Generic realizations of conformal and de Sitter algebras
New generic realizations of conformal Lie algebra and two de Sitter algebras are obtained.Deformation of the Poincare algebra to the de Sitter ones is constructed.
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Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872552856552210432 |
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| author | Myronova, М. Nesterenko, M. Миронова, M. Нестеренко, М. |
| author_facet | Myronova, М. Nesterenko, M. Миронова, M. Нестеренко, М. |
| author_institution_txt_mv | [
{
"author": "М. Myronova",
"institution": "University of Montreal"
},
{
"author": "M. Nesterenko",
"institution": "Institute of Mathematics"
}
] |
| author_sort | Myronova, М. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2020-08-13T08:52:53Z |
| description | New generic realizations of conformal Lie algebra and two de Sitter algebras are obtained.Deformation of the Poincare algebra to the de Sitter ones is constructed. |
| first_indexed | 2026-08-04T01:06:05Z |
| format | Article |
| fulltext |
Çáiðíèê ïðàöü Iíñòèòóòó ìàòåìàòèêè ÍÀÍ Óêðà¨íè 2019, ò. 16, � 1, 100�112
ÓÄÊ 517.912:512.816
Generic realizations of conformal
and de Sitter algebras
M. Myronova †, M. Nesterenko ‡
‡ Universit�e de Montr�eal, Montr�eal, Canada
E-mail: maria.myronova@gmail.com
‡ Iíñòèòóò ìàòåìàòèêè ÍÀÍ Óêðà¨íè, Êè¨â
E-mail: maryna@imath.kiev.ua
Îòðèìàíî íîâi ïîðîäæóþ÷i ðåàëiçàöi¨ êîíôîðìíî¨ àëãåáðè Ëi òà äâîõ
àëãåáð äå Ñiòòåðà. Ïîáóäîâàíî äåôîðìàöiþ àëãåáðè Ïóàíêàðå äî àë-
ãåáð äå Ñiòòåðà.
New generic realizations of conformal Lie algebra and two de Sitter algebras
are obtained. Deformation of the Poincar�e algebra to the de Sitter ones is
constructed.
1. Introduction. Each well-established physical theory has its own
certain fundamental invariance group and, therefore, realizations (repre-
sentations by first-order differential operators) of their Lie algebras are
effectively used for reduction, integration, differential invariants, etc., see
e.g. [1, 2, 3, 5, 8].
In this work we consider three types of conformal groups: standard
conformal group C(3, 1) and two conformal groups of pseudoeuclidian
spaces C(3, 0) and C(2, 1). For the respective Lie algebras c(3, 1), c(3, 0)
and c(2, 1) we construct the maximal possible (generic) realizations using
the algebraic approach proposed in [7]. Some covariant realizations of the
conformal and de Sitter algebras are well known, but we first represent
realizations in fifteen and ten essential variables respectively. Realiza-
tions in smaller number of variables can be obtained from the given ones
by means of projection with respect a subalgebra.
The paper is arranged as follows. Fist we outline the algorithm for
construction of realizations and define the conformal Lie algebra. Then
we obtain it’s generic realization and we do the same for the both de
Sitter Lie algebras so(4, 1) and so(3, 2). And, finally, we include naturally
Generic realizations of conformal and de Sitter algebras 101
the contraction parameters to de Sitter algebras in such a way, that
contraction results are the Poincaré algebra.
2. Definitions and conventions.Let V be an n-dimensional vector
space over the field of real numbers. Consider a Lie algebra g on V
spanned by a basis {e1, e2, . . . , en} with the structure constants Ckij ∈ R,
here and below i, j, k = 1, 2, . . . , n. We denote an open domain of Rm
as M and Vect(M) is the Lie algebra of smooth vector fields on M with
the Lie product defined as commutator (i.e., the Lie algebra of first-order
linear differential operators with analytical function coefficients).
A realization of a Lie algebra g in vector fields on M is a homomor-
phism R(g) = R : g → Vect(M). The realization is called faithful if
kerR = {0} and unfaithful otherwise.
In Lie theory realizations are considered locally at some neighborhood
Ux ⊂M ⊂ Rm of a point x ∈M and in most of the cases without loss of
generality the realization can be considered in a neighborhood of a zero
point x = 0.
Denote local coordinates of a point x ∈ M as (x1, . . . , xm), then in
coordinate form a realization R(g) is performed by the images Ξi(x) of
the basis elements ei of a general form
Ξi(x) = R(ei) =
m∑
l=1
ξil(x1, x2, . . . , xm)∂l, (1)
hereafter ∂l = ∂
∂xl
and the coefficients ξil(x1, x2, . . . , xm) are smooth
(analytic) functions.
Let us fix a point x ∈ M and let Rx be a realization of g at this
point. Consider the linear map Rx : g → Vect(M)(x) that transforms
a vector v ∈ g to it’s image R(v(x)) at x. The matrix that corresponds
to this linear map is the n by m matrix ξ formed by the coefficients of
the realization (1)
ξ(x) =
ξ11(x) ξ12(x) . . . ξ1m(x)
ξ21(x) ξ22(x) . . . ξ2m(x)
...
...
. . .
...
ξn1(x) ξn2(x) . . . ξnm(x)
.
The rank of the linear map Rx, or, equivalently, the rank of the
matrix ξ(x) at a point x is called a rank of realization R at point x
and is denoted rankRx. The realization rank value possess the obvious
102 M. Myronova, M. Nesterenko
inequality 0 ≤ rankRx ≤ n, where n is the dimension of a Lie algebra g.
The second inequality is dictated by the number of rows in matrix ξ,
which is equal to the number of basis vector fields of g.
A realization R of a Lie algebra g is called transitive if the action of
the local Lie group corresponding to R is transitive. Or, equivalently
(see [4]), a realization R of a Lie algebra g is called transitive if rank
Rp = m for all p ∈M .
For many practical applications it is necessary to decide if two given
sets of first order differential operators (with the isomorphic commuta-
tion relations) can be transformed to each other or not. This task is
rather complicated even in the case of small number of operators and
variables.
Roughly speaking, two realizations are equivalent, if they can be
transformed to the identical form by means of non-singular automorphic
basis changes (ei 7→ ẽi) and 1 to 1 changes of variables (xl 7→ yl = ϕl(x))
with non-zero Jacobi determinant.
Let us have a diffeomorphism of M such that for the corresponding
x, y ∈ M we have y1 = ϕ1(x1, . . . , xm), y2 = ϕ2(x1, . . . , xm), . . . , ym =
ϕm(x1, . . . , xm). Then the realization of the form (1) transforms to the
following:
R̃(ei) =
m∑
l=1
ξ̃il(y)∂yl =
m∑
l=1
(
m∑
l′=1
ξ̃il′(x)
∂ϕl(x)
∂xl′
)
∂yl .
Note, that the coefficients ξ̃il(y) are written in terms of y using the
inverse transformation ϕ−1.
It is obvious that application of transformations from Aut(g) to the
realization R does not change the rank of R, and none of diffeomorphisms
of M can change the realization rank either. Therefore the equivalent
realizations have the same ranks.
Let a realization R(x) : g → Vect(M) has a rank r = rankR < m at
a regular point x ∈ M , where m = dimM . Then there exists a locally
equivalent realization R̃(y) : g→ Vect(M) at a regular point y ∈M such
that the coefficients of basic vector fields ξ̃il(y) = 0 for all i = 1, . . . , n,
l = r + 1, . . . ,m. To prove this let us construct the desired diffeo-
morphism. Since the realization rank is equal to r it is known from
the theory of invariants [9] that there are m − r functionally indepen-
dent invariants J1(x1, . . . , xm), . . . , Jm−r(x1, . . . , xm) of the realization
R. The diffeomorphism of the form ya = xa, a = 1, . . . , r; yr+b = Jb,
Generic realizations of conformal and de Sitter algebras 103
b = 1, . . . ,m− r gives the following zero coefficients of the realization R̃:
ξ̃i(r+b)(y) = R(ei)(Jb) = 0 for all i = 1, . . . , n, b = 1, . . . ,m− r.
The above variables y1, . . . , yr are called essential and the rest of
non-zero variables from yr+1, . . . , ym are called additional.
Example 1. Consider two-dimensional abelian Lie algebra 2A1. It is
well-known that the basis elements of this algebra can be realized by two
operators of translations
R1(e1) = ∂1, R1(e2) = ∂2.
It was shown in [10] that there are exactly two inequivalent realiza-
tions of 2A1, and the second one is
R2(e1) = ∂1, R2(e2) = x2∂1.
In these cases rankR1 = 2 and rankR2 = 1.
Consider the formal sum of these realizations R3 = R1 +R2 (R1 for
the variables (x1, x2) and R2 for the variables (x3, x4)), namely
R3(e1) = ∂1 + ∂3, R3(e2) = ∂2 + x4∂3.
As far as [∂1 + ∂3, ∂2 + x4∂3] = 0, then R3 do realize the Lie algebra
2A1 in the space of four variables (x1, x2, x3, x4) and rankR3 = 2, what
means that the number of essential variables is equal to 2.
Indeed, the diffeomorphism ϕ given by the non-singular functions
ϕ1(x1, . . . , x4) = x1, ϕ2(x1, . . . , x4) = x2,
ϕ3(x1, . . . , x4) = x1 − x3 + x2x4, ϕ4(x1, . . . , x4) = x4
transforms the realization R3 to the equivalent realization R1 in 2 essen-
tial variables.
In case of transitive realizations all variables are essential and, since
rankR ≤ n, any transitive realization of a Lie algebra is realized in not
more then n variables.
A recent paper [7] establishes the one-to-one correspondence between
inequivalent transitive realizations of a Lie algebra g and Int-inequivalent
subalgebras of g. Moreover, this relation was extended to the non-tran-
sitive case as well, see [4].
104 M. Myronova, M. Nesterenko
The coefficients ξik(x) of the generic realization
Ξi =
n∑
k=1
ξik(x)
∂
∂xk
, i = 1, 2, . . . , n,
can be recovered from the left-invariant differential one-forms
Ωi=
n∑
l=1
ωli(x)dxl
using the duality ωli(x)ξik(x) = δlk and the coefficients ωli(x) of the dif-
ferential one-forms are constructed as follows:
ωli(x) =
(
A(1)
(
x1
)
A(2)
(
x2
)
· · ·A(i−1)
(
xi−1
))l
i
,
where i = 2, 3, . . . , n, l = 1, 2, . . . , n, ωl1 = δl1, and the matrices A(p),
p = 1, 2, . . . , n, are the exponential solutions of the system
Ȧ(p)(t) = − adep A
(p)(t), A(p)(0) = I.
All the rest of transitive realizations of a fixed Lie algebra are cons-
tructed by means of projection of the generic realization using the known
set of Aut(g)-inequivalent subalgebras and the following rule.
Let h = 〈em+1, . . . , en〉 be a subalgebra of g = 〈e1, . . . , en〉 with
a complementary space {e1, . . . , em}, then, using the above approach and
the shortcut ∂i = ∂
∂xi
, we will obtain the realization of basis elements in
the form
R(ei) = ξ1
i (x1, x2, . . . , xm)∂1 + · · ·+ ξmi (x1, x2, . . . , xm)∂m
+ ξm+1
i (x1, x2, . . . , xn)∂m+1 + · · ·+ ξni (x1, x2, . . . , xn)∂n.
The realization projected on the coordinates x1, x2, . . . , xm is well
defined and has the form
prhR(ei) = ξ1
i (x1, x2, . . . , xm)∂1 + · · ·+ ξmi (x1, x2, . . . , xm)∂m.
The subalgebra that corresponds to the given realization is the kernel
of its linear map at the origin of coordinates. In other words at the point
x = 0 ∈ Rm the realization vectors that form a basis of corresponding
subalgebra are identically equal to zero.
Generic realizations of conformal and de Sitter algebras 105
Example 2. Consider the realizations
R1 : e1 = ∂1, e2 = x2∂1, e3 = x1∂1 + 2x2∂2,
R2 : e1 = ∂1, e2 = x1∂1 − x2∂2, e3 = ∂2.
At the origin of coordinates x = 0 their basis vectors have the form
R1(x = 0): e1 = ∂1, e2 = 0, e3 = 0,
R2(x = 0): e1 = ∂1, e2 = 0, e3 = ∂2.
Therefore the realization R1 corresponds to the subalgebra 〈e2, e3〉
and R2 corresponds to 〈e2〉.
The structure of realizations constructed by means of the algebraic
method reminds a tree diagram, namely: a realization corresponding to
a subalgebra h1 can be constructed by means of projection from a real-
ization corresponding to a subalgebra h2 if h2 ⊂ h1.
Note that all inequivalent realizations of a fixed Lie algebra can be
obtained by the above method, as far as any realization corresponds
to a quotient group G/H that acts effectively on some subspace M ,
where H is a subgroup that corresponds to some subalgebra h.
In this paper we use the above method to construct the realizations
of three conformal Lie algebras in maximal possible number of essen-
tial variables, that is we construct realizations that correspond to zero
subalgebras.
3. Conformal Lie algebra. First of all we consider a conformal
group and it’s 15-dimensional Lie algebra c(3, 1). The conformal Lie
group C(3, 1) = SO(4, 2) = SU(2, 2) of the Minkowski space is the max-
imal invariance group of the Maxwell equations in the flat space-time.
This group in many aspects unite all physical groups. It is generated by
10 Poincaré generators Pµ, Jµν , dilatation generator D and generators of
special conformal transformations Kµ, hereafter µ, ν = 1, 2, . . . , 4. The
non-zero commutation relations of the Lie algebra are
[Jµν , Jρσ] = gµρJνσ − gνρJµσ + gµσJρν − gνσJρµ, (2)
[Jµν , Pρ] = gµρPν − gνρPµ, (3)
[Jµν ,Kρ] = gµρKν − gνρKµ, (4)
[Pµ,Kν ] = 2(gµνD + Jµν), (5)
[Pµ, D] = Pµ, (6)
106 M. Myronova, M. Nesterenko
[Kµ, D] = −Kµ. (7)
Here gµν is the metric tensor of the Minkowski space g11 = g22 = g33 =
−g44 = 1.
It is possible to consider conformal groups C(p, q) of the pseudoeu-
clidian spaces with metric tensors
g11 = g22 = · · · = gpp = −gp+1,p+1 = · · · = −gp+q,p+q = 1 (8)
and µ, ν = 1, . . . , p+ q = n.
Consider the group SO(p + 1, q + 1) = span{Iab}, Iab = −Iba with
the commutators
[Iab, Icd] = gacIbd − gbcIad + gadIcb − gbdIca,
where gab are from (8) and gn+1,n+1 = −gn+2,n+2 = 1. Then matching
Jµν = Iµν , Pµ = Iµ,n+1 − Iµ,n+2, Kµ = Iµ,n+1 + Iµ,n+2,
D = In+1,n+2
we get the isomorphism C(p, q) ' SO(p+ 1, q + 1). Therefore a number
of well-known groups (like de Sitter groups) are conformal groups of
pseudoeuclidian spaces. Consider the well-known realization of the con-
formal group
Pµ = ∂µ, Jµν = xν∂µ − xµ∂ν , D = xν∂ν ,
Kµ = 2xµxν∂ν − x2∂µ;
hereafter the summation with respect to the repeated indices is implied
and x2 = x2
1 + · · ·+ x2
n.
Let us define the subalgebra that corresponds to the given realiza-
tion. To do this we study the realization at the point x = (0, 0, 0, 0)
and see that the kernel of this linear map coincides with the subalgebra
span{Jµν , D,Kµ}. Indeed, this is proven by the construction and projec-
tion of the generic realization of c(3, 1) with the following complemen-
tary part {Pµ, Jµν ,Kµ, D} taken in the lexicographical order. To make
formula more readable we have introduced the shortcuts:
sinxi = si, cosxi = ci, tanxi = ti, sinhxi = shi,
coshxi = chi, tanhxi = thi, i = 1, 10.
Generic realizations of conformal and de Sitter algebras 107
Rgeneric(c(3, 1)) :
P1 = ∂1, P2 = ∂2, P3 = ∂3, P4 = ∂4,
J12 = x2∂1 − x1∂x2 + ∂5,
J13 = x3∂1 − x1∂3 − th6s5∂5 + c5∂6 + 2
s5
ch6
∂8,
J14 = −x4∂1 − x1∂4 + 2
t7s5
ch6
∂5 + t7s6c5∂6 + c5c6∂7
+
s9s6c5c8 + th6s7c9s5 + s9s8s5
c7c9
∂8
− c5s6s8 − s5c8
c7
∂9 +
s5s8 + c5s6c8
c7c9
∂10,
J23 = x3∂2 − x2∂3 +−th6c5∂5 − s5∂6 +
c5
ch6
∂8,
J24 = −x4∂2 − x2∂4 +
t7c5
c7ch6
∂5 − t7s5s6∂6 − s5c6∂7
+
th6s7c9c5 − s9s6c8s5 + s9c5s8
c7c9
∂8
+
c5c8 + s5s6s8
c7
∂9 −
s5s6c8 − c5s8
c7c9
∂10,
J34 = −x4∂3 − x3∂4 + c6t7∂6 − s6∂7 +
t9c8c6
c7
∂8
− s8c6
c7
∂9 +
c6c8
c7c9
∂10,
K1 =
(
x2
1 − x2
2 − x2
3 + x2
4
)
∂1 + 2x1x2∂2 + 2x1x3∂3 + 2x1x4∂4
− 2
(
x2 +
x4t7s5
ch6
− x3th6s5
)
∂5
− 2(x4t7s6 + x3)c5∂6 − 2x4c5c6∂7
− 2
(
x4t9s6c5c8
c7
+ x4t7th6s5 +
x3s5
ch6
+
x4t9s5s8
c7
)
∂8
+ 2
(c5s6s8 − s5c8)x4
c7
∂9
− 2
(s5s8 + c5s6c8)x4
c7c9
∂10 − (2x1x11 − ch7c5c6)∂11
+ (sh7sh9c5c6 + ch9(s5c8 − s6s8c5)− 2x1x12)∂12
+ (sh9sh10(s5c8 − c5s6s8) + ch10(s6c8c5 + s5s8)
108 M. Myronova, M. Nesterenko
+ sh7ch9sh10c5c6 − 2x1x13)∂13
+ (sh9ch10(s5c8 + c5s6s8) + sh10(s6c8c5 + s5s8)
+ sh7ch9ch10c5c6 − 2x1x14)∂14 + 2x1∂15,
K2 = 2x1x2∂1 +
(
−x1
2 + x2
2 − x3
2 + x4
2
)
∂x2
+ 2x2x3∂3
+ 2x2x4∂4 + 2
(
x4c5t7
ch6
− x1 − x3c5th6
)
∂5
+ 2(x3 + x4t7s6)s5∂6 + 2x4s5c6∂7
+ 2
(
x4t9
c7
(s6c8s5 − c5s8)− x4th7c5 −
x3c5
ch6
)
∂8
− 2
(c5c8 + s5s6s8)x4
c7
∂9 + 2
(s5s6c8 − c5s8)x4
c7c9
∂10
− (2x2x11 + ch7s5c6)∂11 +
(
ch9(c5c8 + s5s6s8)
− sh7sh9s5c6 − 2x2x12
)
∂12 +
(
sh9ch10c5c8
+ ch10c5s8 − sh7sh10ch9s5c6 − ch10s5s6c8
+ sh9sh10s5s6s8 − 2x2x13
)
∂13
+
(
sh9ch10(c5c8 + s5s6c8) + sh10(s8c5 − s5s6c8)
− sh7ch9ch10s5c6 − 2x2x14
)
∂14 + 2x2∂15,
K3 = 2x1x3∂1 + 2x2x3∂x2
+
(
−x2
1 − x2
2 + x2
3 + x2
4
)
∂3 + 2x3x4∂4
− 2th6(x1s5 + x2c5)∂5 − 2(x4t7c6 − x1c5 + x2s5)∂6
+ 2x4s6∂7 +
(
x1s5 + x2c5
ch6
− 2
x4t9c6c8
c7
)
∂8
+ 2
s8c6x4
c7
∂9 − 2
c6c8x4
c7c9
∂10 − (ch7s6 + 2x3x11)∂11
− (ch9c6s8 + 2x3x12 + sh7sh9s6)∂12
+
(
ch10c6c8 − sh9sh10c6s8 − sh7ch9sh10s6
− 2x3x13
)
∂13 +
(
ch10c6c8 − sh9ch10c6s8
− sh7ch9ch10s6 − 2x3x14
)
∂14 + 2x3∂15,
K4 = −2x1x4∂1 − 2x2x4∂x2
− 2x4x3∂3 −
(
x2
1 + x2
2 + x2
3 + x2
4
)
∂4
+ 2
t7(x1s5 + x2c5)
ch6
∂5 + 2t7(x1s6c5 − x2s6s5 + x3c6)∂6
− 2(x2c6s5 − x1c6c5 + x3s6)∂7
Generic realizations of conformal and de Sitter algebras 109
+ 2
(
t9
c7
(x2s5s6c8 − x1s6c5c8 − x3c6c8 − x1s5s8
− x2s8c5)− th6t7(x1s5 + x2c5)
)
∂8
− 2
c7
(x2 − s5s6s8 + x1c5s6s8 + x3c6s8 − x1s5c8
− x2c5c8)∂9 +
2
c7c9
(x1c5s6c8 − x2s5s6c8 + x3c6c8
+ x1s5s8 + x2c5s8)∂10 + (2x4x11 + sh7)∂11
+ (2x4x12 + ch7sh9)∂12 + (2x4x13 + ch7ch9sh10)∂13
+ (2x4x14 + ch7ch9ch10)∂14 − 2x4∂15,
D = x1∂1 + x2∂2 + x3∂3 + x4∂4 − x11∂11 − x12∂12 − x13∂13
− x14∂14 + ∂15.
4. De Sitter Lie algebras. Consider de Sitter groups SO(4, 1)
and SO(3,2) that are the groups of isometry transformations of pseu-
doeuclidean spaces with metric forms x2
1 + x2
2 + x2
3 − x2
4 + x2
5 and x2
1 +
x2
2 + x2
3 − x2
4 − x2
5 respectively. Them are the movement groups of 4-
dimensional Riemann spaces of a constant curvature (de Sitter spaces).
Both de Sitter spaces describe the expanding Universe, where the ra-
dial velocities of galaxies are approximately proportional to distances
from any space point. For the de Sitter Lie algebras we can use the
isomorphisms c(3, 0) ∼ so(4, 1) and c(2, 1) ∼ so(3, 2) with the conformal
commutation relations (2)–(7) for the metric tensors g11 = g22 = g33 = 1
and g11 = g22 = −g33 = 1 respectively. Then, constructing the generic
realization by the method given in second section (with the complemen-
tary part {Pµ, Jµν ,Kµ, D} taken in the lexicographical order), we have
got two following realizations. Note that it is possible to construct one
realization for both de Sitter algebras (putting the parameter to the
commutation relations that changes the tensor sign), but this essentially
complicates calculations and appearance of realizations.
Rgeneric(c(3, 0)) :
P1 = ∂1, P2 = ∂2, P3 = ∂3, J12 = x2∂1 − x1∂2 + ∂4,
J13 = x3∂1 − x1∂3 − th5s4∂4 + c4∂5 +
s4
ch5
∂6,
J23 = x3∂2 − x2∂3 − th5c4∂4 − s4∂5 +
c4
ch5
∂6,
110 M. Myronova, M. Nesterenko
K1 =
(
x2
1 − x2
2 − x2
3
)
∂1 + 2x1x2∂2 + 2x1x3∂3
+ 2(x3s4th5 − x2)∂4 − 2x3c4∂5 − 2
x3s4
ch5
∂6
+ (c4c5 − 2x1x7)∂7 + (s4ch6 − c4sh5sh6 − 2x1x8)∂8
+ (c4s5c6 + s4s6 − 2x1x9)∂9 + 2x1∂10,
K2 = 2x1x2∂1 +
(
x2
2 − x2
1 − x2
3
)
∂2 + 2x2x3∂3
+ 2(x1 + x3c4th5)∂4 + 2x3s4∂5 − 2
x3c4
ch5
∂6
− (s4c5 + 2x2x7)∂7 + (s4s5s6 + c4c6 − 2x8x2)∂8
+ (c4s6 − s4s5c6 − 2x2x9)∂9 + 2x2∂10,
K3 = 2x1x3∂1 + 2x2x3∂2 +
(
x2
3 − x2
1 − x2
2
)
∂3
− 2(x1s4 + x2c4)th5∂4 + 2(x1c4 − x2s4)∂5
+ 2
x1s4 + x2c4
ch5
∂6 − (s5 + 2x3x7)∂7
− (c5s6 + 2x8x3)∂8 + (c5c6 − 2x9x3)∂9 + 2x3∂10,
D = x1∂1 + x2∂2 + x3∂3 − x7∂7 − x8∂8 − x9∂9 + ∂10.
Rgeneric(c(2, 1)) :
P1 = ∂1, P2 = ∂2, P3 = ∂3, J12 = x2∂1 − x1∂2 + ∂4,
J13 = −x3∂1 − x1∂3 + s4t5∂4 + c4∂5 +
s4
c5
∂6,
J23 = −x3∂2 − x2∂3 + c4t5∂4 − s4∂5 +
c4
c5
∂6,
K1 =
(
x2
1 − x2
2 + x2
3
)
∂1 + 2x1x2∂2 + 2x1x3∂3
− 2(x2 + x3s4t5)∂4 − 2x3c4∂5 − 2
x3s4
c5
∂6
+ (c4ch5 − 2x1x7)∂7 + (s4ch6 + c4sh5sh6 − 2x1x8)∂8
+ (s4sh6 + c4sh5ch6 − 2x1x9)∂9 + 2x1∂10,
K2 = 2x1x2∂1 +
(
x2
2 + x2
3 − x2
1
)
∂2 + 2x2x3∂3
− 2(x3c4t5 − x1)∂4 + 2x3s4∂5 − 2
x3c4
c5
∂6
− (2x2x7 + s4ch5)∂7 + (c4ch6 − s4sh5sh6 − 2x2x8)∂8
+ (c4sh6 − s4sh5ch6 − 2x2x9)∂9 + 2x2∂10,
K3 = −2x1x3∂1 − 2x2x3∂2 −
(
x2
1 + x2
2 + x2
3
)
∂3
Generic realizations of conformal and de Sitter algebras 111
+ 2(t5(x1s4 + x2c4)∂4 + 2(x1c4 − x2s4)∂5
+ 2
x1s4 + x2c4
c5
∂6 + (2x3x7 + sh5)∂7
+ (ch5sh6 + 2x3x8)∂8 + (ch5ch6 + 2x3x9)∂9 − 2x3∂10,
D = x1∂1 + x2∂2 + x3∂3 − x7∂7 − x8∂8 − x9∂9 + ∂10.
5. Connection to the Poincaré Lie algebra. Classical Poincaré
algebra p(1, 3) is ten-dimensional and formed by the operators {Pµ, Jµν}
with the commutation relations (2) and (3). Extending this set of com-
mutation relations by the following ones
[Pν , Pµ] = τJµν , τ ∈ R (9)
we get the well-defined 10-dimensional Lie algebra which is the deforma-
tion pτ (1, 3) of p(1, 3) to the both de Sitter algebras at the same time.
Indeed, for τ = 0 pτ (1, 3) coincides with the Poincaré algebra, for τ ≥ 0
pτ (1, 3) ∼ so(4, 1) and for τ ≤ 0 pτ (1, 3) ∼ so(3, 2). So, one can construct
uniform realizations for the both de Sitter and Poincaré algebras apply-
ing the algebraic method to the structure constants from the deformed
relations (2), (3) and (9). The inverse connection between de Sitter and
Poincaré algebras is provided by standard Inönü–Wigner contraction [6]
with respect to the six-dimensional subalgebra so(3, 1).
The result of the paper can be used for construction of differential
invariants and respective invariant differential equations [9].
[1] Fushchych W., Nikitin A., Symmetries of equations of quantum mechanics,
Allerton Press Inc., New York, 1994.
[2] Fushchych W., Tsyfra I., Boyko V., Nonlinear representations for Poincaré and
Galilei algebras and nonlinear equations for electromagnetic field, J. Nonlinear
Math. Phys. 1 (1994), 210–221.
[3] Fushchych V., Zhdanov R., Symmetries and exact solutions of nonlinear Dirac
equations, Mathematical Ukraina Publisher, Kyiv, 1997.
[4] Gromada D., Pošta S., On classification of Lie algebra realizations,
arXiv:1703.00808.
[5] Hernádez Heredero R., Olver P., Classification of invariant wave equations,
J. Math. Phys. 37 (1996), 6419–6438.
[6] Inönü E., Wigner E.P., On the contraction of groups and their representations,
Proc. Nat. Acad. Sci. USA 39 (1953), 510–524.
[7] Magazev A., Mikheyev V., Shirokov I., Computation of composition functions
and invariant vector fields in terms of structure constants of associated Lie
algebras, SIGMA 11 (2013), 066, 17 pp.
112 M. Myronova, M. Nesterenko
[8] Olver P., Applications of Lie groups to differential equations, Springer, New
York, 1993.
[9] Olver P.J., Differential invariants and invariant differential equations, Lie
Groups Appl. 1 (1994), 177–192.
[10] Popovych R., Boyko V., Nesterenko M., Lutfullin M., Realizations of real
low-dimensional Lie algebras, J. Phys. A: Math. Gen. 36 (2003), 7337–7360,
arXiv:math-ph/0301029.
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| id | oai:trim.imath.kiev.ua:article-362 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:06:05Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/ed/77d7409f553e785cd74e0d8fcb6e1ced.pdf |
| spelling | oai:trim.imath.kiev.ua:article-3622020-08-13T08:52:53Z Generic realizations of conformal and de Sitter algebras Породжуючі реалізації алгебр де Сіттера та конформної Myronova, М. Nesterenko, M. Миронова, M. Нестеренко, М. New generic realizations of conformal Lie algebra and two de Sitter algebras are obtained.Deformation of the Poincare algebra to the de Sitter ones is constructed. Отримано нові породжуючі реалізації конформної алгебри Лі та двох алгебр де Сіттера.Побудовано деформацію алгебри Пуанкаре до алгебр де Сіттера. Інститут математики НАН України 2019-09-03 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/362 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 16 No. 1 (2019): Symmetry and Integrability of Equations of Mathematical Physics; 100-112 Сборник Трудов Института математики НАН Украины; Том 16 № 1 (2019): Симетрія та інтегровність рівнянь математичної фізики; 100-112 Збірник Праць Інституту математики НАН України; Том 16 № 1 (2019): Симетрія та інтегровність рівнянь математичної фізики; 100-112 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/362/366 Авторське право (c) 2019 M. Миронова, М. Нестеренко |
| spellingShingle | Myronova, М. Nesterenko, M. Миронова, M. Нестеренко, М. Generic realizations of conformal and de Sitter algebras |
| title | Generic realizations of conformal and de Sitter algebras |
| title_alt | Породжуючі реалізації алгебр де Сіттера та конформної |
| title_full | Generic realizations of conformal and de Sitter algebras |
| title_fullStr | Generic realizations of conformal and de Sitter algebras |
| title_full_unstemmed | Generic realizations of conformal and de Sitter algebras |
| title_short | Generic realizations of conformal and de Sitter algebras |
| title_sort | generic realizations of conformal and de sitter algebras |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/362 |
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