 and r3 in (3.39) to the
Jordanian r matrix r J := r1 by using Eq. (3.11).
In the case ?02 +?+?? = 0 in (3.4) we have seven versions
of sl(2; R)real classical r matrices. Five of them are with
negative values of ?i , (i = 1, 2, . . . , 5):
7 The r matrix rqst is connected with r3 (3.33) in the following way.
Substituting ?+ = ?+e?? in r3 (3.33) and using the su(1, 1)real
rescaling automorphism ?(E?) = e?i? E?, ?(H ) = H we obtain rqst
with ? = ?+.
where all parameters ? are purely imaginary, and ?1 = ?2 =
?3 = ?02 < 0, ?4 = ?+?? < 0, ?5 = ?0 + ?+?? < 0; is
the sl(2; R)invariant element8: = (4E? ? H ? E+. The
remaining two r matrices ri (i = 6, 7) have positive values
of ?i :
r6 := ?+ E+ ? H + ?? H ? E?,
r7 := ?+ E+ ? H + ?0 E+ ? E? + ?? H ? E?, (3.42)
[[ri , ri ]] := ?i (i = 6, 7),
where ?6 = ?+ ?? > 0 and ?7 = ?0 + ?+ ?? > 0.
Let the classical r matrices (3.15) be sl(2; R)antireal,
i.e. with their parameters satisfying the reality conditions
(3.38). In such a way the functions (3.17) for real ? are real,
i.e. ??? = ??0, ???? = ???, and we see that the automorphism
0
(3.14) with such parameters is sl(2; R)real. We can conclude
that for the case of the negative ? characteristics ?i < 0
(i = 1, . . . , 5) all r matrices ri (i = 2, . . . , 5) in (3.41) are
reduced to the standard formula rst := r1 and in the case of
the positive ? characteristics ?i > 0 (i = 6, 7) the classical
r matrix r7 in (3.42) is reduced to the quasistandard r matrix
rqst := r6.
Let us show that the r matrix rqst effectively depends only
on one positive parameter. Indeed, it is easy to see that
rqst =
E+ ? H +
H ? E??
where ? is the sl(2, R)real automorphism: ?(E?) =
?+?? is positive.
???? E?, ?(H ) = H , and ? =
?+ ??
Finally for sl(2, R) we obtain the following result:
For the noncompact real form sl(2, R) there exist up to
sl(2, R) automorphisms three solutions of CYBE, namely
Jordanian r J , standard rst and quasistandard rqst :
r J = ? ? E+ ? H , [[r J , r J ]] = 0,
rst = ? ? E+ ? E?, [[rst , rst ]] = ??2 ,
rqst = ? ? (E+ + E?) ? H , [[rqst , rqst ]] = ?2
4 Explicit isomorphism between su(1, 1) and sl(2; R)
bialgebras and its application to o(2, 1) quantizations
Using Eqs. (2.7) and (2.8) we express the triplets of the
classical su(1, 1) and sl(2; R) r matrices in terms of the o(2, 1)
basis (2.3), (2.5). We get the following results.
8 Using (2.11) it is easy to check that
(i) The su(1, 1) case:
= ??(? I1 ? I2) ? I3, [[rq J , rq J ]] = 0,
where the o(2, 1)invariant element expressed in terms of
the Cartesian basis (2.3) satisfying the reality condition (2.5)
looks as follows:
= ?8I1 ? I2 ? I3.
(ii) The su(2; R) case:
We see that the quasiJordanian r matrix rq J in the su(1, 1)
basis is the same as the Jordanian r matrix r J in the sl(2; R)
basis, and the standard r matrix rst in the su(1, 1) basis
becomes the quasistandard r matrix rqst in the sl(2; R)
basis. Further, the quasistandard r matrix rqst in the su(1, 1)
basis is the same as the standard r matrix rst in the sl(2; R)
basis.
Equations (4.8)?(4.10) show that the su(1, 1) and sl(2; R)
bialgebras are isomorphic. This result finally resolves the
doubts about isomorphism of these two bialgebras (for
example, see [14]).
Using the isomorphisms of the su(1, 1) and sl(2; R)
bialgebras we take as basic r matrices for the D = 3 Lorentz
algebra o(2, 1) the following ones:
The first two r matrices rst and rst with the effective
positive parameter ? correspond to the qanalogs of su(1, 1) and
sl(2; R) real algebras, the third r matrix r J presents the
Jordanian twist deformation of sl(2; R). In the next section we
shall show how to quantize the r matrices (4.11)?(4.13) in
an explicit form.
5 Quantizations of the D = 3 Lorentz symmetry
The qanalogs of the universal enveloping algebras U (g) for
the real Lie algebras g = su(1, 1), sl(2; R) were already
considered (see e.g. [7,28,29,41]) and they are given as follows.
The quantum deformation (qanalog) of U (g) is an unital
associative algebra Uq (g) with generators X?, q?X0 and the
defining relations:
q X0 q?X0 = q?X0 q X0 = 1,
q ? q?1
with the reality conditions:
(i) X ?? = ?X?, (q X0 )? = q X0 , q := e? for Uq (su(1, 1)),
(ii) X ?? = ?X?, (q X0 )? = q X0 , q := e??
where ? is real in accordance with (4.11) and (4.12).
A Hopf structure on Uq (g) (g = su(1, 1), sl(2; R)) is
defined with the help of three additional operations:
coproduct (comultiplication) q , antipode Sq and counit q :
q (q?X0 ) = q?X0 ? q?X0 ,
q (X?) = X? ? q X0 + q?X0 ? X?,
Sq (q?X0 ) = q?X0 , Sq (X?) = ?q?1 X?,
q (q?X0 ) = 1,
q (X?) = 0
with the reality conditions9:
q? (X ) =
q (X ?), Sq?(X ) = Sq?1(X ?),
q?(X ) = q (X ?)
for any X ? Uq (g). The quantum algebra Uq (g) is endowed
also with the opposite Hopf structure: opposite coproduct
? q ,10 corresponding antipode S?q and counit ?q .
9 q? (X ) := ( q (X ))???.
10 The opposite (transformed) coproduct ? q (?) is a coproduct with
permuted components, i.e. ? q (?) = ? ? q (?) where ? is the flip operator:
? ? X(1) ? X(2) = X(2) ? X(1).
Rq q (X ) = ? q (X )Rq , ?X ? Uq (g),
( q ? id)Rq = Rq13 Rq23, (id ?
Rq12 Rq13 R23 q
q = Rq23 Rq13 R12
is called the universal Rmatrix. Let Uq (b+) and Uq (b?)
be quantum Borel subalgebras of Uq (g), generated by X+,
q?X0 and X?, q?X0 , respectively. We denote by Tq (b+ ?b?)
the Taylor extension of Uq (b+) ? Uq (b?).11 One can show
(see [42,43]) that there exists a unique solution of equations
(5.5) in the space Tq (b+ ? b?) and such a solution has the
following form:
Rq (g) := Rq
= expq?2 ((q ? q?1)X+q?X0 ? q X0 X?)q2X0?X0
= q2X0?X0 expq?2 ((q ? q?1)X+q X0 ? q?X0 X?),
where q = e? for Uq (su(1, 1)) and q = e?? for Uq (sl(2; R)).
Here we use the standard definition of the qexponential:
expq (x ) :=
Analogously, there exists a unique solution of equations (5.5)
in the space Tq (b? ? b+) = ? ? Tq (b+ ? b?) and such a
solution is given by the formula
= expq2 ((q?1 ? q)X?q?X0 ? q X0 X+)q?2X0?X0
= q?2X0?X0 expq2 ((q?1 ? q)X?q X0 ? q?X0 X+),
where q satisfies the conditions (5.2).
As formal Taylor series the solutions (5.7) and (5.9) are
independent and they are related by
It should be noted also that
(Rq )?1 = Rq?1 , (Rq?)?1 = Rq??1 .
11 Tq (b+ ? b?) is an associative algebra generated by formal Taylor
series of the monomials X +n ? X?m with coefficients which are rational
functions of q?X0 , q?X0?X0 , provided that all values n ? m for each
formal series are bounded, n ? m < N .
From the explicit forms (5.7) and (5.9) we see that
(Rq?)? = ? ? Rq? = (Rq )?1 for Uq (su(1, 1)),
(Rq )? = (Rq )?1,
(Rq?)? = (Rq?)?1 for Uq (sl(2; R)),
i.e. in the case Uq (sl(2; R)) the two Rmatrices Rq , Rq? are
unitary and in the case Uq (su(1, 1)) they can be called
?flipHermitian? or ?? Hermitian?.
In the limit ? ? 0 (q ? 1) we obtain for the Rmatrix
(5.5)
Here rB D is the classical Belavin?Drinfeld r matrix:
rB D = 2?(X+ ? X? + X0 ? X0),
where ? = ?, X? = E?, X0 = H for the case g = su(1, 1),
and ? = ? ?, X? = E?, X0 = H for the case g = sl(2; R).
The r matrix rB D is not skewsymmetric and it satisfies the
standard CYBE
[r B12D, r B13D + r B23D] + [r B13D, r B23D] = 0,
which is obtained from QYBE (5.6) in the limit (5.13). The
standard r matrix (4.11) or (4.12) is the skewsymmetric part
of rB D, namely
where r?st = r B12D ? r B21D is the standard r matrix (4.11) or
(4.12) and C? 2 = 2?C2 = r B12D + r B21D where C2 is the split
Casimir element of su(1, 1) or sl(2; R).
We can introduce the quantum Cartesian generators by the
formulas X? = ? J1 ? J3, q?X0 = q?? J2 .12 In terms of these
generators the quantum algebra Uq (su(1, 1)), which will be
denoted by U(rst )(o(2, 1)), can be reformulated as follows.
The quantum deformation of U (o(2, 1)), corresponding to
the classical r matrix (4.11), is an unital associative algebra
U(rst )(o(2, 1)) with the generators { J1, J3, q?? J2 } and the
defining relations (k = 1, 3):
q? J2 q?? J2 = q?? J2 q? J2 = 1, [ J1, J3] =
? (q2? J2 ? q?2? J2 )
2(q ? q?1)
with the reality condition J1? = J1, J3? = J3, (q?? J2 )? =
q?? J2 , q? = q (q := e?, ? ? R). These relations are the
qanalog of Eqs. (2.3) with the reality condition (2.5). The
q ( Jk ) = Jk ? q? J2 + q?? J2 ? Jk ,
q (q?? J2 ) = q?? J2 ? q?? J2 , Sq (q?? J2 ) = q?? J2 ,
1 ?
Sq ( Jk ) = ? 2 (q + q?1) Jk + 2 (q ? q?1)?k2l Jl ,
q (q?? J2 ) = 1,
q ( Jk ) = 0.
Substituting in Eqs. (5.7) and (5.9) the expressions X? =
? J1 ? J3, q?X0 = q?? J2 we obtain the universal Rmatrix in
the terms of the quantum Cartesian generators Ji (i = 1, 2, 3)
with the defining relations (5.17).
We can also introduce other quantum Cartesian
generators, by the formulas X? = ? J1 ? J2, q?X0 = q?? J3 .13 In
terms of these generators the quantum algebra Uq (sl(2; R),
which will be denoted by U(rst )(o(2, 1)), can be
reformulated as follows. The quantum deformation of U (o(2, 1)),
corresponding to the classical r matrix (4.12), is an
unital associative algebra U(rst )(o(2, 1)) with the generators
{ J1, J2, q?? J3 } and the defining relations (k = 1, 2):
q? J3 q?? J3 = q?? J3 q? J3 = 1,
[ J1, J2] = ?
? (q2? J3 ? q?2? J3 )
2(q ? q?1)
1 ?
q?? J3 Jk = 2 (q + q?1) Jk q?? J3 ? 2 (q ? q?1)?3kl Jl q?? J3
with the reality conditions J1? = J1, J2? = ? J2, (q? J3 )? =
q? J3 , q? = q?1 (q := e??, ? ? R). The Hopf structure on
U(rst )(o(2, 1)) are provided by the formulas (k = 1, 2):
q ( Jk ) = Jk ? q? J3 + q?? J3 ? Jk ,
q (q?? J3 ) = q?? J3 ? q?? J3 , Sq (q?? J3 ) = q?? J3 ,
1 ?
Sq ( Jk ) = ? 2 (q + q?1) Jk + 2 (q ? q?1)?k3l Jl ,
q (q?? J3 ) = 1,
q ( Jk ) = 0.
Substituting in Eqs. (5.7) and (5.9) the expressions X? =
? J1 ? J2, q?X0 = q?? J3 we obtain the universal Rmatrix
in terms of the quantum physical generators Ji (i = 1, 2, 3)
with the defining relations (5.19).
The quantization of U (sl(2; R)) corresponding to the
classical Jordanian r matrix (4.13) has well been known for a
long time [44?46] and it is defined by the twist F (see [45]):
F = exp(H ? ? ), ? = ln(1 + ? ? E+).
The twotensor F satisfies the 2cocycle condition
? id)(F ) = F 23(id ?
12 The generators Ji = (?1)i?1 Ji? (i = 1, 2, 3) are qanalogs of the
Cartesian basis (2.3), (2.5) (limq?1 Ji ? Ii ).
13 The generators Ji (i = 1, 2, 3) are also the qanalog of the Cartesian
basis given by (2.3), (2.5) (limq?1 Ji ? Ii ).
and the ?unital? normalization
( ? id)(F ) = (id ? )(F ) = 1.
It is evident that the twist (5.21) is unitary
F ? = F ?1.
The twisting element F defines a deformation of the
universal enveloping algebra U (sl(2; R)) considered as a Hopf
algebra. The new deformed coproduct and antipode are given
as follows:
(F)(X ) = F (X )F ?1, S(F)(X ) = u S(X )u?1
for any X ? U (sl(2; R)), where (X ) and S(X ) are the
coproduct and the antipode before twisting: (X ) = X ?
1 + 1 ? X , S(X ) = ?X ; and
u = m(id ? S)(F ) = exp(?? ? H E+).
It is easy to see that we get the ?Hopf algebra, i.e.
( (F)(X ))? =
(F)(X ?), (S(F)(X ))? = S(F)(X ?)
for any X ? U (sl(2; R)). One can calculate the following
formulas for the deformed coproducts (F) (see [45]):
Using (5.25) and (5.26) one gets the formulas for the
deformed antipode S(F):
S(F)(H ) = ?H e?? , S(F)(E+) = ?E+e?? ,
S(F)(E?) = ?E?e? + 2? ? H 2e? ? ?2 H (H ? 1)E+e? .
(5.29)
It is easy to see that the universal Rmatrix R(F) for this
twisted deformation looks as follows:
R(F) = F? F ?1, (R(F))? = (R(F))?1.
where r J is the classical Jordanian r matrix (4.13). Using
Eqs. (2.8) we can express all the formulas (5.28)?(5.30) in
terms of the Cartesian basis (2.3) and (2.5).
We add that the Jordanian deformation has been described
as well in a deformed sl(2; R) algebra basis [47?49].
6 Short summary and outlook
By using the threefold isomorphism of classical Lie algebras
o(2, 1) sl(2; R) su(1, 1) one can express the
infinitesimal versions of the D = 3 Lorentz quantum deformations in
terms of classical o(2, 1), sl(2; R) and su(1, 1) r matrices.
The first aim of our paper was to derive o(2, 1), su(1, 1)
and sl(2; R) bialgebras using a representationindependent
purely algebraic method (see Sect. 3) and further to
provide the explicit maps which relate them (see Sect. 4). We
start in Sect. 3 with the derivation of a wellknown pair of
inequivalent complex o(3; C) sl(2; C) r matrices ? the
Jordanian (nonstandard) one and the Drinfeld?Jimbo
(standard) r matrix. Passing from sl(2; C) to sl(2; R) we obtain
three independent sl(2; R) r matrices. The first two of them
are the real forms of two basic complex sl(2; C) r matrices,
the third sl(2; R) r matrix, which we called quasistandard
(see (3.46)), is the sum of two skewsymmetric 2tensors.
We do not know, however, how to obtain directly the
universal Rmatrix from the third r matrix. We show that there
is, however, a way out: the quasistandard r matrix (3.46))
(see also (3.7)) can be transformed by the map (2.11) into the
standard r matrix in su(1, 1) basis, with a wellknown
universal Rmatrix (see e.g. [7]). In such a way we can derive the
effective quantization of all three D = 3 Lorentz r matrices;
however, we recall that for such a purpose it is necessary to
use both sl(2; R) and su(1, 1) bases.
In the second part of the Introduction we mentioned main
applications of D = 3 Lorentz symmetries and their
deformations, but still more important for the description of
noncommutative D = 3 spacetime geometry and D = 3
quantum gravity are the quantum deformations of the D = 3
Poincar? algebra, with a noncommutative translation sector.
These quantum deformations were classified (see e.g. [50])
in terms of classical r matrices, but systematic studies of
their Hopf quantizations still should be completed. Also the
quantum deformations were considered of D = 3 de Sitter
(d S) and antide Sitter ( Ad S) spacetimes, with nonvanishing
cosmological constant . In the D = 3 d S case ( > 0) all
Hopfalgebraic quantizations are known, because they were
studied as the quantum deformations of D = 4 Lorentz
algebra o(3, 1) [51]. In D = 3 Ad S case ( < 0) with o(2, 2)
symmetry some Hopfalgebraic quantum deformations were
also given, but recently a complete classification was
presented of real o(2, 2) r matrices.14
For physical applications it is very important to
consider subsequently the quantum spacetime deformations for
14 See [8] and the addendum (to be published); for earlier efforts to
describe o(2, 2) quantum deformations see e.g. [52]. We recall that
o(D, D) algebras describe the symmetries of double geometry [53?55],
which were used recently e.g. in the description of selfdual models in
socalled metastring theory [56,57].
D = 4, 5, 6. The deformations of physical D = 4 spacetime
and D = 4 Poincar? algebra have been extensively studied
for more than a quarter of the century [58?62], but it should be
observed that the complete list of D = 4 Poincar? r matrices
( D = 4 Poincar? bialgebras) has not been obtained.15 The
next task could be to describe all deformations of D = 4
spacetimes with constant curvature and arbitrary signature,
which would classify all possible D = 4 quantum d S and
Ad S algebras as well as the quantumdeformed D = 5
Euclidean o(5) symmetries. For such a purpose one can look
for the extension of algebraic methods used to classify the
deformations of o(4; C) and its real forms (see [8]) to the
case of o(5; C) and the real forms o(5), o(4, 1) and o(3, 2).
Finally the systematic study of deformations of o(6; C) is
another important challenge, in particular because the
deformations of its real form o(4, 2) su(2, 2) will provide the
list of quantum D = 4 conformal algebras.
Acknowledgements The authors would like to thank A. Borowiec
for discussions. JL would like to acknowledge the financial support of
NCN (Polish National Science Center) Grants 2013/09/B/ST2/02205,
2014/13/B/ST2/04043 and by European Project COST, Action MP1405
QSPACE. VNT is supported by RFBR Grant No. 160100562a.
Open Access This article is distributed under the terms of the Creative
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