Acta Polytechnica


https://doi.org/10.14311/AP.2025.65.0554
Acta Polytechnica 65(5):554–561, 2025 © 2025 The Author(s). Licensed under a CC-BY 4.0 licence

Published by the Czech Technical University in Prague

ON REALIZATIONS OF LIE ALGEBRAS

Maryna Nesterenkoa,c, Severin Poštab,∗, Mykola Staryia

a Institute of Mathematics of NASciences of Ukraine, 3 Tereshchenkivska St., 01004 Kyiv, Ukraine
b Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering, Trojanova 13,

120 00 Prague, Czech Republic
c Kyiv School of Economics, 3 Mykoly Shpaka St., 03113 Kyiv, Ukraine
∗ corresponding author: severin.posta@fjfi.cvut.cz

Abstract. We discuss and compare the main methods for constructing of Lie vector fields from the
given Lie algebra structure constants. Generic realizations of three conformal algebras are obtained by
the algebraic method and all realizations of the three-dimensional complex special linear algebra are
obtained by the general method and compared with the finite-dimensional weight representations.

Keywords: Realization, representation, local group.

1. Introduction
Despite the fact that Sophus Lie himself began con-
structing realizations, this problem still remains un-
solved for many important cases. The description of
Lie algebra representations by vector fields is of great
interest and widely applicable, e.g. to the integration
of ordinary differential equations. Some new trends in
this area are indicated in [1, 2]. Realizations are also
used in the group classification of partial differential
equations and in the classification of gravity fields of
a general form under the motion groups or groups of
conformal transformations [3]. The construction of
realizations of Lie algebras is also a necessary prereq-
uisite for finding differential invariants and construct-
ing mathematical models with nontrivial symmetry.
An exhaustive description of nonequivalent realiza-
tions of given Lie algebras by vector fields is also an
independent fundamental mathematical problem.

In this work, we review the main methods of real-
ization construction and apply them to special linear
and conformal Lie algebras. In particular, we present
the realization of sl(2,C) that is not equivalent to
a finite-dimensional representation.

The paper is arranged as follows. We first review
the basic definitions and notations in Section 2, then
in Section 3, we describe and compare the main meth-
ods representing an abstract Lie algebra by vector
fields. In Section 4, we compare realizations of sl(2,C)
constructed by the direct method and from the weight
representations. And, in Section 5, we apply the Shi-
rokov’s method to three conformal Lie algebras, with
the resulting generic realizations being presented in
Appendix A.

2. Definitions and statement of
the problem

Consider a Lie algebra g = (V, [·, ·]), where V is
an n-dimensional complex or real vector space with
a bilinear antisymmetric operation [·, ·] : V × V → V

that satisfies the Jacobi identity, and is usually called
a Lie bracket or a commutator.

Fixing the basis e1, . . . , en of V , we can define the
Lie algebra g by its commutation relations:

[ei, ej ] =
n∑

k=1
ck

ijek,

or by the structure constant tensor c, with the com-
ponents ck

ij ∈ C or ck
ij ∈ R.

Hereafter, we assume that the indices i, j, k, ĩ, j̃
and k̃ run from 1 to n, and we will imply the summa-
tion over the repeating indices.

The general linear group acts on the variety of n-
dimensional Lie algebras as follows. Let A ∈ GLn(V )
and B = A−1, then the components of the initial
structure constant tensor c and the resulting tensor c̃
are connected by the formula:

c̃k̃
ĩj̃

= Ai
ĩ
Aj

j̃
Bk̃

kck
ij .

Denote the whole automorphism group of g by
Aut(g) ⊆ GLn(V) and the group of the inner au-
tomorphisms by Inn(g).

In this work, we mostly follow the definitions pro-
posed in [4] with some minor modern modifications
and generalization to the case of complex field. Note
that we work only locally. The definitions given below
are similar for the field of real and complex numbers,
but differ in some details.

Let M ⊂ Rm, m ∈ N be an m-dimensional smooth
manifold and Vect(M) denote the Lie algebra of
smooth vector fields on M .

Definition 1. A realization of Lie algebra g in vector
fields on M is a homomorphism R: g → Vect(M).

The realization is called faithful if ker R = {0}, and
unfaithful otherwise.

554

https://doi.org/10.14311/AP.2025.65.0554
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vol. 65 no. 5/2025 On realizations of Lie algebras

Loosely speaking, we consider Lie algebras of ho-
mogeneous first-order differential operators:

ξ1(x) ∂

∂x1
+ ξ2(x) ∂

∂x2
+ · · · + ξm(x) ∂

∂xm
,

with the coefficients ξα that are smooth functions
on M , hereafter indices α, β and γ run from 1 to m,
∂α = ∂xα = ∂

∂xα
and x = (x1, x2, . . . , xm).

In the case of a complex Lie algebra, we denote a do-
main of Cm as M and Vect(M) are the vector fields
on M with analytical coefficients. The formulation of
the realization definition is the same.

The main problem considered in this work is the
construction of all possible realizations of an abstract
Lie algebra g given by its commutation relations. Note
that for the fixed Lie algebra g, it is reasonable to look
for the faithful realizations only, since all unfaithful
realizations can be investigated as faithful realizations
of some lower-dimensional algebra.

As it follows from the definition, realization is a par-
tial case of representations and we will discuss connec-
tions between realizations and weight representations
in the next section.

To find exhaustive lists, we need to define which
realizations of a given Lie algebra we will consider
different, or, conversely, equivalent.

Definition 2 (R). The realizations R1: g → Vect(M1)
and R2: g → Vect(M2) are called weakly equiva-
lent if there exist φ ∈ Aut(g) and a diffeomorphism
f : M1 → M2 such that R2(v) = f∗ R1(φ(v)) for all
v ∈ g. Here, f∗ is the pushforward from Vect(M1) to
Vect(M2).

If φ is the identical transformation, the realizations
are called strongly equivalent.

Definition 3 (C). The realizations R1: g → Vect(M1)
and R2: g → Vect(M2) are called weakly equivalent if
there exist φ ∈ Aut(g) and a biholomorphic mapping
f : M1 → M2 such that R2(v) = f∗ R1(φ(v)) for all
v ∈ g. Here, f∗ is the pullback from Vect(M1) to
Vect(M2).

If φ is the identical transformation, the realizations
are called strongly equivalent.

The strong equivalence is verified in a simpler way
than the weak one and it can be used for construction
of realizations using realizations of subalgebras, but in
general case, we should classify all realizations of the
given Lie algebra with respect to the weak equivalence.
However, the algebraic method of constructing realiza-
tions (Shirokov’s method) gives grounds to consider
the hypothesis that for two weakly equivalent real-
izations, it is possible to construct a non-degenerate
change of variables that transforms them from one to
another, at least this hypothesis may hold when the
full automorphism group coincides with the group of
inner automorphisms.

3. Construction methods
There are two main approaches to the construction
and classification of realizations of Lie algebras:
(1.) Construction of basis vector fields that satisfy

the given structure constants of the Lie algebra;
(2.) construction of finite-dimensional spaces of vec-

tor fields closed with respect to a multiplication
(commutation) that satisfies the definition of a Lie
bracket.

Regarding the second approach, the classification of
realizations was started by Sophus Lie himself and
was done for the case of one variable over real and
complex fields and for two variables in the complex
case. Within the framework of the second approach,
there are several interesting methods for constructing
realizations, in particular for nilpotent Lie algebras.
Certain results have also been obtained for spaces of
a small number of variables, but this approach is not
relevant to the problem we are considering in this
paper, so we will not provide a detailed overview of
the methods and results.

3.1. The direct method
As for the first approach to constructing realizations
of Lie algebras, the most obvious is the direct method
(which is technically quite complex). First, it was
formulated in [4], here we present it in detail since
this is the only general method known to us today.
The direct method consists of three main steps:
(1.) Take n linearly independent vector fields of the

general form ei = ξiα(x)∂α, and require them to
satisfy the given commutation relations of g.

(2.) By comparing coefficients near different partial
differentiation operators, we obtain a system of first-
order PDEs for the coefficients ξia. Integrate this
system considering all the possible cases.

(3.) Transform the solution into the simplest form,
using nondegenerate transformations of the coordi-
nates on M and automorphism transformations of g.

The direct method is quite difficult to apply, primarily
due to the need to solve a system of partial differential
equations and the requirement to check a large num-
ber of branching cases. But we can propose several
approaches that make the procedure less cumbersome.
First, we formulate the ideas, then we introduce all the
definitions and formulate the necessary statements:
• To transform one of basis elements to the shift

operator, for example, e1 = ∂
∂x1

. Note that the
choice of the basis element that we transform to the
shift operator significantly affects the subsequent
complexity of the calculations and the form of the
operators that we obtain as a result.

• To classify sequential realizations of a series of
nested subalgebras of g, starting with a one-
dimensional subalgebra and ending with g. Inequiva-

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M. Nesterenko, S. Pošta, M. Staryi Acta Polytechnica

lence of the obtained realizations can be guaranteed
if we consider a chain of megaideals.

• To split all possible cases into the groups with the
different realization ranks.
Let us fix x ∈ M and let Rx be a realization of g

in this point. Consider the linear map Rx : g →
Vect(M)(x). The matrix that corresponds to this
linear map is the n by m matrix ξ formed by the
coefficients of the realization:

ξ(x) =


ξ11(x) ξ12(x) . . . ξ1m(x)
ξ21(x) ξ22(x) . . . ξ2m(x)

...
...

. . .
...

ξn1(x) ξn2(x) . . . ξnm(x)

.

Definition 4. The general (maximal possible) rank
of the linear map Rx is called a rank of realization R
and is denoted as rank R.

The realization rank value possesses the obvious
inequality 0 ≤ rank Rx ≤ n, where n is the dimension
of the 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.

If there exists a subset g0 ⊂ g such that
rank R1(g0) ̸= rank R2(g0), then the realizations R1
and R2 are strongly inequivalent.

Definition 5. A megaideal m of g is such a vector
subspace m ⊂ g, that it is invariant under any trans-
formation from Aut(A).

It is clear that any megaideal is a subalgebra and,
moreover, an ideal in g. But there exist ideals which
are not megaideals. Moreover, any megaideal is in-
variant with respect to all the derivations, i.e. it is
a characteristic subalgebra.

Let m be a megaideal and R1 and R2 be realizations
of the algebra g. If R1

∣∣
m

and R2
∣∣
m

are inequivalent,
then R1 and R2 are inequivalent too.

Moreover, if there exists a megaideal m of g such
that rank R1(m) ̸= rank R2(m) then the realizations
R1 and R2 are weakly inequivalent.

The notion of megaideal allows us to construct real-
izations starting from the known lists of realizations
for the low-dimensional algebras, but only in the case
when we can find a nested chain of megaideals. Such
a chain of megaideals cannot be constructed for simple
Lie algebras, so constructing realizations for them is
a particularly difficult task.

For low-dimensional Lie algebras, the direct method
is quite effective and has allowed us to describe real-
izations of real Lie algebras of dimensions no higher
than four [4].

3.2. Blattner’s method
Let us consider the method proposed by Blattner [5]
in 1969. It constructs transitive realizations of Lie
algebras starting from their structure constants. For

further explanation, we need to define transitive real-
izations.

By the third Lie theorem, there exists a local n-
parametric transformation group G corresponding to
the vector fields ei, i.e. ei are infinitesimal generators
of the action of the group G on M . Note that such
a correspondence between the group and the Lie alge-
bra is given by the tangent space at the unit of the
group (the zero of the Lie algebra). A Lie group G
can act on M in several different ways: primitively
(only one orbit is formed when G acts on M) or im-
primitively (M is split into several orbits); transitively
(orbits may exist but there are no stationary points),
and intransitively (G decomposes M into orbits that
are invariant under the action of G). If the local group
G corresponding to a realization acts transitively on
M , then the realization is also called transitive.

Let {e1, . . . , ek, ϵ1, . . . , ϵm} be the basis of a finite-
dimensional Lie algebra g, where s = ⟨e1, . . . , ek⟩ is
a subalgebra of co-dimension m, and U(g) is the uni-
versal enveloping algebra of g. Then, using the set of
vectors {ϵ1, . . . , ϵm} complementary to the subalgebra
s, we can construct the transformation φϵ, defined
for v ∈ g, that gives a transitive realization of the
algebra g by the formula:

φϵ(v) =
m∑

i=1

(∑
l∈Nm

χi(ϵlv)xl

l!

)
∂i,

where l is multi-index and l! :=
m∏

i=1
li!. Here ϵlv ∈

U(g), and χi — the Poincaré–Birkhoff–Witt coeffi-
cient before the monomial ϵi. A Poincaré–Birkhoff–
Witt coefficient is the scalar that appears when a prod-
uct of Lie algebra generators is rewritten in terms of
the ordered PBW basis of the universal enveloping
algebra. It is the coefficients χi that significantly
complicate the application of the Blattner’s formula,
since no regular procedure has been found for their
calculation.

3.3. Shirokov’s method
Let’s consider another approach that is algebraic and
does not require solving differential equations. This
method was first proposed in 1997 by I. Shirokov for
left-invariant vector fields, and it was extended to
the transitive case in 2013 [6]. The main ideas of
this method are to construct vector fields as duals
(inverses) to differential one-forms, and to construct
differential one-forms using adjoint representations.

Let the local Lie group that corresponds to g is
parametrized by the canonical coordinates of the sec-
ond kind, i.e., group elements are represented as or-
dered products of exponentials, each corresponding
to a separate basis element of g. In this case, we can
construct components ωj

i (x) of differential one-forms
by the formula:

ωj
i (x) = (exp(−x1ade1) · · · exp(−xiadei

))j
i .

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vol. 65 no. 5/2025 On realizations of Lie algebras

Now the problem of constructing left-invariant vec-
tor fields reduces to finding the inverse transformation
ξi

k(x) =
(
(ωj

l (x))−1)i

k
. Note that this construction

of realization corresponds to the primitive action of
the local group, we will call such a realization generic.
All other realizations that correspond to the tran-
sitive group action can be obtained as projections
of the generic realizations to the spaces of variables
complementary to subalgebras. In the case of a non-
transitive action of the corresponding local group, the
method becomes more complicated. It was discussed
in the thesis [7] of D. Gromada, but, for practical
application, it is necessary to find the classification
of subalgebras that satisfies certain conditions. This
subtask is rather complicated itself.

Nevertheless, this algorithm is extremely efficient in
constructing generic realizations and can be success-
fully applied to physically interesting high-dimensional
algebras. Appendix A contains generic realizations of
three important conformal Lie algebras c(3, 1), c(3, 0)
and c(2, 1), which are constructed using the algebraic
method.

3.4. Representations and realizations
with linear coefficients

For each Lie algebra, there are always two representa-
tions: trivial and adjoint. Trivial representation has
no essential applications but adjoint representation
can always give us the realization with linear coeffi-
cients. Indeed, if the Lie algebra g has a representation
φ with the corresponding (m × m)-matrices Φi, then
the set of operators:

ei =
m∑

α=1

 m∑
β=1

(Φi)β
αxβ

∂α

generates the realization g and vice versa: any real-
ization with linear homogeneous coefficients generates
the matrix representation of the Lie algebra.

This method of constructing realizations is particu-
larly interesting for the case of simple and semisimple
Lie algebras, since it is difficult to apply the direct
method to them due to the lack of megaideals. It is
also difficult to classify subalgebras of simple algebras
for the application of the Shirokov method, but, at
the same time, weighted representations are known
for simple Lie algebras.

It is clear that some realizations with non-linear
coefficients will be equivalent to those obtained from
representations, but the more interesting question is
whether all realizations can be obtained from the rep-
resentations. To investigate this connections between
realizations and representations, we considered the
smallest simple Lie algebra sl(2,C) in Section 4. Note
that this case has been studied by many authors in
the past, for example [8].

4. Realizations and
representations of sl(2,C)

Consider the Lie algebra sl(2,C) of 2 × 2 traceless
complex matrices, the standard choice of basis is:

e =
(

0 1
0 0

)
, f =

(
0 0
1 0

)
, h =

(
1 0
0 −1

)
, (1)

[e, f ] = h, [h, e] = 2e, [f, h] = 2f. (2)

Let us also consider the basis e1, e2, e3, where:

e = e1, h = −2e2, f = −e3,

and commutation relations have the form:

[e1, e2] = e1, [e2, e3] = e3, [e1, e3] = 2e2. (3)

Irreducible finite-dimensional weight representations
of sl(2,C) in the basis e, f, h are:

Φ1 =

 0 1 0 ··· 0
0 0 2 ··· 0
...

...
. . . . . .

...
0 0 ··· 0 d
0 0 ··· 0 0

, Φ2 = 1
2

−d 0 ... 0
0 −d+2 ... 0
...

...
. . . . . .

0 0 ... d

,

Φ3 =


0 0 ··· 0 0

−d 0 ··· 0 0
0 −d+1 ··· 0 0
...

...
. . .

...
...

0 0 ··· −1 0

.

The realizations that correspond to these representa-
tions are (d ∈ N):

e1 =
d+1∑
α=2

(α − 1)xα−1∂α,

e2 =
d+1∑
α=1

(
−d

2 − 1 + α

)
xα∂α,

e3 =
d∑

α=1
(−d + α − 1)xα+1∂α.

Applying the direct method we obtain the exhaus-
tive list of inequivalent realizations of sl(2,C):
(1.) ∂1, x1∂1 + x2∂2, x2

1∂1 + 2x1x2∂2 + x2∂3,

(2.) ∂1, x1∂1 + x2∂2, (x2
1 + x2

2)∂1 + 2x1x2∂2,
(3.) ∂1, x1∂1 + x2∂2, x2

1∂1 + 2x1x2∂2,
(4.) ∂1, x1∂1, x2

1∂1.
Note that this list contains one less realization than the
similar list obtained for the case of real numbers [4].

To establish the correspondence between the realiza-
tions and representations, we consider different values
of d = 1, d = 2, d = 3, . . . transform one of the opera-
tors to the form ∂x1 and look for the locally invertible
transformations for the rest of the variables. We also
compare the ranks of the realizations. In this way we
have shown that the case d = 1 is transformed to the
realization (3.) by the change of variables x̃1 = x1

x2
,

x̃2 = 1
x2

1
.

557



M. Nesterenko, S. Pošta, M. Staryi Acta Polytechnica

And the case d = 2 is transformed to the realiza-
tion (2.) by the change of variables x̃1 = x2+1

x1
− x2

2 +
x1x3, x̃2 = x

− 1
2

1
(
x2

2 − x1x3
)
, x̃3 = 1

x1
.

The cases d ≥ 3 are equivalent to the realization (1.)
since all of them are of the rank three.

The only realization that was not obtained from the
irreducible weight representations is the realization of
the rank one (4.).

To investigate whether it is possible to linearize this
realization we look for non-degenerate transformations
x̃α = fα(x), α = 1, . . . , m, x = (x1, x2, . . . , xm) such
that for some complex matrices a, b and c:

ẽ1 = (aα1x̃1 + · · · + aαmx̃m)∂x̃α
,

ẽ2 = (bα1x̃1 + · · · + bαmx̃m)∂x̃α
,

ẽ3 = (cα1x̃1 + · · · + cαmx̃m)∂x̃α
.

Solving the obtained system of PDEs, we come to the
functional system that is linear and homogeneous with
respect to the functions fα(x), therefore, the Jacobian
determinant is zero and non-degenerate transforma-
tions do not exist.

This simple example allows us to draw an impor-
tant conclusion that irreducible representations do not
allow us to obtain all differential equations that are
invariant under a given group or Lie algebra.

5. Realizations of conformal
algebras

Consider three important conformal groups: the stan-
dard conformal group C(3, 1) and two conformal
groups of pseudo-Euclidean spaces C(3, 0) and C(2, 1).
Their Lie algebras are denoted c(3, 1), c(3, 0) and
c(2, 1), respectively. Some covariant realizations of
conformal algebras and de Sitter algebras are already
known, but we constructed realizations in spaces of
fifteen and ten variables, respectively. This is the
highest possible dimensions of spaces of essential vari-
ables for these algebras. By essential variables we
mean variables that cannot be eliminated through
a nondegenerate change of variables, i.e., they cannot
be replaced by invariants of vector fields. Realizations
with fewer variables, in particular “classical” realiza-
tions, can be obtained from the presented realizations
by a projection to the spaces of lower dimensions.

The 15-dimensional Lie algebra c(3, 1) of the confor-
mal group is the Lie algebra of the maximal invariance
group of the Maxwell’s equations in flat space-time.
This group, from many points of view, unites all phys-
ical groups (Lorentz, Poincaré, de Sitter, orthogonal,
etc.). It is generated by ten Poincaré operators Pµ,
Jµν , dilation operator D and special conformal trans-
formation operators Kµ, where µ, ν = 1, 2, 3, 4.

The nonzero commutation relations of the Lie alge-

bra have the form:

[Jµν , Jρσ] = gµρJνσ − gνρJµσ

+ gµσJρν − gνσJρµ, (4)
[Jµν , Pρ] = gµρPν − gνρPµ, (5)
[Jµν , Kρ] = gµρKν − gνρKµ, (6)
[Pµ, Kν ] = 2(gµνD + Jµν), (7)
[Pµ, D] = Pµ, (8)
[Kµ, D] = −Kµ. (9)

here gµν is the metric tensor of Minkowski space
g11 = g22 = g33 = −g44 = 1.

If we consider the well-known realization (10) of the
conformal Lie algebra, we can see that it is the projec-
tion of the generic realization that corresponds to the
subalgebra span{Jµν , D, Kµ} with the complementary
part {Pµ, Jµν , Kµ, D}:

Pµ = ∂µ, Jµν = xν∂µ − xµ∂ν ,

D = xν∂ν , Kµ = 2xµxν∂ν − x2∂µ;
(10)

where x2 = x2
1 + · · · + x2

n.
We also consider de Sitter algebras that correspond

to the transformation groups of isometry of pseudo-
Euclidean spaces with the metric forms x2

1 +x2
2 +x2

3 −
x2

4 + x2
5 and x2

1 + x2
2 + x2

3 − x2
4 − x2

5. They are the
groups of motion of 4-dimensional Riemannian spaces
of constant curvature (de Sitter spaces). Both de
Sitter spaces describe an expanding universe where
the radial velocities of galaxies are proportional to
the distances to points in space. For de Sitter al-
gebras, we can use isomorphisms c(3, 0) ∼ so(4, 1)
and c(2, 1) ∼ so(3, 2) with conformal commutation re-
lations (4)–(9) for µ, ν = 1, 2, 3 and metric tensors
g11 = g22 = g33 = 1 and g11 = g22 = −g33 = 1, respec-
tively. Below, we present all their generic realizations
that we obtained by the Shirokov’s method with the
complementary part {Pµ, Jµν , Kµ, D} in other words,
this means that the basis elements are ordered exactly
in the same way as they are presented in the appen-
dices. The theory allows us to construct one realiza-
tion for the both de Sitter algebras by parametrizing
the structure constants with a parameter that changes
the sign, but this significantly complicates the calcu-
lations and the appearance of the realizations.

In the future, we plan to construct projections of
the obtained generating realizations onto spaces of
lower dimensions, find differential invariants for them,
and write invariant partial differential equations.

Acknowledgements
This work was supported by a grant from the Simons
Foundation (SFI-PD-Ukraine-00014586, M.N., M.S.).

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vector fields in terms of structure constants of
associated Lie algebras. Symmetry, Integrability and
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https://doi.org/10.3842/SIGMA.2015.066

[7] D. Gromada. Classification of realizations of
low-dimensional Lie algebras. Master’s thesis, Czech
Technical University in Prague, Faculty of Nuclear
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2017/dp_mf_17_gromada.pdf

[8] M. Havlíček, W. Lassner. Canonical realizations of the
Lie algebras gl(n, R) and sl(n, R). I. Formulae and
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https://doi.org/10.1016/0034-4877(75)90081-6


M. Nesterenko, S. Pošta, M. Staryi Acta Polytechnica

Appendix A.
In this appendix we present generic realizations of Lie algebras of three important conformal groups: the
standard conformal group C(3, 1) and two conformal groups of pseudo-Euclidean spaces C(3, 0) and C(2, 1).

A.1. Generic realization of the conformal algebra c(3, 1)
To simplify the form of the formulas, we introduce the notations:

sin xi = si, cos xi = ci, tan xi = ti, sinh xi = shi, cosh xi = chi, tanh xi = thi, i = 1, . . . , 10.

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 + 2t7s5

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) + 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
− 2x4t9c6c8

c7

)
∂8 + 2s8c6x4

c7
∂9 − 2c6c8x4

c7c9
∂10 − (ch7s6 + 2x3x11)∂11

− (ch9c6s8 + 2x3x12 + sh7sh9s6)∂12 +
(
ch10c6c8 − sh9sh10c6s8 − sh7ch9sh10s6 − 2x3x13

)
∂13

+
(
ch10c6c8 − sh9ch10c6s8 − sh7ch9ch10s6 − 2x3x14

)
∂14 + 2x3∂15,

560



vol. 65 no. 5/2025 On realizations of Lie algebras

K4 = − 2x1x4∂1 − 2x2x4∂x2 − 2x4x3∂3 −
(
x2

1 + x2
2 + x2

3 + x2
4
)
∂4

+ 2t7(x1s5 + x2c5)
ch6

∂5 + 2t7(x1s6c5 − x2s6s5 + x3c6)∂6

− 2(x2c6s5 − x1c6c5 + x3s6)∂7

+ 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.

A.2. Generic realizations of de Sitter algebras
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,

K1 =
(
x2

1 − x2
2 − x2

3
)
∂1 + 2x1x2∂2 + 2x1x3∂3 + 2(x3s4th5 − x2)∂4 − 2x3c4∂5 − 2x3s4

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 − 2x3c4

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

+ 2x1s4 + 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 − 2x3s4

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 − 2x3c4

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 + 2(t5(x1s4 + x2c4)∂4 + 2(x1c4 − x2s4)∂5

+ 2x1s4 + 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.

561


	Acta Polytechnica 65(5):554–561, 2025
	1 Introduction
	2 Definitions and statement of the problem
	3 Construction methods
	3.1 The direct method
	3.2 Blattner's method
	3.3 Shirokov's method
	3.4 Representations and realizations with linear coefficients

	4 Realizations and representations of sl(2,C)
	5 Realizations of conformal algebras
	Acknowledgements
	References
	A 
	A.1 Generic realization of the conformal algebra c(3,1)
	A.2 Generic realizations of de Sitter algebras