1. Introduction
A
Lie system is a first-order system of ordinary differential equations (ODEs)whose general solution can be written as a function, a so-called
superposition rule, of a generic family of particular solutions and some constants to be related to initial conditions [
1,
2,
3,
4,
5,
6,
7]. Lie systems can be characterised by the Lie–Scheffers theorem [
1,
2,
3,
4,
7,
8], which states that a Lie system is equivalent to a
t-dependent vector field taking values in a finite-dimensional Lie algebra of vector fields, called a
Vessiot–Guldberg (VG) Lie algebra of the Lie system [
3,
8].
Every Lie system on
can be endowed with a VG Lie algebra that is, around a genericpoint, locally diffeomorphic to one of the twenty-eight possible classes of VG Lie algebras on
described in References [
4,
8,
9]. In particular, only twelve classes can be considered, again locally around a generic point, as VG Lie algebras of Hamiltonian vector fields relative to a symplectic form (see References [
8,
10] and
Table 1). Lie systems admitting a VG Lie algebra of Hamiltonian vector fields relative to a Poisson bivector are called
Lie–Hamilton systems [
8,
10,
11]. In the case of Lie–Hamilton systems on
, the role played by the Poisson bivector is accomplished, at a generic point on the plane, by a symplectic form [
12].
Although Lie–Hamilton systems on
are the exception rather than the rule among general differential equations (cf. References [
8,
10]), they admit a plethora of geometric properties and relevant applications [
8,
10,
13,
14,
15], which motivates their analysis. For instance, Smorodinsky–Winternitz oscillators [
16] or certain diffusion equations can be analysed via Lie–Hamilton systems on
(see References [
8,
10,
15] and references therein).
Among Lie systems, a relevant role is played by
automorphic Lie systems, that is, a class of Lie systems described by a
t-dependent vector field on a Lie group
G taking values in its Lie algebra of right-invariant vector fields (or in its Lie algebra of left-invariant vector fields either) [
1,
3,
8]. All Lie systems can be described via automorphic Lie systems (see References [
1,
3,
8,
17,
18]). Nevertheless, there exists no prototypical geometric model to describe Lie–Hamilton systems. Moreover, many works on Lie–Hamilton systems try to derive or to explain the existence of a Poisson bivector or symplectic form turning the elements of a VG Lie algebra into Hamiltonian vector fields [
10,
19]. This has been done by solving systems of partial differential equations (PDEs) [
12] or using other algebraic and geometric techniques [
10,
14,
19].
The aim of this work is to provide prototypical geometric models for Lie–Hamilton systems on
admitting
locally transitive VG Lie algebras, that is, Lie algebras of vector fields whose elements span, at every point, the tangent space to the manifold where they are defined on. More specifically, we focus on the classes P
, P
, P
, P
, I
, I
, I
, I
, I
and I
given in
Table 1. We use two types of geometric models for studying Lie–Hamilton systems on
and we propose methods to derive geometrically their associated symplectic structures.
Our first model describes Lie–Hamilton systems on
as the restriction of a Lie–Hamilton system on the dual,
, of a Lie algebra
to an even-dimensional symplectic leaf on
relative to the Poisson structure given by the
Kirillov-Kostant-Souriau (KKS) bracket [
21]. Our methods significantly extend and clarify the procedure briefly sketched in Reference [
14] Theorem 2, which was focused on studying only a particular simple example and it did not consider several difficulties for the application of the technique. Moreover, we now obtain as a byproduct that certain Lie–Hamilton systems on
are endowed with a pseudo–Riemannian metric that is invariant relative to the Lie derivative with respect to the vector fields of the VG Lie algebra of the Lie–Hamilton system. This explains previous results relative to the existence of Lie systems on
admitting a VG Lie algebra of Killing vector fields relative to a pseudo-Riemannian metric given in Reference [
19].
Our second model is a projection onto a quotient space of an automorphic Lie system whose VG Lie algebra leaves invariant a Poisson bracket, or a general bivector field that it is also projectable. This model extends the methods in Reference [
22], where invariant tensor fields relative to VG Lie algebras for Lie systems were constructed via
-invariant elements of tensor algebras on the elements of a Lie algebra.
It is remarkable that all presented geometric models can be immediately extended to describe Lie–Hamilton systems on higher-dimensional manifolds. Moreover, our results also provide additional information on the existence of symplectic leaves for KKS brackets related to general Lie algebras, for example, the existence of two-dimensional symplectic leaves relative to the KKS bracket on . Finally, our techniques give rise to the conjecture that the compatible structures for Lie systems can be obtained very generally via the projections onto quotient spaces of automorphic Lie systems admitting VG Lie algebras of right-invariant vector fields and associated left-invariant tensor fields.
2. Fundamentals
Let us provide a brief account of the theory of Lie–Hamilton systems needed to understand our work and to make our exposition almost self-contained. To highlight our main ideas and to avoid minor technical details, we assume, if not otherwise stated, that mathematical objects are smooth and well defined globally (see References [
1,
3,
8,
21,
23] for further details). Hereafter,
N stands for a connected
n-dimensional manifold and
denotes an abstract finite-dimensional Lie algebra.
We call a
t-dependent vector field on
N a map
satisfying that
, where
is the canonical tangent bundle projection and we define
. An
integral curve of
X is a solution
to
Then,
becomes an integral curve of the
autonomisation (also called
suspension) of
X, that is, the vector field on
of the form
[
3,
23]. Conversely, a section
of the bundle
that is additionally an integral curve of
leads to a solution
to (
1). This one-to-one correspondence enables us to identify system (
1) with its associated
t-dependent vector field
X. This also allows us to shorten the terminology of the paper.
Each
t-dependent vector field
X on
N amounts to a
t-parametric family of standard vector fields on
N of the form
. We call
smallest Lie algebra of
X (also called
minimal Lie algebra or
irreducible Lie algebra in the literature [
8]) the smallest Lie algebra (in the sense of inclusion) of vector fields,
, containing
. Let us denote by
the generalised distribution on
N spanned by the elements of a Lie algebra of vector fields
V. Then,
is regular in the connected components of an open and dense subset of
N (see References [
21,
24] for details). Special attention will be hereafter paid to finite-dimensional Lie algebras of vector fields, the so-called
Vessiot–Guldberg (VG) Lie algebras.
A
superposition rule [
2,
3,
7,
8] for a system
X on
N is a map
satisfying that the general solution,
, to
X can be expressed as
for a fixed family of particular solutions
to
X and a parameter
to be related to the initial conditions of
. We call
Lie system a system of ODEs admitting a superposition rule. Although the term superposition rule has been used in the literature with different meanings and there exist different approaches to each notion, our definition in this paper is the predominant in the literature on nonlinear differential equations (cf. References [
1,
3,
8,
25,
26] and references therein).
Theorem 1. (The Lie–Scheffers theorem [1,2,3,4,7]) A system X on N admits a superposition rule if and only if for a set of vector fields on N generating an r-dimensional Lie algebra of vector fields, called a VG Lie algebra of X and a family of t-dependent functions. A prototypical type of Lie system (see References [
1,
3,
8]) is given by the system of differential equations on a Lie group
G associated with a
t-dependent vector field
where
stand for a basis of right-invariant vector fields on
G and
are arbitrary
t-dependent functions. Indeed,
span an
r-dimensional VG Lie algebra on
G [
23]. The Lie–Scheffers theorem yields then that (
2) admits a superposition rule, which turns
into a Lie system. In fact, the right-invariance of (
2) relative to the action of
G on itself from the right ensures that if
is any particular solution to (
2) and
, then
is another particular solution to (
2). Since initial conditions to (
2) are in one-to-one correspondence with its particular solutions, the general solution to (
2), let us say
, can be written as
where
h is an arbitrary element of
G. Then,
possesses a superposition rule
. Lie systems taking the form (
2) are called
automorphic Lie systems [
3,
22]. Every Lie system can be solved through one particular solution to a related automorphic Lie system [
1,
3,
8]. Automorphic Lie systems play a relevant role in the determination of superposition rules for special classes of Lie systems [
17,
18].
Let us turn to studying Lie systems with compatible Poisson structures (see Reference [
8] for details). We recall that a
Poisson structure on
N is a map
that is antisymmetric, bilinear and it satisfies the Leibniz property and the Jacobi identity. The pair
is called a
Poisson manifold. We just say that
N is a Poisson manifold if
is understood by context. Since
is a derivation on each entry, it amounts to a bivector field
on
N satisfying
, where
stands for the
Schouten-Nijenhuis bracket. Conversely, a bivector field
on
N satisfying
leads to a Poisson structure
for every
. Hence,
and its associated
can be considered as equivalent.
A vector field X on a Poisson manifold N is Hamiltonian if there exists a certain such that for every . A Casimir function of is a function satisfying that for every .
The use of Poisson structures in the study of Lie systems appeared very succinctly in Reference [
1] but its usefulness, for instance so as to obtain superposition rules and constants of motion for Lie–Hamilton systems, was shown posteriorly in Reference [
11]. In particular, the use of Poisson geometry in the study of Lie systems is based on the definition below [
8,
11].
Definition 1. A system X on N is called a Lie–Hamilton system if is a VG Lie algebra of Hamiltonian vector fields relative to a Poisson bivector on N.
Reference [
12] showed that every VG Lie algebra
V of Hamiltonian vector fields on a two-dimensional Poisson manifold
N is, around each point
where
, locally diffeomorphic to a VG Lie algebra detailed in just one of the twelve classes of
Table 1. Note that I
, I
, I
and I
are additionally subdivided into subclasses depending on the index
r, which fixes the dimension of the VG Lie algebras of each subclass.
VG Lie algebras may be isomorphic as Lie algebras without being locally diffeomorphic, for example, this concerns the VG Lie algebras of the classes P
, I
and I
(see
Table 1). Meanwhile, VG Lie algebras belonging to one of the classes I
, I
, I
and I
, may not be isomorphic between themselves as Lie algebras. For instance,
and
belong to I
(for
) but they are not isomorphic as Lie algebras because the first one is isomorphic to
while the second is isomorphic to
for a different, not equivalent to
, semi-direct product
.
If a Poisson bivector
satisfies that the map
is an isomorphism, then
becomes a symplectic form on
N. Every Poisson manifold
is such that
N can be decomposed into the sum of non-intersecting submanifolds (generally of different dimension), a so-called
stratification [
23], in such a way that
is tangent to each submanifold and its restriction induces a symplectic structure on it. Moreover, the tangent space to each submanifold of the stratification is spanned by the restriction to it of the Hamiltonian vector fields relative to
(see Reference [
21] for details).
Let us now define a new structure for the study of Lie systems. Let G be a Lie group and let be the space of left-invariant contravariant tensor fields on G. It is clear that if is a basis of left-invariant vector fields on G, then every element of can be considered as a linear combination of tensor products of . Moreover, if stands for the space of left-invariant vector fields on G, then one can define a Lie algebra morphism , where denotes the Lie derivative relative to X.
Let us briefly analyse the space of invariants,
relative to
. Let us show how Casimir elements of a Lie algebra
and the
-invariant elements of the Grassmann algebra,
, of
can be considered as elements in
(see Reference [
27] for details). A Lie algebra
admits a so-called
universal enveloping algebra,
, which is linearly isomorphic to the space of symmetric tensor products of elements of
. The adjoint representation
, can be extended to a Lie algebra morphism
by considering that the extension of each
with
to
, namely
, is a derivation relative to the tensor product. Then, a
Casimir element of
is a
satisfying that
for every
. Similarly,
is linearly isomorphic to the space of skew-symmetric tensor products of elements of
. Likewise,
can be extended to a new Lie algebra morphism
by extending each
, with
, to a derivation in
relative to the tensor product. A
-invariant element is an
such that
for every
.
Since every abstract Lie algebra is isomorphic to the Lie algebra of left-invariant vector fields of a connected and simply connected Lie group G, Casimir elements and -invariant elements amount to elements of . A similar construction to the above one can be accomplished by considering right-invariant objects on G. We denote then by the space of right-invariant tensor fields on G.
Our previous construction is similar to the tensor algebra structure given in Reference [
22]. In this paper, we will use such an approach to study Lie–Hamilton systems on the plane or higher-dimensional manifolds.
There exist other kinds of Lie systems admitting compatible geometric structures, for example, a Lie system
X on
N is called a
pseudo-Riemannian Lie system when it possesses a VG Lie algebra of Killing vectors relative to a pseudo-Riemannian metric on
N (see Reference [
8] and references therein).
Finally, we detail a lemma that, although being rather immediate, it seems to be absent in the literature. It will be useful to study the description of Lie–Hamilton systems through automorphic Lie systems with a compatible geometric structure.
Lemma 1. A k-vector field on a Lie group G, that is, an antisymmetric k-contravariant tensor field on G, can be projected onto the quotient space of left cosets of a connected Lie subgroup if and only if , where is the canonical projection onto the quotient space, for every left-invariant vector field tangent to H.
Proof. The projection of
to
exists if and only if
is the same for every
projecting onto the same coset in
. This implies that every integral curve
with
of a vector field
tangent to
H satisfies that
Hence, for every
and
, we have
Then, . Hence, for every tangent to H. This shows that the latter condition is necessary. Since the vector fields, , tangent to H span the whole tangent space to H and H is connected, the previous reasoning can be reversed to get that the given condition is sufficient. □
3. Lie–Hamilton Systems on Related to Simple VG Lie Algebras
Let us show how a Lie–Hamilton system on
admitting a simple VG Lie algebra can be considered as the restriction of a certain type of Lie–Hamilton system on the dual to a Lie algebra to even-dimensional symplectic leaves of the KKS bracket on such a dual [
21]. Our approach extends the results and applications given in Reference [
14] Theorem 2.
Let
be a Lie algebra with a Lie bracket
. If
, where
is the dual space to
, the canonical isomorphisms
and
, for any
, allow us to consider
as a vector in
. This leads to a Poisson bracket on
, called the
KKS bracket [
21], given by
where
stands for the value of the linear form
at
.
Let us consider a basis
of
. The isomorphism
enables us to understand the elements of this basis as a coordinate system on
. Let us now define a differential equation on
given by
where
for
and
are arbitrary
t-dependent functions. The vector fields
for every
and
, span a finite-dimensional Lie algebra,
, since
where
, with
, are the structure constants of
in the chosen basis. Hence,
is a Lie algebra and
is a Lie algebra morphism. Moreover,
, namely
, where
denotes the centre of
.
In view of the above comments, (
4) is the system of differential equations associated with
which is a Lie system admitting a VG Lie algebra
. Moreover, since
due to the assumption on the simplicity of
, one obtains that
. By construction,
are Hamiltonian relative to the KKS bracket (
3) on
. Hence, (
4) becomes a Lie–Hamilton system.
Recall that a Poisson structure on
N gives rise to a stratification of
N into symplectic submanifolds [
21]. In particular,
admits a stratification,
, induced by the KKS bracket. The tangent space to each leaf of the stratification is spanned by the Hamiltonian vector fields of the KKS bracket [
21]. Consequently,
are tangent to each leaf
of
and system (
4) can be restricted to each
. We write
for the restriction of
to
. We also denote by
the restrictions of
to
, respectively.
Since the elements of
can be restricted to each
, such restrictions give rise to a VG Lie algebra
for the system
. It may happen that
is not isomorphic to
, since some restrictions of the vector fields of
to
may vanish. Nevertheless, the restriction mapping
is a surjective Lie algebra morphism. Using the above facts and since
is simple by assumption, we obtain that the kernel of
must be equal to zero or
, because these are the only ideals of
. Let us prove that
. Every Hamiltonian vector field on
is of the form
for a certain
. Since
are coordinates on
, one obtains that
h is a function of
, that is,
and, using the properties of Poisson brackets, one gets that
Hence, the vector fields
span the tangent space to the symplectic leaves of
. In other words, the elements of
span the tangent space to each
. If we assume that
, then the elements of
span the tangent space to
and
. Hence,
and
. Thus, (
4) can be restricted to each leaf,
, giving rise to a Lie–Hamilton system with a VG Lie algebra of Hamiltonian vector fields
.
Summarising, we have proved the following result.
Theorem 2. Every simple Lie algebra defines a Lie–Hamilton system (4) on with respect to the KKS Poisson bracket. The restriction of (4) to each non-zero dimensional symplectic leaf gives rise to a new Lie–Hamilton system with a VG Lie algebra isomorphic to . In particular, if
, then
is locally diffeomorphic to one of the classes of Lie–Hamilton systems on
admitting a VG Lie algebra isomorphic to
. In view of
Table 1 and since
and
have two-dimensional symplectic leaves (cf. Reference [
28,
29]), Theorem 2 potentially allows one to obtain the Lie–Hamilton systems on
related to the VG Lie algebras isomorphic to
and
, that is, P
, P
, I
and I
. According to
Table 1, there exist three classes of VG Lie algebras of Hamiltonian vector fields on
isomorphic to
: P
, I
and I
. To check that Theorem 2 allows us to recover all of them, we have to verify all the restrictions of (
4) on
to each symplectic leaf relative to the KKS bracket in detail (see
Figure 1). On the other hand, it is immediate that Theorem 2 gives rise to a Lie–Hamilton system on
admitting a VG Lie algebra locally diffeomorphic to P
, since it is the only class of Hamiltonian VG Lie algebras on
isomorphic to
(see
Table 1).
Although Lie–Hamilton systems on
related to VG Lie algebras isomorphic to
and
were very briefly studied in Reference [
14], our analysis here is much more detailed and it additionally shows, as a bonus, the existence of additional features of such Lie–Hamilton systems, which retrieves in a more natural and general manner results given in References [
19,
22].
• VG Lie algebras isomorphic to (classes P, I and I):
Let
be a basis of
satisfying the commutation relations
In the given basis, system (
4) takes the form
Hence,
admits a Casimir function given by
(see References [
28,
29]). The symplectic leaves of the KKS bracket on
are therefore given by the submanifolds where the function
takes a constant value. Then, Hamiltonian vector fields are tangent to such submanifolds. This implies that we have three main types of two-dimensional symplectic leaves where
is positive (two-sheeted hyperboloids), negative (one-sheeted hyperboloids) or zero (a two-sided cone without vertex). Three types of symplectic leaves are illustrated in the diagram aside in orange, green and blue colour, respectively.
In the coordinate system
one obtains that the restriction of (
5) to each leaf of the KKS bracket, reads
To determine to which class of the
Table 1 systems (
6) belong to, one can use the so-called
Casimir tensor field (see Reference [
10])
This tensor field appears naturally as the restriction to each of on . In fact, since the previous tensor field is constructed as a linear combination of tensor products of Hamiltonian vector fields, it is then tangent to each and it gives rise to .
Recall that the two-dimensional symplectic leaves in
relative to the KKS bracket are given by its symplectic leaves in
. Hence, one can use ([
10], Theorem 4.4), which claims that the sign,
, of the matrix of the coefficients of
at a single point of
classifies to which class of
Table 1, the Lie algebra
is locally diffeomorphic to. In particular, if
, then
is locally diffeomorphic to P
; while
gives that
is locally diffeomorphic to I
; and finally
indicates that
is locally diffeomorphic to I
.
To check which class of Lie algebras on the plane is
diffeomorphic to, it is enough to see that
are local coordinates at a certain open subset of each two-dimensional leaf
. Hence,
for
. Then,
Thus, each one of the main three types of symplectic leaves leads to a VG Lie algebra of a different type. In particular,
yields a VG Lie algebra locally diffeomorphic to P
on a two-sheeted hyperboloid;
leads to a VG Lie algebra locally diffeomorphic to I
on a one-sheeted hyperboloid; and
induces a VG Lie algebra locally diffeomorphic to I
on a two-sided cone without vertex. In turn, this makes the Lie system
into a Lie–Hamilton system possessing a VG Lie algebra locally diffeomorphic to P
, I
and I
, when
,
or
, respectively. Recall that the symplectic form related to (
9) is the one associated with the restriction to
of the Poisson bivector on
.
For the sake of completeness, we provide the coordinate expression for (
9). In particular the system can be mapped, for
, onto the system
through the change of variables
and
. In the case
, system (
6) can be mapped, via the change of variables
and
, onto the system
Finally, the change of variables
maps (
6) into
It is worth noting that our approach also explains other aspects of the Lie–Hamilton systems (
9). In fact, the Casimir tensor field (
7) can also be understood as a symmetric contravariant tensor field on each leaf
that is invariant relative to the Lie derivative with the elements of
. When
, the matrix of the coefficients of (
7) becomes non-degenerate by (
8) and
admits an inverse
that is a metric invariant relative to the Lie derivaties with the elements of
. Then, the Lie–Hamilton systems on the leaves with
become pseudo-Riemannian Lie systems as proved in Reference ([
19], Table 1). In particular, the metric is positive definite for
, that is, Riemannian, while it is pseudo-Riemannian for
.
• VG Lie algebra isomorphic to (class P):
Consider the basis of
of the form
admitting the structure constants (We recall that these commutation relations are not the same as the ones of the basis of P
given in
Table 1).
Then, system (
4) takes in this case the form
In this case, the Casimir function of the KKS Poisson structure on
reads [
28,
29]
The two-dimensional submanifolds where
becomes a constant
, that is, the symplectic surfaces associated with the KKS bracket, are therefore spheres,
, with different radius
. As previously explained in the general theory of this section, the Lie algebra
defined on each sphere becomes a VG Lie algebra isomorphic to
. In view of
Table 1, the Lie algebra
is locally diffeomorphic to P
.
For the sake of completeness, we can describe the restriction process in detail. If one consider the set of variables
the restriction of (
10) to a symplectic leaf
reads
As stated previously, there must exist a change of variables mapping this system of differential equations into a Lie system with the VG Lie algebra P
. In fact, the change of variables
maps (
11), that is, the restriction of (
10) to
, onto
where
.
It is worth noting that the Casimir tensor field relative to
, namely
is tangent to
. The coefficients of
in the local coordinates
, which are well defined on a point with
of every leaf
for
, read
and
becomes a non-degenerate tensor field. Its inverse,
, becomes a Riemmanian metric since the determinant of its coefficients is positive. As a similar result can be obtained in any local coordinate system around points of each
, the leaf
becomes a Riemmanian manifold. Since (
12) satisfies that
for
, then
consists of Killing vector fields with respect to
, as proved in Reference [
19] and the Lie–Hamilton systems on
are also Riemannian–Lie systems.
To finish this section, we stress that Theorem 2 implies that the Lie algebra
or other simple Lie algebras have no two-dimensional symplectic leaves. Otherwise, such Lie algebras of vector fields would give rise to Lie algebras of Hamiltonian vector fields on
, which are absent in
Table 1. Moreover, the method exposed here can be employed to obtain Lie–Hamilton systems on higher-dimensional manifolds.
4. Non-Simple VG Lie Algebras and Symplectic Foliations
It is immediate that the method showed in the last section can be slightly modified to give rise to Lie–Hamilton systems on the non-zero-dimensional symplectic leaves of the symplectic stratifications of the KKS bracket of general, not necessarily simple, Lie algebras
. Nevertheless, the induced VG Lie algebras on each leaf,
, do not need to be isomorphic neither to
nor to
. The difficulties due to the lack of an isomorphism between
and
were not addressed in Reference [
14]. Despite above mentioned problems, we prove in this section that the method given in
Section 3 can be extended to the Lie algebras
and
to obtain Lie–Hamilton systems on their symplectic leaves related to VG Lie algebras locally diffeomorphic to elements of the class I
for
.
• VG Lie algebra isomorphic to (class I for ):
Let
be a basis of the Lie algebra
, that is, the abstract Lie algebra isomorphic to a VG Lie algebra of the class I
, with commutation relations
The Poisson structure on
given by the KKS bracket admits a Casimir function [
28,
29] given by
Consequently, the symplectic stratification on
is given by the cylinders
, with
and the set of points with
. We focus on the two-dimensional symplectic leaves,
with
, of this stratification. Using (
14), we obtain that the vector fields
for
read
on the coordinates
of
. The restrictions of
to the two-dimensional symplectic leaves
can be expressed on each leaf by restricting the cylindrical coordinates
on
to each leaf, that is, via
. Then,
Therefore, one obtains that are Hamiltonian and span a three-dimensional Lie algebra . Since , and the Lie brackets of with elements of belong also to the latter linear space, must be locally diffeomorphic to a Lie algebra of the class I for .
• VG Lie algebras isomorphic to (class I for ):
Let
be a basis of
with the structure constants given by
Then, a Casimir function is given by
Hence, the two-dimensional symplectic leaves associated with the KKS Poisson bracket on are given by surfaces or the four semiplanes with or . We now write for the symplectic leaf for a certain .
Let us derive the restrictions of the vector fields
on
, that is,
to the symplectic leaves
by defining the coordinates
and
, when they will have sense. This gives rise to the expressions
Therefore, the restriction to a leaf with is a Lie system with a VG Lie algebra belonging to the class I for .
It is immediate that for gives rise to a VG Lie algebra on a semi-plane isomorphic to , that is, a non-Abelian two-dimensional Lie algebra and locally diffeomorphic to I for .
5. An Approach through Automorphic Lie Systems and Left-Invariant Bivector Fields
The method established in
Section 3 and
Section 4 cannot be easily applied to all Lie–Hamilton systems on
. On the one hand, this is due to the fact that the abstract Lie algebras isomorphic to the VG Lie algebras in
Table 1 may not posses a sufficient number of Casimir functions or invariants to reduce the system to a two-dimensional submanifold with a compatible symplectic structure, for example, the Lie algebra I
for
has no Casimir function (cf. References [
28,
29]). On the other hand, one could consider other Lie algebras
not given in
Table 1 and expect that the restriction of the vector fields
on
onto certain two-dimensional symplectic leaves (relative to the KKS bracket) will give rise to Lie–Hamilton systems with the remaining VG Lie algebras in
Table 1. But there is no hint about the form of the general Lie algebras
to be used.
Above problems motivate our following alternative model for Lie–Hamilton systems on (and other possible ones on higher-dimensional manifolds). Our approach suggests that Lie–Hamilton systems on can be recovered as projections of automorphic Lie systems with appropriate projectable invariants in or, more generally, elements of .
Let us provide our new approach. Consider a Lie system
on a two-dimensional connected manifold
N with a VG Lie algebra
. The integration of
V allows us to obtain a Lie group action
whose Lie algebra of fundamental vector fields is equal to
V and
. Assume that
, which yields that
V is a locally transitive VG Lie algebra. Then,
is surjective for every
. If
is the isotropy group of
, then one gets the commutative diagram
where
is a diffeomorphism. Moreover, if
is the right-invariant vector field on
G with
for
and
is the fundamental vector field (We choose the fundamental vector field
to be defined as
for every
) of
associated with
v, then
In particular,
, with
, for a certain basis
of
. Due to the commutativity of the diagram (
15), one obtains that
where
, as usually, stands for the Lie algebra of right-invariant vector fields on
G.
Our aim now is to obtain a Poisson bivector on G that can be projected onto , for example, is a left-invariant bivector field that is invariant relative to the left-invariant vector fields tangent to . This shows that exists. Then, induces a Poisson bivector on that is invariant relative to . Hence, this allows us to define a Lie–Hamilton system on given by , which is diffeomorphic to the one, X, on N via . If does not vanish, then X is a Lie–Hamilton system relative to a symplectic structure.
To illustrate the above method, let us analyse Lie–Hamilton systems related to P, I, P, and I.
• VG Lie algebra isomorphic (class P):
We consider the Lie group
as the direct product of the matrix Lie group
of
unimodular matrices with
relative to the multiplication
Let us analyse now the automorphic Lie system on
given by
for a certain basis
of right-invariant vector fields on
. Let us use a local coordinate system
defined close to the neutral element of
and related to the description of elements of
in the form
In this coordinate system, a basis of left-invariant vector fields on
reads
They span indeed a VG Lie algebra isomorphic to the VG Lie algebra of the class P
. Meanwhile,
Since is a exterior product of left-invariant vector fields, it is then invariant relative to the Lie derivatives with respect to the elements of .
If we consider the quotient space
of left-cosets of
, we obtain the projection
which satisfies that
. Since
and all the elements of
are invariant with respect to the Lie derivatives relative to the elements of
, one obtains that
and
are projectable onto
. The projections of the elements of
onto
are spanned by linear combination over the reals of the elements of the basis
Therefore, the projection of the vector fields (
16) onto
span, as proved in our general theory, a VG Lie algebra isomorphic to
. Moreover, the projection of
onto
takes the form
and
admits a VG Lie algebra
. Moreover, one has that the projection,
, of
onto
reads
which becomes equivalent to a symplectic form. Since
is invariant relative to the elements of
, it follows that
is invariant with respect to the elements of
, and (
17) becomes a Lie–Hamilton system relative to
. Recall that the integration of the VG Lie algebra P
gives rise to a Lie group action
. In view of the basis of P
in
Table 1, one sees that
can be considered as the isotropy Lie group of
. Then, the mapping
gives rise to a diffeomorphism
that maps (
17) onto a Lie–Hamilton system on
of the form
where
is a basis of P
.
• VG Lie algebra isomorphic to (class I):
The Lie algebra,
, which is isomorphic to the VG Lie algebra I
for the fixed
r, has an associated Lie group given by
. This Lie group acts on
having a set of fundamental vector fields given in the class I
of
Table 1. The Lie algebra of vector fields vanishing at
is given by
. Then, the isotropy group at
is
, and its corresponding Lie algebra is isomorphic to
.
Let us choose a basis of
of the form
obeying the same commutation relations as the basis of I
in
Table 1. In other words, the chosen basis has non-vanishing commutation relations
for
. If
is a basis of left-invariant vector fields on
such that
for
, one can consider the bivector field on
of the form
This bivector field is invariant relative to the Lie derivatives with right-invariant vector fields. Moreover,
can be projected onto
together with the automorphic Lie system on
of the form
If
is the quotient map,
is a Poisson bivector that is invariant relative to
, for
, and this leads to a Lie–Hamilton system on
given by
In view of our previous general comments, the VG Lie algebra of the projection is locally diffeomorphic to I, and it represents geometrically this class of Lie–Hamilton systems on .
• VG Lie algebras P and I:
These cases follow exactly the same ideas given in previous examples, hence we will just sketch the procedure. In the case of the VG Lie algebra P
, its vector fields can be integrated to define an action of the Lie group
on
admitting a set of fundamental vector fields given by P
. One then considers an automorphic Lie system
on the Lie group
, where
are assumed to admit the same commutation relations as the basis
of P
given in
Table 1. The isotropy group of the point
is given by a Lie subgroup
whose Lie algebra is spanned by
.
Then, is a Poisson bivector on the Lie group that is also invariant relative to the Lie derivatives with respect to the VG Lie algebra . Thus, and are invariant relative to and they can be therefore projected simultaneously onto via , giving rise to a Lie–Hamilton system on the plane relative to with a VG Lie algebra locally diffeomorphic to P. In the case of I, the procedure is absolutely analogous.
Let us provide a slight generalisation of the above procedure that will allow us to retrieve the Lie–Hamilton systems on the plane admitting a VG Lie algebra locally diffeomorphic to any of the VG Lie algebras of the classes I and I.
• VG Lie algebras isomorphic to (classes I and I):
Any VG Lie algebra V of the classes I or I can be integrated to give rise to a Lie group action of a Lie group on whose fundamental vector fields are given by V. The Lie algebra of the isotropy group of 0 is given by an -dimensional Lie subalgebra of , where we can assume without loss of generality that does not vanish at 0 and therefore . Thus, is isomorphic to the Abelian Lie algebra with a Lie subgroup H. Then, is diffeomorphic to the quotient space with the Abelian Lie group . We write for the Lie algebra of the left-invariant vector fields of the isotropy group of 0, which is isomorphic to .
We define the automorphic Lie system on
of the form
where
are assumed to close the opposite structure constants than the basis
of the VG Lie algebra on
under inspection. Let us also define
, which is not necessarily a Poisson bivector as (See Reference [
21] for our convention on the Schouten-Nijenhuis bracket).
and
, which may different for the VG Lie algebras of the classes I
and I
.
Let us try to project and J onto the two dimensional quotient space . Since is invariant relative to left-invariant vector fields, it will be projectable onto via the canonical projection . Meanwhile, for every , one gets that . Since for a certain constant c and an element , one obtains that , which projects onto zero in . By Lemma 1, the J is projectable onto . Moreover, is such that and is a Poisson bivector. Hence, one obtains that the projection of onto is a Lie–Hamilton system relative to that is locally diffeomorphic to a Lie–Hamilton system on the plane with a VG Lie algebra locally diffeomorphic to V.
It is now very simple to prove that the above generalisation allows us to recover the Lie–Hamilton systems on the plane related to the VG Lie algebras P, P, I, and I.
It seems to us that the previous procedure could be extended to study Lie–Hamilton systems on higher-dimensional manifolds. The idea is to construct from a Lie–Hamilton system X on a manifold N with a locally transitive VG Lie algebra V of Hamiltonian vector fields an automorphic Lie system on a certain Lie group G with a VG Lie algebra of right-invariant vector fields isomorphic to V. Then, a left-invariant bivector field must become projectable onto the quotient space with an isotropy group, which can be tested by using the algebraic properties of and/or using as previously.