Jacobi-Lie Hamiltonian systems on real low-dimensional Jacobi-Lie groups and their Lie symmetries

TL;DR

This paper investigates Jacobi-Lie Hamiltonian systems on low-dimensional Jacobi-Lie groups, exploring their Lie symmetries and providing concrete examples on two- and three-dimensional groups.

Contribution

It introduces the study of Jacobi-Lie Hamiltonian systems on low-dimensional groups and identifies their Lie symmetries with explicit examples.

Findings

Examples of Jacobi-Lie Hamiltonian systems on 2D and 3D groups.

Identification of Lie symmetries for these systems.

Analysis of Hamiltonian vector fields related to Jacobi structures.

Abstract

We study Jacobi-Lie Hamiltonian systems admitting Vessiot-Guldberg Lie algebras of Hamiltonian vector fields related to Jacobi structures on real low-dimensional Jacobi-Lie groups. Also, we find some examples of Jacobi-Lie Hamiltonian systems on real two- and three- dimensional Jacobi-Lie groups. Finally, we present Lie symmetries of Jacobi-Lie Hamiltonian systems on some three-dimensional real Jacobi-Lie groups.

Equations

$$X:\mathbb{R}\times M⟶TM,(t,x)\mapsto X(t,x),$$

$$\tilde{X}:\mathbb{R}\times M⟶T(\mathbb{R}\times M)\simeq T\mathbb{R}\oplus TM,(t,x)\mapsto \frac{\partial }{\partial t}+X(t,x),$$

$$x(t)=\Psi (x_{(1)}(t),...,x_{(n)}(t);k),$$

$$X(t,x)=\sum _{i=1}^r{b}_i(t)X_i(x)$$

$$[\Lambda ,\Lambda ]=2E\wedge \Lambda ,L_E\Lambda =[E,\Lambda ]=0,$$

$$\{f,g{\}}_{\Lambda ,E}=\Lambda (df,dg)+fEg-gEf\forall f,g\in C^{\infty }(M).$$

$$d_{\ast X_0}[X,Y{]}^{\mathfrak{g}}=[X,d_{\ast X_0}Y{]}_{{\varphi }_0}^{\mathfrak{g}}-[Y,d_{\ast X_0}X{]}_{{\varphi }_0}^{\mathfrak{g}},$$

$${\varphi }_0(X_0)=0,$$

$$i_{{\varphi }_0}(d_{\ast }X)+[X_0,X{]}^{\mathfrak{g}}=0,$$

$$i_{{\varphi }_0}:{\wedge }^k\mathfrak{g}⟶{\wedge }^{k-1}\mathfrak{g},P\mapsto i_{{\varphi }_0}P;$$

$${\Lambda }^{\#}(d\sigma )={\tilde{X}}_0-e^{-\sigma }{\bar{X}}_0,$$

$$d_{\ast X_0}X=a{d}_{({\varphi }_0,1)}(X)(r)\text{for all}X\in \mathfrak{g}$$

$$\Lambda =\tilde{r}-e^{-\sigma }\bar{r},E=-{\tilde{X}}_0,$$

$$\Lambda =\frac{1}{2}{r}^{ij}(X_i^{R\mu }{X}_j^{R\nu }-e^{-\sigma }{X}_i^{L\mu }{X}_j^{L\nu }){\partial }_{\mu }\wedge {\partial }_{\nu },$$

$$E=-{\alpha }^i{X}_i^{R\mu }{\partial }_{\mu },$$

$$X_f=[\Lambda ,f]+fE={\Lambda }^{\#}(df)+fE,$$

$$X_{f}g=\{f,g{\}}_{\Lambda ,E}\forall g\in C^{\infty }(M).$$

$${\Lambda }_{{\mathbb{A}}_2}=(1-e^{-(b+1)y}){\partial }_x\wedge {\partial }_y,E_{{\mathbb{A}}_2}=b{\partial }_x.$$

$$[{\Lambda }_{{\mathbb{A}}_2},{\Lambda }_{{\mathbb{A}}_2}]=0=2E_{{\mathbb{A}}_2}\wedge {\Lambda }_{{\mathbb{A}}_2},[E_{{\mathbb{A}}_2},{\Lambda }_{{\mathbb{A}}_2}]=0,$$

$$X_1^H=b{e}^{\frac{-x}{b}}{\partial }_x+\frac{(-1+e^{-(1+b)y})e^{\frac{-x}{b}}}{b}{\partial }_y,X_2^H=b{\partial }_x,$$

$$\frac{d{\alpha }_2}{dt}=\sum _{i=1}^2{b}_i(t)X_i^H({\alpha }_2),\forall {\alpha }_2\in {\mathbb{A}}_2,$$

$${\Lambda }_{\text{@slowromancap}iii@}=\frac{1}{2}(1-e^{b(y-z)}){\partial }_x\wedge {\partial }_y-\frac{1}{2}(1-e^{b(y-z)}){\partial }_x\wedge {\partial }_z+(y+z)e^{b(y-z)}{\partial }_y\wedge {\partial }_z,E_{\text{@slowromancap}iii@}=-b{\partial }_x.$$

$$X_{1}^{H}=-b\,\partial_{x},\quad X_{2}^{H}=\Big{(}-\frac{1}{2}+\frac{1}{2}e^{b(y-z)}-by\Big{)}\partial_{x}+(y+z)e^{b(y-z)}\,\partial_{z},$$

$$X_3^H=Ei(1,-b(y+z))e^{-b(y+z)}b{\partial }_x-{\partial }_y+\left(2b\left(y+z\right)Ei\left(1,-b\left(y+z\right)\right)e^{-b\left(y+z\right)}+1\right){\partial }_z$$

$$\frac{d\gamma }{dt}=\sum _{i=1}^3{b}_i(t)X_i^H(\gamma ),\forall \gamma \in \text{@slowromancap}iii@,$$

$${\Lambda }_{\text{@slowromancap}iii@}=\frac{e^{2x}-1}{4}{\partial }_x\wedge {\partial }_y+\frac{e^{2x}-1}{4}{\partial }_x\wedge {\partial }_z-\frac{1}{2}(y+z+1-e^{2x}){\partial }_y\wedge {\partial }_z,E_{\text{@slowromancap}iii@}=-\frac{1}{2}{\partial }_y+\frac{1}{2}{\partial }_z,$$

$$[{\Lambda }_{\text{@slowromancap}iii@},{\Lambda }_{\text{@slowromancap}iii@}]=\frac{e^{2x}-1}{2}{\partial }_x\wedge {\partial }_y\wedge {\partial }_z=2E_{\text{@slowromancap}iii@}\wedge {\Lambda }_{\text{@slowromancap}iii@},[E_{\text{@slowromancap}iii@},{\Lambda }_{\text{@slowromancap}iii@}]=0.$$

$$\frac{d\gamma }{dt}=\sum _{i=1}^3{b}_i(t)X_i^H(\gamma ),\forall \gamma \in \text{@slowromancap}iii@,$$

$${\Lambda }_{\text{@slowromancap}iv@}=-x{e}^x{\partial }_x\wedge {\partial }_z+e^x(z-y-1+e^x){\partial }_y\wedge {\partial }_z,E_{\text{@slowromancap}iv@}=e^x{\partial }_y+e^x(1-x){\partial }_z.$$

$$[{\Lambda }_{\text{@slowromancap}iv@},{\Lambda }_{\text{@slowromancap}iv@}]=2x{e}^{2x}{\partial }_x\wedge {\partial }_y\wedge {\partial }_z=2E_{\text{@slowromancap}iv@}\wedge {\Lambda }_{\text{@slowromancap}iv@},[E_{\text{@slowromancap}iv@},{\Lambda }_{\text{@slowromancap}iv@}]=0.$$

Full text

keywords: Jacobi-Lie group, Jacobi manifold, Lie system, Jacobi-Lie Hamiltonian system, Lie symmetry.

1 Introduction

A Lie system is a non-independent system of first-order ODEs that possesses a superposition rule. In other words, a Lie system amounts to a non-autonomous vector field that takes values in a finite-dimensional real Lie algebra of vector fields, referred to as Vessiot-Guldberg Lie algebra (VG Lie algebra) of the system, with respect to a geometric structure.

At the end of the 19th century, the study of Lie systems was carried out by Sophus Lie, who pioneered the study of systems of ODEs admitting superposition rules [References]. Then about a century, the study of this problem was silent. After the work of Winternitz [References], many authors have recently investigated this problem [References-References]. Some results have been obtained for Lie systems admitting a VG Lie algebra of Hamiltonian vector fields relative to symplectic and Poisson structures [References].

A particular class of Lie systems on Poisson manifolds, the so-called Lie-Hamilton systems, that admit a VG Lie algebra of Hamiltonian vector fields with respect to a Poisson structure was studied in [References]. Recently Lie systems possessing a VG Lie algebras of Hamiltonian vector fields with respect to Jacobi structures [References, References, References] were referred to as the Jacobi-Lie systems are studied and exactly introduced in [References].

It is well known that the symplectic manifold is a particular case of the Poisson manifold so that the Poisson bracket is not necessarily assumed to be non-degenerate. In order that the Jacobi bracket is not necessarily assumed to be derivation, the Jacobi manifold is a generalization of the Poisson manifold [References, References]. To wrap up the discussion, Lie-Hamiltonian systems are a generalization of Hamiltonian systems and a particular case of the Jacobi-Lie Hamiltonian systems.

In this work, we study Lie systems with VG Lie algebras of Hamiltonian vector fields with respect to Jacobi-Lie groups [References], especially on real two and three-dimensional Jacobi-Lie groups [References, References].

The outline of the paper is as follows: In section 2, we recall several definitions and results in Lie systems and Jacobi structures on Jacobi-Lie groups [References] and Jacobi-Lie Hamiltonian system on a Jacobi manifold [References]. In Section 3 we exemplify results of Sections 2 on real two- and three-dimensional Jacobi-Lie groups [References, References].

Finally, in Section 4 we study Lie symmetry of Jacobi-Lie Hamiltonian system on some three-dimensional real Jacobi-Lie groups.

2 Definitions and Notations on Lie and Jacobi-Lie Hamiltonian system

For simplicity, all functions and geometric structures throughout this paper are assumed to be real, smooth, and globally defined. In order to highlight the main aspects of our results, let us omit minor technical problems. Here, for self-containment of the paper, we review some basic concepts of Lie, Lie-Hamiltonian [References, References] and Jacobi-Lie Hamiltonian systems [References].

2.1 Lie systems and Lie-Hamiltonian systems

Let $\mathfrak{a}$ and $\mathfrak{b}$ be two vector subspaces of Lie algebra $\mathfrak{g},$ and let $[\mathfrak{a},\mathfrak{b}]$ denote the vector space spanned by the Lie brackets between elements of $\mathfrak{a}$ and $\mathfrak{b},$ respectively. We define $Lie(\mathfrak{a},\mathfrak{g},[.,.])$ to be the smallest Lie subalgebra of $(\mathfrak{g},[.,.])$ containing $\mathfrak{a}$ and represent it with $Lie(\mathfrak{a})$.

Definition 2.1

A t-dependent vector field on a manifold $M$ is a map

[TABLE]

satisfying $\tau_{M}\tiny{o}X=\pi_{2},$ where $\pi_{2}$ and $\tau_{M}$ are the projections from $\mathbb{R}\times M$ and $TM$ onto $M,$ respectively.

Using this definiition, we can identify every $t$-dependent vector field with a family $\{X_{t}\}_{t\in\mathbb{R}}$ of vector fields $X_{t}:M\longrightarrow TM,\quad x\mapsto X_{t}(x)=X(t,x),$ and vice versa.

Definition 2.2

The minimal Lie algebra of $X$ on a manifold $M$ is the smallest real Lie algebra, $\mathfrak{g}^{X},$ containing the vector fields $\{X_{t}\}_{t\in\mathbb{R}}.$ In other words, $\mathfrak{g}^{X}=Lie(\{X_{t}\}_{t\in\mathbb{R}})$.

Definition 2.3

An integral curve of $X$ is an integral curve $\alpha:\mathbb{R}\longrightarrow\mathbb{R}\times M,\quad t\mapsto(t,x(t)),$ of the suspension of $X$ [References], i.e. for the vector field

[TABLE]

we have $\dfrac{dx(t)}{dt}=X(t,x).$

Definition 2.4

A Lie system is a system $X$ on a manifold $M$ whose $\mathfrak{g}^{X}$ is finite-dimensional [References].

Definition 2.5

A superposition rule depending on n particular solutions for a system $X$ on a manifold $M$ is a function $\Psi:M^{n}\times M\longrightarrow M,$ $(x_{(1)},...,x_{(n)};k)\longmapsto x,$ such that the general solution $x(t)$ of $X$ can be written as

[TABLE]

where $x_{(1)}(t),...,x_{(n)}(t)$ is any generic collection of particular solutions to $X$ and $k=(k_{1},...,k_{m})$ is a point of M to be related to initial conditions of $X$ [References].

Theorem 2.6

*(The Lie-Scheffers Theorem) *** A system $X$ on $M$ admits a superposition rule if and only if it can be written in the form

[TABLE]

for a set $b_{1}(t),...,b_{r}(t)$ of t-dependent functions and a family of vector fields $X_{1},...,X_{r}$ on M spanning an r-dimensional real Lie algebra: a VG Lie algebra of X. That is to say, a system X on $M$ possesses a superposition rule if and only if it is a Lie system [References,References].

Definition 2.7

A manifold M endowed with bivector structure $P\in\Gamma(\large{\bigwedge}^{2}TM)$ satisfying that $[P,P]=0$ is called a Poisson manifold $(M,P),$ where [., .] is the Schouten-Nijenhius bracket 444$[X_{1}\wedge...\wedge X_{p},Y_{1}\wedge...\wedge Y_{q}]=\sum\limits_{{\rm{i=1}}}^{\rm{p}}\sum\limits_{{\rm{j=1}}}^{\rm{q}}(-1)^{i+j}[X_{i},X_{j}]\wedge X_{1}\wedge...\wedge\hat{X_{i}}\wedge...\wedge X_{p}\wedge Y_{1}\wedge...\wedge\hat{Y_{j}}\wedge...\wedge Y_{q}$ and $\Gamma(\bigwedge^{2}TM)$ is the space of sections of $\bigwedge^{2}TM$ [References].

The bivector $P$ has a bundle morphism $P^{\#}:TM^{*}\longrightarrow TM$ is defined by $\beta(P^{\#}\alpha)=P(\alpha,\beta)\quad\forall\alpha,\beta\in TM^{*}.$

Definition 2.8

A vector field X on M with the bivector $P$ is said to be a Hamiltonian vector field if it can be written as $X={P}^{\#}(df)$ where f is a function on M, called the Hamiltonian; conversely, every function f is called the Hamiltonian function of a unique Hamiltonian vector field $X_{f}$.

Definition 2.9

A Lie-Hamiltonian system on a Poisson manifold $M$ is a Lie system X whose $\mathfrak{g}^{X}$ consists of Hamiltonian vector fields relative to a Poisson bivector $P$ [References].

2.2 Jacobi and Jacobi-Lie structures

The study of the Jacobi manifolds was made by Lichnerowicz and Kirillov [References, References].

Definition 2.10

A Jacobi manifold is a triple $(M,\Lambda,E)$, where $\Lambda$ is a $2$-vector field and $E$ is a vector field on $M$ (called the Reeb vector field) such that

[TABLE]

where the bracket is that of Schouten-Nijenhuis bracket.

The space $(C^{\infty}(M),\{.,.\}_{\Lambda,E})$ is a local Lie algebra in the sense of Kirillov [References] with the Jacobi bracket

[TABLE]

This Lie bracket is a Poisson bracket if and only if the vector field $E$ identically vanishes.

Iglesias and Marrero have proved in [References] that if Lie group $\mathbb{G}$ is a connected simply connected with Lie algebra $\mathfrak{g}$ and the pair$((\mathfrak{g},\phi_{0}),(\mathfrak{g}^{*},X_{0}))$ is a Jacobi-Lie bialgebra then $\mathbb{G}$ is Jacobi-Lie group and has a special Jacobi structure.

Definition 2.11

A Jacobi-Lie bialgebra is a pair $((\mathfrak{g},\phi_{0}),(\mathfrak{g}^{*},X_{0})),$ where $\mathfrak{g}$ is a finite dimensional real Lie algebra with Lie bracket $[,]^{\mathfrak{g}}$ and $\mathfrak{g}^{*}$ is dual space of $\mathfrak{g}$ with Lie bracket $[,]^{\mathfrak{g}^{*}}$; also, $\phi_{0}\in\mathfrak{g}^{*}$ and $X_{0}\in\mathfrak{g}$ are 1-cocycles on $\mathfrak{g}$ and $\mathfrak{g}^{*}$, respectively, such that for all $X,Y\in\mathfrak{g}$ satisfying the following properties:

[TABLE]

[TABLE]

[TABLE]

where

[TABLE]

moreover, $d_{*}$ is the Chevalley-Eilenberg differential of $\mathfrak{g}^{*}$ and $d_{*X_{0}}$ is the $X_{0}$-differential of $\mathfrak{g}^{*}$ as well as the operation $[,]_{\phi_{0}}^{\mathfrak{g}}$ is the $\phi_{0}$-Schouten-Nijenhuis bracket [References].

Theorem 2.12

Let the pair $((\mathfrak{g},\phi_{0}),(\mathfrak{g}^{*},X_{0}))$ be a Jacobi-Lie bialgebra and Lie group $\mathbb{G}$ be a connected simply connected admitting Lie algebra $\mathfrak{g}$. Then, there exists a unique multiplicative function $\sigma:\mathbb{G}\longrightarrow\mathbb{R}$ and a unique $\sigma$-multiplicative bivector $\Lambda$ on $\mathbb{G}$ satisfying $(d\sigma)(e)=\phi_{0}$ and the intrinsic derivative of bivector at $e$ is $-d_{*X_{0}}$, i.e., $d_{e}\Lambda=-d_{*X_{0}}.$ Furthermore, the following identity holds

[TABLE]

where $E=-\tilde{X}_{0}$ and the pair $(\Lambda,E)$ is a Jacobi structure on $\mathbb{G}$; in addition, both $\tilde{X}$ and $\bar{X}$ are the right and left invariant vector field such that $\tilde{X}_{0}(e)=\bar{X}_{0}(e)=X_{0}$ [References].

Definition 2.13

A Coboundary Jacobi-Lie bialgebra is a Jacobi-Lie bialgebra such that $d_{*X_{0}}$ is a 1-coboundary that is, there exists $r\in\wedge^{2}\mathfrak{g}$ satisfying that

[TABLE]

(for more details, see [References]).

A Jacobi structure on $\mathbb{G}$ was determined by Iglesias and Marrero in [References], as follows:

[TABLE]

Where both $\tilde{r}$ and $\bar{r}$ are right and left invariant 2-vector on the Lie group $\mathbb{G}$ as well as $\tilde{X}_{0}$ is right invariant vector field on $\mathbb{G}$. Furthermore, $(d\sigma)(e)=\phi_{0}$.

The relation (12) can be expressed in terms of coordinate on M as follows [References]:

[TABLE]

and

[TABLE]

Where both $X_{i}^{R}$ and $X_{i}^{L}$ are the $i$th coordinates of the right and left invariant vector fields on the Lie group $\mathbb{G}$. Moreover, $r^{ij}$ is calculated from the relation $r=\frac{1}{2}r^{ij}X_{i}\wedge Xj$ , and multiplicative function $\sigma:\mathbb{G}\longrightarrow\mathbb{R}$ is defined as $(d\sigma)(e)=\phi_{0}$ as well as $\alpha^{i}$ is obtained from the relation $X_{0}=\alpha^{i}X_{i}$ where $\{X_{i}\}$ is the basis of the Lie algebra $\mathfrak{g}$. Now, using the results in [References] and relations (13) and (14), one can calculate the vector field $E$ and 2-vector $\Lambda$ related to real two- and three- dimensional Jacobi-Lie bialgebras. The results are listed in Tables 1-3. The first column gives the names of the real two- and three-dimensional Jacobi-Lie bialgebras according to [References], and the second column gives the Jacobi structure on Lie group $\mathbb{G}$ related to Lie algebra $\mathfrak{g}$. Note that in [References] the bi-r-matrix Jacobi-Lie bialgebras as Jacobi-Lie bialgebras being coboundary in two directions, that is, $((\mathfrak{g},\phi_{0}),(\mathfrak{g^{*}},X_{0}))$ and $((\mathfrak{g^{*}},X_{0}),(\mathfrak{g},\phi_{0}))$ having classical r-matrices r and $\tilde{r}$ have been classified. Nevertheless, they classified coboundary Jacobi-Lie bialgebras having r or $\tilde{r}.$

2.3 Jacobi-Lie Hamiltonian systems

Jacobi-Lie Hamiltonian systems were introduced in [References]. In other words, Lie systems possessing a Vessiot-Guldberg Lie algebra of Hamiltonian functions with respect to a Jacobi structure.

One of the most useful constructions on Jacobi manifolds is in a sense analogue of the gradient, defined as follows.

Definition 2.14

A vector field X on the manifold M with the Jacobi structure is said to be a Hamiltonian if there exists a function f, referred to as Hamiltonian, such that

[TABLE]

where the bracket is that of Schouten-Nijenhuis bracket[References].

If $Ef=0$ (that is, the derivative of the function $f$ in the direction of the vector field $E$ is equal to zero), then the function $f$ is called a good Hamiltonian function and $X_{f}$ a good Hamiltonian vector field [References].

Let $f$ be a smooth function on the Jacobi manifold $M$. There exists a unique vector field $X_{f}$ on $M$, referred to as Hamiltonian vector field associated with $f,$ such that the following equality is satisfied

[TABLE]

However, a vector field $X_{f}$ can possess several Hamiltonian functions. Obviously, $(Ham(M,\Lambda,E),[.,.])$ is a Lie algebra, where the bracket is that of vector fields bracket and $Ham(M,\Lambda,E)$ is the space of Hamiltonian vector fields of Jacobi manifold.

Definition 2.15

A Jacobi-Lie system is a quadruple $(M,\Lambda,E,X)$ where $(M,\Lambda,E)$ is a Jacobi manifold and a Lie system $X$ such that $\mathfrak{g}^{X}\subset Ham(M,\Lambda,E)$ [References].

Definition 2.16

A Jacobi-Lie Hamiltonian system is a quadruple $(M,\Lambda,E,f)$ where $(M,\Lambda,E)$ is a Jacobi manifold and $f:\mathbb{R}\times M\longrightarrow M,\quad(t,x)\mapsto f_{t}(x)$ is a t-dependent function and $Lie(\{f_{t}\}_{t\in\mathbb{R}},\{.,.\}_{\Lambda,E})$ is finite-dimensional. The vector field X on M is said to be Jacobi-Lie Hamiltonian system $(M,\Lambda,E,f)$ if $X_{f_{t}}$ is a Hamiltonian vector field with Hamiltonian function $f_{t}$ (relative to Jacobi manifold) $\forall t\in\mathbb{R}$ [References].

Theorem 2.17

If $(M,\Lambda,E,f)$ is a Jacobi-Lie Hamiltonian system, then the system X of the form $X_{t}=X_{f_{t}},\forall t\in\mathbb{R},$ is a Jacobi-Lie system $(M,\Lambda,E,X).$ If X is a Lie system whose $\{X_{t}\}_{t\in\mathbb{R}}$ are good Hamiltonian vector fields, then X possesses a Jacobi-Lie Hamiltonian [References].

3 Jacobi-Lie Hamiltonian systems on

real low-dimensional Jacobi-Lie groups

Now, we consider some Jacobi structures obtained by using the real two and three dimensional Jacobi-Lie groups related to Jacobi-Lie bialgebras (see table 1, 3 below). In these examples, we consider the Lie group $\mathbb{G}$ related to the Jacobi-Lie bialgebra $((\mathfrak{g},\phi_{0}),(\mathfrak{g^{*}},X_{0}))$. For this propose we use the formalisms mentioned in the previous section for calculation of Jacobi-Lie Hamiltonian systems on real low dimensional Jacobi-Lie groups.

Example 1. Real two-dimensional bi-r-matrix Jacobi-Lie bialgebra $((A_{2},b\tilde{X}^{2}),(A_{2}.i,-bX_{1}))$[References]:

Consider the Lie group $\mathbb{A}_{2}$ (with the coordinates $x,y$) related to Lie algebra $A_{2}$. For this example, the Jacobi structure and Reeb vector have the following forms (see table 1):

[TABLE]

Simple calculations show that they satisfying in

[TABLE]

where [.,.] is the Schouten-Nijenhuis bracket. Thus, $(\mathbb{A}_{2},\Lambda_{\mathbb{A}_{2}},E_{\mathbb{A}_{2}})$ is a Jacobi manifold.

It is easy to check that

[TABLE]

span Lie algebra $A_{2}\,(i.e.,[X_{1}^{H},X_{2}^{H}]=X_{1}^{H})$ of Hamiltonian vector fields on $\mathbb{A}_{2}.$

Consider now the system on $\mathbb{A}_{2}$ defined by

[TABLE]

for arbitrary t-dependent functions $b_{i}(t).$

$X^{\mathbb{A}_{2}}$ is a Lie system since the associated t-dependent vector field $X^{\mathbb{A}_{2}}=\sum_{i=1}^{2}b_{i}(t)X_{i}^{H}$ takes values in the Lie algebra $[X_{1}^{H},X_{2}^{H}]=X_{1}^{H}$,namely, Lie algebra $A_{2}$.

We now illustrate that the Lie system (18) is a Jacobi-Lie system.

Indeed, $X_{1}^{H}$ and $X_{2}^{H}$ are Hamiltonian vector fields with respect to $(\mathbb{A}_{2},\Lambda_{\mathbb{A}_{2}},E_{\mathbb{A}_{2}})$ with Hamiltonian functions $f_{1}=e^{\frac{-x}{b}}$ and $f_{2}=1$, respectively ( i.e., $X_{i}^{H}={\Lambda}^{\#}(df_{i})+f_{i}E)$; consequently, $(\mathbb{A}_{2},\Lambda_{\mathbb{A}_{2}},E_{\mathbb{A}_{2}},X^{\mathbb{A}_{2}})$ is a Jacobi-Lie system.

Since $f=\sum_{i=1}^{2}b_{i}(t)f_{i}=b_{1}(t)e^{\frac{-x}{b}}+b_{2}(t)$ is a Hamiltonian function of $X^{\mathbb{A}_{2}}$ for every $t\in\mathbb{R}$ and the functions $f_{1}$ and $f_{2}$ satisfy the commutation relation $\{f_{1},f_{2}\}_{\Lambda_{\mathbb{A}_{2},E_{\mathbb{A}_{2}}}}=f_{1},$ then the functions $\{f_{t}\}_{t\in\mathbb{R}}$ span a finite-dimensional real Lie algebra of functions with respect to the Lie bracket induced by (17). Consequently, $X^{\mathbb{A}_{2}}$ admits a Jacobi-Lie Hamiltonian system $(\mathbb{A}_{2},\Lambda_{\mathbb{A}_{2}},E_{\mathbb{A}_{2}},f).$

Example 2. Real three-dimensional bi-r-matrix Jacobi-Lie bialgebra $((\@slowromancap iii@,-b\tilde{X}^{2}+b\tilde{X}^{3}),(\@slowromancap iii@.iv,bX_{1}))$ [References]:

Consider the Lie group $\mathbb{\@slowromancap iii@}$ (with the coordinates $x,y,z$) related to Lie algebra $\@slowromancap iii@$. For this example, the Jacobi structure and Reeb vector have the following forms (see table 2):

[TABLE]

Obviously, $[\Lambda_{\mathbb{\@slowromancap iii@}},\;\Lambda_{\mathbb{\@slowromancap iii@}}]=-2b(y+z)e^{b(y-z)}\partial_{x}\wedge\partial_{y}\wedge\partial_{z}=2E_{\mathbb{\@slowromancap iii@}}\wedge\Lambda_{\mathbb{\@slowromancap iii@}}$  and  $[E_{\mathbb{\@slowromancap iii@}},\;\Lambda_{\mathbb{\@slowromancap iii@}}]=0.$

As a result, $(\mathbb{\@slowromancap iii@},\Lambda_{\mathbb{\@slowromancap iii@}},E_{\mathbb{\@slowromancap iii@}})$ is a Jacobi manifold. It is straightforward to verify that the Lie algebra, $\@slowromancap ii@,$ of Hamiltonian vector fields on $\mathbb{\@slowromancap iii@}$ is spanned by

[TABLE]

[TABLE]

where ${\it Ei}\left(1,-b\left(y+z\right)\right)=\int_{-b\left(y+z\right)}^{\infty}\!{\frac{{{\rm e}^{-x}}}{x}}{dx}$

The system on $\mathbb{\@slowromancap iii@}$ can be defined as

[TABLE]

for arbitrary t-dependent functions $b_{i}(t).$

So that the associated t-dependent vector field $X^{\mathbb{\@slowromancap iii@}}=\sum_{i=1}^{3}b_{i}(t)X_{i}^{H}$ takes values in the Lie algebra $\@slowromancap ii@$ ,that is, $[X_{2}^{H},X_{3}^{H}]=X_{1}^{H}$ ; so $X^{\mathbb{\@slowromancap iii@}}$ is a Lie system.

We now manifest that $(\mathbb{\@slowromancap iii@},\Lambda_{\mathbb{\@slowromancap iii@}},E_{\mathbb{\@slowromancap iii@}},X^{\mathbb{\@slowromancap iii@}})$ is a Jacobi-Lie system. Infact, $X_{1}^{H}$ and $X_{2}^{H}$,and $X_{3}^{H}$ are Hamiltonian vector fields relative to $(\mathbb{\@slowromancap iii@},\;\Lambda_{\mathbb{\@slowromancap iii@}},\;E_{\mathbb{\@slowromancap iii@}})$ with good Hamiltonian functions $f_{1}=1$ and $f_{2}=y$, and $f_{3}=-e^{-2by}Ei(1,\;-b(y+z)$ respectively ( i.e., $X_{i}^{H}={\Lambda}^{\#}(df_{i})+f_{i}E)$; subsequently, $(\mathbb{\@slowromancap iii@},\;\Lambda_{\mathbb{\@slowromancap iii@}},\;E_{\mathbb{\@slowromancap iii@}},X^{\mathbb{\@slowromancap iii@}})$ is a Jacobi-Lie system. Because $f=\sum_{i=1}^{2}b_{i}(t)f_{i}=b_{1}(t)+b_{2}(t)y-b_{3}(t)e^{-2by}Ei(1,-b(y+z)$ is a Hamiltonian function of $X^{\mathbb{\@slowromancap iii@}}$ for every $t\in\mathbb{R}$, and the functions $f_{1},f_{2}$ and $f_{3}$ satisfy the commutation relations $\{f_{2},f_{3}\}_{\Lambda_{\mathbb{\@slowromancap iii@},\,E_{\mathbb{\@slowromancap iii@}}}}=f_{1}$, then the functions $\{f_{t}\}_{t\in\mathbb{R}}$ span a finite-dimensional real Lie algebra of functions with respect to the Lie bracket induced by (19). Consequently, $X^{\mathbb{\@slowromancap iii@}}$ admits a Jacobi-Lie Hamiltonian system $(\mathbb{\@slowromancap iii@},\Lambda_{\mathbb{\@slowromancap iii@}},E_{\mathbb{\@slowromancap iii@}},f).$

Example 3. Real three-dimensional bi-r-matrix Jacobi-Lie bialgebra $((\@slowromancap iii@,0),(\@slowromancap iii@.v,\frac{1}{2}X_{2}-\frac{1}{2}X_{3}))$ [References]:

Consider the Lie group $\mathbb{\@slowromancap iii@}$ (with the coordinates $x,y,z$) related to Lie algebra $\@slowromancap iii@$. For this example, the Jacobi structure and Reeb vector have the following forms (see table 2):

[TABLE]

one can show that

[TABLE]

So $(\mathbb{\@slowromancap iii@},\Lambda_{\mathbb{\@slowromancap iii@}},E_{\mathbb{\@slowromancap iii@}})$ is a Jacobi manifold. A simple calculation shows that

X_{1}^{H}=-\frac{1}{2}\,\left(2\,x-3\,y-z-2\right)\left(-1+{e}^{2\,x}\right)\partial_{x}+\Big{(}\frac{3}{2}\,{e}^{2\,x}y+\frac{1}{2}\,{e}^{2\,x}z+{e}^{2\,x}+\frac{y}{2}-\frac{z}{2}-1-{e}^{2\,x}x-x\Big{)}\partial_{y}+\Big{(}-\frac{3}{2}\,{e}^{2\,x}y-\frac{1}{2}\,{e}^{2\,x}z-{e}^{2\,x}+\frac{3}{2}\,y+\frac{5}{2}\,z+1+{e}^{2\,x}x+{y}^{2}+2\,yz+{z}^{2}+x\Big{)}\partial_{z},\,X_{2}^{H}=\Big{(}\frac{1}{2}-\frac{1}{2}e^{2x}\Big{)}\partial_{x}-\frac{1}{2}e^{2x}\,\partial_{y}+\frac{1}{2}\,e^{2x}\,\partial_{z} and $X_{3}^{H}=-\frac{1}{2}\partial_{y}+\frac{1}{2}\partial_{z}$ span the Lie algebra $\@slowromancap iii@$ of Hamiltonian vector fields on $\mathbb{\@slowromancap iii@}.$

The system on $\mathbb{\@slowromancap iii@}$ can be written as

[TABLE]

for arbitrary t-dependent functions $b_{i}(t).$

Since the associated t-dependent vector field $X^{\mathbb{\@slowromancap iii@}}=\sum_{i=1}^{3}b_{i}(t)X_{i}^{H}$ takes values in the Lie algebra $\@slowromancap iii@$ ,that is, $[X_{1}^{H},X_{2}^{H}]=-(X_{2}^{H}+X_{3}^{H}),[X_{1}^{H},X_{3}^{H}]=-(X_{2}^{H}+X_{3}^{H})$ , then $X^{\mathbb{\@slowromancap iii@}}$ is a Lie system.

We now prove that $(\mathbb{\@slowromancap iii@},\Lambda_{\mathbb{\@slowromancap iii@}},E_{\mathbb{\@slowromancap iii@}},X^{\mathbb{\@slowromancap iii@}})$ is a Jacobi-Lie system. As a matter of fact, $X_{1}^{H}$ and $X_{2}^{H}$,and $X_{3}^{H}$ are Hamiltonian vector fields relative to $(\mathbb{\@slowromancap iii@},\Lambda_{\mathbb{\@slowromancap iii@}},E_{\mathbb{\@slowromancap iii@}})$ with Hamiltonian functions $f_{1}=2\,\left(y+z+2\right)\left(-y+x\right)$ and $f_{2}=1+y+z$, and $f_{3}=1$ respectively ( i.e., $X_{i}^{H}={\Lambda}^{\#}(df_{i})+f_{i}E);$ subsequently, $(\mathbb{\@slowromancap iii@},\Lambda_{\mathbb{\@slowromancap iii@}},E_{\mathbb{\@slowromancap iii@}},,X^{\mathbb{\@slowromancap iii@}})$ is a Jacobi-Lie system.

Using the Lie bracket induced by $\Lambda_{\mathbb{\@slowromancap iii@}}$ and $E_{\mathbb{\@slowromancap iii@}}$ of Lie group $\mathbb{\@slowromancap iii@}$, we can write $\{f_{1},f_{2}\}_{\Lambda_{\mathbb{\@slowromancap iii@},\,E_{\mathbb{\@slowromancap iii@}}}}=-f_{2}-f_{3},\quad\{f_{1},f_{3}\}_{\Lambda_{\mathbb{\@slowromancap iii@},\,E_{\mathbb{\@slowromancap iii@}}}}=-f_{2}-f_{3}.$ Therefore, $(\mathbb{\@slowromancap iii@},\Lambda_{\mathbb{\@slowromancap iii@}},E_{\mathbb{\@slowromancap iii@}},f=\sum_{i=1}^{3}b_{i}(t)f_{i})$ for $X^{\mathbb{\@slowromancap iii@}}$ is a Jacobi-Lie Hamiltonian system, where $\Lambda_{\mathbb{\@slowromancap iii@}}$ and $E_{\mathbb{\@slowromancap iii@}}$ are those appearing in (21).

Example 4. Real three-dimensional bi-r-matrix Jacobi-Lie bialgebra $((\@slowromancap iv@,-\tilde{X}^{1}),(\@slowromancap iii@.vi,-X_{2}-X_{3}))$ [References]:

Consider the Lie group $\mathbb{\@slowromancap iv@}$ (with the coordinates $x,y,z$) related to Lie algebra $\@slowromancap iv@$. For this example, the Jacobi structure and Reeb vector have the following forms (see table 2):

[TABLE]

Then one can show that

[TABLE]

so, $(\mathbb{\@slowromancap iv@},\Lambda_{\mathbb{\@slowromancap iv@}},E_{\mathbb{\@slowromancap iv@}})$ is a Jacobi manifold. A short calculation shows that

[TABLE]

and

[TABLE]

span the Lie algebra $,\@slowromancap iv@,$ of Hamiltonian vector fields on $\mathbb{\@slowromancap iv@}.$ The system on $\mathbb{\@slowromancap iv@}$ can be considered as

[TABLE]

for arbitrary t-dependent functions $b_{i}(t).$ Since the associated t-dependent vector field $X^{\mathbb{\@slowromancap iv@}}=\sum_{i=1}^{3}b_{i}(t)X_{i}^{H}$ takes values in the Lie algebra $\@slowromancap iv@$ ,that is, $[X_{1}^{H},X_{2}^{H}]=-(X_{2}^{H}-X_{3}^{H}),[X_{1}^{H},X_{3}^{H}]=-X_{3}^{H}$ ; then, $X^{\mathbb{\@slowromancap iv@}}$ is a Lie system. We now exhibits that $(\mathbb{\@slowromancap iv@},\Lambda_{\mathbb{\@slowromancap iv@}},E_{\mathbb{\@slowromancap iv@}},X^{\mathbb{\@slowromancap iv@}})$ is a Jacobi-Lie system. Actually, $X_{1}^{H}$ and $X_{2}^{H}$,and $X_{3}^{H}$ are Hamiltonian vector fields relative to $(\mathbb{\@slowromancap iv@},\Lambda_{\mathbb{\@slowromancap iv@}},E_{\mathbb{\@slowromancap iv@}})$ with Hamiltonian functions $f_{1}={\frac{\left(2\,y-z\right){e}^{-x}}{x}}$ and $f_{2}=1$, and $f_{3}=-\frac{1}{x}$ respectively ( i.e., $X_{i}^{H}={\Lambda}^{\#}(df_{i})+f_{i}E)$; subsequently $(\mathbb{\@slowromancap iv@},\Lambda_{\mathbb{\@slowromancap iv@}},E_{\mathbb{\@slowromancap iv@}},X^{\mathbb{\@slowromancap iv@}})$ is a Jacobi-Lie system.

Using the Lie bracket induced by $\Lambda_{\mathbb{\@slowromancap iv@}}$ and $E_{\mathbb{\@slowromancap iv@}}$ of Lie group $\mathbb{\@slowromancap iv@}$, the functions $f_{1},f_{2}$ and $f_{3}$ satisfy the commutation relations $\{f_{1},f_{2}\}_{\Lambda_{\mathbb{\@slowromancap iv@},\,E_{\mathbb{\@slowromancap iv@}}}}=-f_{2}+f_{3},\quad\{f_{1},f_{3}\}_{\Lambda_{\mathbb{\@slowromancap iv@},\,E_{\mathbb{\@slowromancap iv@}}}}=-f_{3}.$ Therefore, $(\mathbb{\@slowromancap iv@},\Lambda_{\mathbb{\@slowromancap iv@}},E_{\mathbb{\@slowromancap iii@}},f=\sum_{i=1}^{3}b_{i}(t))$ for $X^{\mathbb{\@slowromancap iv@}}$ is a Jacobi-Lie Hamiltonian system, where $\Lambda_{\mathbb{\@slowromancap iv@}}$ and $E_{\mathbb{\@slowromancap iv@}}$ are those appearing in (23).

4 Lie symmetry for Jacobi-Lie Hamiltonian systems

We now present some examples of Jacobi-Lie Hamiltonian systems on three-dimensional real Jacobi-Lie groups whose distribution associated with their systems is of dimension two. Then, there exists a constant of motion for their systems. We obtain a time-independent Lie symmetry [References] for their systems to illustrate our procedure.

Let $X$ be a t-dependent vector field on M, the associated distribution of $X$ is the generalised distribution $\Delta^{X}$ on M spanned by the vector fields of $\mathfrak{g}^{X}.$ In other words,

[TABLE]

and the associated co-distribution of $X$ is the generalised co-distribution ${(\Delta^{X}})^{\perp}$ on M of the form

[TABLE]

where ${(\Delta_{x}^{X}})^{\perp}$ is the annihilator of $\Delta_{x}^{X}$.

The function $\rho^{X}:M\longrightarrow\mathbb{N}\cup\{0\},\quad x\mapsto dim\,\Delta_{x}^{X}$ is a lower semicontinuous at $x$ since it cannot decrease in a neighbourhood of $x$. In addition, $\rho^{X}(x)$ is constant on the connected components of dense and an open subset $U^{X}$ of $M$ ( cf. [References, p. 19] ), where $\Delta^{X}$ becomes a regular involutive distribution. Also, $(\Delta_{x}^{X})^{\perp}$ becomes a regular co-distribution on each connected component since $dim(\Delta_{x}^{X})^{\perp}=dimM-\rho^{X}(x)$.

Proposition 4.1

A function $h:U^{X}\longrightarrow\mathbb{R}$ is a local t-independent constant of motion for a t-dependent vector field X if and only if $dh\in(\Delta_{x}^{X})^{\perp}|_{U^{X}}$  [References].

Definition 4.2

Let X be a Jacobi-Lie system with a Jacobi-Lie Hamiltonian structure $(M,\Lambda,E,f)$. Then, one can define its symmetry distribution as follows: $(S^{X}_{\Lambda,E})_{x}={\Lambda}^{\#}(dh_{i})+h_{i}E\in T_{x}M,$ where $dh_{i}\in(\Delta_{x}^{X})^{\perp}|_{U}.$

Now using the symmetry distribution, we study the t-independent Lie symmetries of Jacobi-Lie Hamiltonian systems on real low dimensional Jacobi-Lie groups.

The following proposition is the same as the proposition 1 in [References], except that Hamiltonian functions are not necessarily a good.

Proposition 4.3

Let X be a Jacobi-Lie system possessing a Jacobi-Lie Hamiltonian structure $(M,\Lambda,E,f)$. The smooth function h on the Jacobi manifold M is a constant of motion for X if and only if it commutes with all elements of $Lie(\{f_{t}\}_{t\in\mathbb{R}},\{.,.\}_{\Lambda,E})$ relative to $\{.,.\}_{\Lambda,E}$.

Proof. Let $h$ be a t-independent constant of motion for X, i.e.

[TABLE]

Using (25) and the Jacobi identity, we get

$\{h,\{f_{t},f_{t^{\prime}}\}_{\Lambda,E}\}_{\Lambda,E}=\{f_{t},\{h,f_{t^{\prime}}\}_{\Lambda,E}\}_{\Lambda,E}+\{\{h,f_{t}\}_{\Lambda,E},f_{t^{\prime}}\}_{\Lambda,E}=0\quad\forall t,t^{\prime}\in\mathbb{R}.$

By the inductive procedure, $h$ commutes with the whole successive Jacobi brackets of elements of $\{f_{t}\}_{t\in\mathbb{R}}$ relative to $\{.,.\}_{\Lambda,E}$ and their linear combinations. Since $\{f_{t}\}_{t\in\mathbb{R}}$ span $Lie(\{f_{t}\}_{t\in\mathbb{R}},\{.,.\}_{\Lambda,E})$, we obtain that h commutes with $Lie(\{f_{t}\}_{t\in\mathbb{R}},\{.,.\}_{\Lambda,E})$ relative to $\{.,.\}_{\Lambda,E}$.

Let us prove the converse. If $h$ commutes with $Lie(\{f_{t}\}_{t\in\mathbb{R}},\{.,.\}_{\Lambda,E})$ relative to $\{.,.\}_{\Lambda,E}$, it commutes with the elements $\{f_{t}\}_{t\in\mathbb{R}}$ relative to $\{.,.\}_{\Lambda,E}$. By applying (25), the function h is a constant of motion of X.

Lemma 4.4

The mappinq

[TABLE]

is a homomorphism Lie algebras, i.e. $\varphi{\{f,g\}_{\Lambda,E}}=[X_{f},X_{g}]$.

Proposition 4.5

Let X be a Jacobi-Lie system admitting a Jacobi-Lie Hamiltonian structure $(M,\Lambda,E,f)$. If h is a t-independent constant of motion for X, then $X_{h}={\Lambda}^{\#}(dh)+hE$ is a t-independent Lie symmetry of X.

Proof. In view of Lemma 4.4, we have

$[X_{h},X_{f_{t}}]=[{\Lambda}^{\#}(dh)+hE,{\Lambda}^{\#}(df_{t})+f_{t}E]=\varphi\{h,f_{t}\}_{\Lambda,E}=-\varphi\{f_{t},h\}_{\Lambda,E}=-\varphi(X_{t}h)=0,\forall t\in\mathbb{R}.$

Example 1. Real three-dimensional bi-r-matrix Jacobi-Lie bialgebra $((\@slowromancap iii@,-b\tilde{X}^{2}+b\tilde{X}^{3}),(\@slowromancap iii@.iv,bX_{1}))$ [References]:

Let us Consider the Lie group $\mathbb{\@slowromancap iii@}$ related to Lie algebra $\@slowromancap iii@$ and the Jacobi structure given by (19). A simple calculation shows that

[TABLE]

and

[TABLE]

span the Lie algebra $,A_{2},$ of Hamiltonian vector fields on $\mathbb{\@slowromancap iii@}.$

The system on $\mathbb{\@slowromancap iii@}$ can be written as

[TABLE]

for arbitrary t-dependent functions $b_{i}(t).$

Since the associated t-dependent vector field $X^{\mathbb{\@slowromancap iii@}}=\sum_{i=1}^{2}b_{i}(t)X_{i}^{H}$ takes values in the Lie algebra $A_{2}$ ,that is, $[X_{1}^{H},X_{2}^{H}]=X_{1}^{H}$ , then $X^{\mathbb{\@slowromancap iii@}}$ is a Lie system.

We now prove that $(\mathbb{\@slowromancap iii@},\Lambda_{\mathbb{\@slowromancap iii@}},E_{\mathbb{\@slowromancap iii@}},X^{\mathbb{\@slowromancap iii@}})$ is a Jacobi-Lie system. As a matter of fact, $X_{1}^{H}$ and $X_{2}^{H}$ are Hamiltonian vector fields relative to $(\mathbb{\@slowromancap iii@},\Lambda_{\mathbb{\@slowromancap iii@}},E_{\mathbb{\@slowromancap iii@}})$ with Hamiltonian functions $f_{1}=x$ and $f_{2}=-{\frac{x\ln\left(x\right)}{b}}$ respectively. Subsequently, $(\mathbb{\@slowromancap iii@},\Lambda_{\mathbb{\@slowromancap iii@}},E_{\mathbb{\@slowromancap iii@}},,X^{\mathbb{\@slowromancap iii@}})$ is a Jacobi-Lie system.

Using the Lie bracket induced by $\Lambda_{\mathbb{\@slowromancap iii@}}$ and $E_{\mathbb{\@slowromancap iii@}}$ of Lie group $\mathbb{\@slowromancap iii@}$, we can write $\{f_{1},f_{2}\}_{\Lambda_{\mathbb{\@slowromancap iii@},E_{\mathbb{\@slowromancap iii@}}}}=f_{1};$ therefore, $(\mathbb{\@slowromancap iii@},\Lambda_{\mathbb{\@slowromancap iii@}},E_{\mathbb{\@slowromancap iii@}},f=\sum_{i=1}^{2}b_{i}(t)f_{i})$ for $X^{\mathbb{\@slowromancap iii@}}$ is a Jacobi-Lie Hamiltonian system.

It is easy check that $h=1-{e}^{-b\left(y-z\right)}$ is a t-independent constant of motion. One can show that

[TABLE]

Then, the function $h$ always Jacobi commutes with the whole Lie algebra $Lie(\{f_{t}\}_{t\in\mathbb{R}},\{.,.\}_{\Lambda_{\mathbb{\@slowromancap iii@}},E_{\mathbb{\@slowromancap iii@}}}),$ as expected. By applying proposition 4.5, $X_{h}={\Lambda}^{\#}(dh)+hE$ must be a Lie symmetry for this system. A short calculation shows that $X_{h}=\left(y+z\right)b\partial_{y}+\left(y+z\right)b\partial_{z}.$ It is easy to check that $X_{h}$ commutes with $X_{1}^{H},X_{2}^{H},$ and thus commutes with every $X_{f_{t}},$ with $t\in\mathbb{R},$ i.e. $X_{h}$ is a Lie symmetry for $X^{\mathbb{\@slowromancap iii@}}$.

Example 2. Real three-dimensional bi-r-matrix Jacobi-Lie bialgebra $((\@slowromancap iii@,0),(\@slowromancap iii@.v,\frac{1}{2}X_{2}-\frac{1}{2}X_{3}))$ [References]:

Let us Consider the Lie group $\mathbb{\@slowromancap iii@}$ related to Lie algebra $\@slowromancap iii@$ and the Jacobi structure given by (21). A simple calculation shows that

[TABLE]

and

[TABLE]

span the Lie algebra $,A_{2},$ of Hamiltonian vector fields on $\mathbb{\@slowromancap iii@}.$

The system on $\mathbb{\@slowromancap iii@}$ can be written as

[TABLE]

for arbitrary t-dependent functions $b_{i}(t).$

Since the associated t-dependent vector field $X^{\mathbb{\@slowromancap iii@}}=\sum_{i=1}^{2}b_{i}(t)X_{i}^{H}$ takes values in the Lie algebra $A_{2},$ that is, $[X_{1}^{H},X_{2}^{H}]=X_{1}^{H}$ , then $X^{\mathbb{\@slowromancap iii@}}$ is a Lie system.

We now prove that $(\mathbb{\@slowromancap iii@},\Lambda_{\mathbb{\@slowromancap iii@}},E_{\mathbb{\@slowromancap iii@}},X^{\mathbb{\@slowromancap iii@}})$ is a Jacobi-Lie system. As a matter of fact, $X_{1}^{H}$ and $X_{2}^{H}$ are Hamiltonian vector fields relative to $(\mathbb{\@slowromancap iii@},\Lambda_{\mathbb{\@slowromancap iii@}},E_{\mathbb{\@slowromancap iii@}})$ with Hamiltonian functions $f_{1}=y$ and $f_{2}=-2\,y\,ln(y)$ respectively. Subsequently, $(\mathbb{\@slowromancap iii@},\Lambda_{\mathbb{\@slowromancap iii@}},E_{\mathbb{\@slowromancap iii@}},,X^{\mathbb{\@slowromancap iii@}})$ is a Jacobi-Lie system.

Using the Lie bracket induced by $\Lambda_{\mathbb{\@slowromancap iii@}}$ and $E_{\mathbb{\@slowromancap iii@}}$ of Lie group $\mathbb{\@slowromancap iii@}$, we can write $\{f_{1},f_{2}\}_{\Lambda_{\mathbb{\@slowromancap iii@},E_{\mathbb{\@slowromancap iii@}}}}=f_{1};$ therefore, $(\mathbb{\@slowromancap iii@},\Lambda_{\mathbb{\@slowromancap iii@}},E_{\mathbb{\@slowromancap iii@}},f=\sum_{i=1}^{2}b_{i}(t)f_{i})$ for $X^{\mathbb{\@slowromancap iii@}}$ is a Jacobi-Lie Hamiltonian system.

It is easy to show that $h=\ln\left({e}^{2\,x}-1\right)-{e}^{-2\,x}\ln\left({e}^{2\,x}-1\right)+y+z$ is a t-independent constant of motion. One can check that

[TABLE]

Then, the function $h$ always Jacobi commutes with the whole Lie algebra $Lie(\{f_{t}\}_{t\in\mathbb{R}},\{.,.\}_{\Lambda_{\mathbb{\@slowromancap iii@}},E_{\mathbb{\@slowromancap iii@}}}),$ as expected.

By applying proposition 4.5, $X_{h}={\Lambda}^{\#}(dh)+hE$ must be a Lie symmetry for this system. A short calculation shows that $X_{h}=\frac{1}{2}(1-e^{2x})\partial_{x}+{\frac{\left({e}^{-2\,x}+{e}^{2\,x}-2\right)\ln\left({e}^{2\,x}-1\right)-2\,{e}^{2\,x}+{e}^{4\,x}+1}{{e}^{2\,x}-1}}\partial_{z}.$ It is easy to check that $X_{h}$ commutes with $X_{1}^{H},X_{2}^{H},$ and hence with every $X_{f_{t}},$ with $t\in\mathbb{R},$ i.e. $X_{h}$ is a Lie symmetry for $X^{\mathbb{\@slowromancap iii@}}$.

Example 3. Real three-dimensional bi-r-matrix Jacobi-Lie bialgebra $((\@slowromancap iv@,-\tilde{X}^{1}),(\@slowromancap iii@.vi,-X_{2}-X_{3}))$[References]:

Let us Consider the Lie group $\mathbb{\@slowromancap iv@}$ related to Lie algebra $\@slowromancap iv@$ and the Jacobi structure given by (23). A simple calculation shows that

[TABLE]

and

[TABLE]

span the Lie algebra $,A_{2},$ of Hamiltonian vector fields on $\mathbb{\@slowromancap iv@}.$

The system on $\mathbb{\@slowromancap iv@}$ can be written as

[TABLE]

for arbitrary t-dependent functions $b_{i}(t).$

Since the associated t-dependent vector field $X^{\mathbb{\@slowromancap iv@}}=\sum_{i=1}^{2}b_{i}(t)X_{i}^{H}$ takes values in the Lie algebra $A_{2}$ ,that is, $[X_{1}^{H},X_{2}^{H}]=X_{1}^{H}$ , then $X^{\mathbb{\@slowromancap iv@}}$ is a Lie system.

We now prove that $(\mathbb{\@slowromancap iv@},\Lambda_{\mathbb{\@slowromancap iv@}},E_{\mathbb{\@slowromancap iv@}},X^{\mathbb{\@slowromancap iv@}})$ is a Jacobi-Lie system. As a matter of fact, $X_{1}^{H}$ and $X_{2}^{H}$ are Hamiltonian vector fields relative to $(\mathbb{\@slowromancap iv@},\Lambda_{\mathbb{\@slowromancap iv@}},E_{\mathbb{\@slowromancap iv@}})$ with Hamiltonian functions $f_{1}={{\rm e}^{-{e}^{-x}y}}$ and $f_{2}=1$ respectively. Subsequently, $(\mathbb{\@slowromancap iv@},\Lambda_{\mathbb{\@slowromancap iv@}},E_{\mathbb{\@slowromancap iv@}},,X^{\mathbb{\@slowromancap iv@}})$ is a Jacobi-Lie system.

Using the Lie bracket induced by $\Lambda_{\mathbb{\@slowromancap iv@}}$ and $E_{\mathbb{\@slowromancap iv@}}$ of Lie group $\mathbb{\@slowromancap iv@}$, we can write $\{f_{1},f_{2}\}_{\Lambda_{\mathbb{\@slowromancap iv@},E_{\mathbb{\@slowromancap iv@}}}}=f_{1};$ therefore, $(\mathbb{\@slowromancap iv@},\Lambda_{\mathbb{\@slowromancap iv@}},E_{\mathbb{\@slowromancap iv@}},f=\sum_{i=1}^{2}b_{i}(t)f_{i})$ for $X^{\mathbb{\@slowromancap iv@}}$ is a Jacobi-Lie Hamiltonian system.

It is easy to show that $h={e}^{x}+\left(x-1\right)y+z-1$ is a t-independent constant of motion. One can check that

[TABLE]

Then, the function $h$ always Jacobi commutes with the whole Lie algebra $Lie(\{f_{t}\}_{t\in\mathbb{R}},\{.,.\}_{\Lambda_{\mathbb{\@slowromancap iv@}},E_{\mathbb{\@slowromancap iv@}}}),$ as expected.

By applying proposition 4.5, $X_{h}={\Lambda}^{\#}(dh)+hE$ must be a Lie symmetry for this system. A short calculation shows that $X_{h}=xe^{x}\partial_{x}+xye^{x}\partial_{y}-x(xye^{x}+e^{2x})\partial_{z}.$ It is easy to check that $X_{h}$ commutes with $X_{1}^{H},X_{2}^{H},$ and subsequently with every $X_{f_{t}},$ with $t\in\mathbb{R},$ i.e. $X_{h}$ is a Lie symmetry for $X^{\mathbb{\@slowromancap iv@}}$.

Example 4. Real three-dimensional bi-r-matrix Jacobi-Lie bialgebra $((\@slowromancap vi@_{0},\tilde{X}^{3}),(\@slowromancap iii@.ix,-X_{1}))$[References]:

Let us Consider the Lie group $\mathbb{\@slowromancap vi@}_{0}$ related to Lie algebra $\@slowromancap vi@_{0}$ and the Jacobi structure given by

[TABLE]

Then one can show that

[TABLE]

so, $(\mathbb{\@slowromancap vi@}_{0},\Lambda_{\mathbb{\@slowromancap vi@}_{0}},E_{\mathbb{\@slowromancap vi@}_{0}})$ is a Jacobi manifold.

A simple calculation shows that $X_{1}^{H}=\partial_{x}$ and $X_{2}^{H}=x\partial_{x}+(1+y-{e}^{-z})\partial_{y}+{e}^{-z}\sinh\left(z\right)\partial_{z}$ span the Lie algebra $,A_{2},$ of Hamiltonian vector fields on $\mathbb{\@slowromancap vi@}_{0}.$

The system on $\mathbb{\@slowromancap vi@}_{0}$ can be written as

[TABLE]

for arbitrary t-dependent functions $b_{i}(t).$

Since the associated t-dependent vector field $X^{\mathbb{\@slowromancap vi@}_{0}}=\sum_{i=1}^{2}b_{i}(t)X_{i}^{H}$ takes values in the Lie algebra $A_{2}$ ,that is, $[X_{1}^{H},X_{2}^{H}]=X_{1}^{H}$ , then $X^{\mathbb{\@slowromancap vi@}_{0}}$ is a Lie system.

We now prove that $(\mathbb{\@slowromancap vi@}_{0},\Lambda_{\mathbb{\@slowromancap vi@}_{0}},E_{\mathbb{\@slowromancap vi@}_{0}},X^{\mathbb{\@slowromancap vi@}_{0}})$ is a Jacobi-Lie system. As a matter of fact, $X_{1}^{H}$ and $X_{2}^{H}$ are Hamiltonian vector fields relative to $(\mathbb{\@slowromancap vi@}_{0},\Lambda_{\mathbb{\@slowromancap vi@}_{0}},E_{\mathbb{\@slowromancap vi@}_{0}})$ with Hamiltonian functions $f_{1}=1$ and $f_{2}=x$ respectively. Subsequently, $(\mathbb{\@slowromancap vi@}_{0},\Lambda_{\mathbb{\@slowromancap vi@}_{0}},E_{\mathbb{\@slowromancap vi@}_{0}},,X^{\mathbb{\@slowromancap vi@}_{0}})$ is a Jacobi-Lie system.

Using the Lie bracket induced by $\Lambda_{\mathbb{\@slowromancap vi@}_{0}}$ and $E_{\mathbb{\@slowromancap vi@}_{0}}$ of Lie group $\mathbb{\@slowromancap vi@}_{0}$, we can write $\{f_{1},f_{2}\}_{\Lambda_{\mathbb{\@slowromancap vi@}_{0},E_{\mathbb{\@slowromancap vi@}_{0}}}}=f_{1};$ therefore, $(\mathbb{\@slowromancap vi@}_{0},\Lambda_{\mathbb{\@slowromancap vi@}_{0}},E_{\mathbb{\@slowromancap vi@}_{0}},f=\sum_{i=1}^{2}b_{i}(t)f_{i})$ for $X^{\mathbb{\@slowromancap vi@}_{0}}$ is a Jacobi-Lie Hamiltonian system.

It is easy to show that $h={{\rm e}^{2\,z}}-1$ is a t-independent constant of motion. One can check that

[TABLE]

Then, the function $h$ always Jacobi commutes with the whole Lie algebra $Lie(\{f_{t}\}_{t\in\mathbb{R}},\{.,.\}_{\Lambda_{\mathbb{\@slowromancap vi@}_{0}},E_{\mathbb{\@slowromancap vi@}_{0}}}),$ as expected.

By applying proposition 4.5, $X_{h}={\Lambda}^{\#}(dh)+hE$ must be a Lie symmetry for this system. A short calculation shows that $X_{h}=(1-{{\rm e}^{2\,z}})\partial y.$ It is easy to check that $X_{h}$ commutes with $X_{1}^{H},X_{2}^{H},$ and subsequently with every $X_{f_{t}},$ with $t\in\mathbb{R},$ i.e. $X_{h}$ is a Lie symmetry for $X^{\mathbb{\@slowromancap vi@}_{0}}$.

5 Concluding remarks

Using the realizations [References] of the complete list of Jacobi structures on real two and three- dimensional Jacobi-Lie groups [References], we attained Hamiltonian vector fields and we achieved Jacobi-Lie Hamiltonian systems on real low dimensional Jacobi-Lie groups. Then we presented the Lie symmetries of these Jacobi-Lie Hamiltonian systems.

Table 1: vector field $E$ and 2-vector $\Lambda$ related to real two-dimensional bi-r-matrix Jacobi-Lie bialgebras.

$((\mathfrak{g},\phi_{0}),(\mathfrak{g^{*}},X_{0}))$

vector field $E$ and 2-vector $\Lambda$

$((A_{1},\tilde{X}^{1}),(A_{1},X_{2}))$

$E=-\partial_{y}$

$\Lambda=(1-e^{-x})\partial_{x}\wedge\partial_{y}$

$((A_{2},b\tilde{X}^{2}),(A_{2}.i,-bX_{1}))$

$E=b\partial_{x}$

$\Lambda=(1-e^{-(b+1)y})\partial_{x}\wedge\partial_{y}$

$((A_{1},0),(A_{2},-X_{2}))$

$E=\partial_{y}$

$\Lambda=0$

Table 2: vector field $E$ and 2-vector $\Lambda$ related to real three-dimensional bi-r-matrix Jacobi-Lie bialgebras.

$((\mathfrak{g},\phi_{0}),(\mathfrak{g^{*}},X_{0}))$

vector field $E$ and 2-vector $\Lambda$

$((\@slowromancap i@,-\tilde{X}^{2}+\tilde{X}^{3}),(\@slowromancap iii@,-2X_{1}))$

$E=2\partial_{x}$

$\Lambda=(-1+e^{y-z})\partial_{x}\wedge\partial_{y}+(1-e^{y-z})\partial_{x}\wedge\partial_{z}$

$((\@slowromancap ii@,0),(\@slowromancap i@,X_{1}))$

$E=-\partial_{x}$

$\Lambda=-y\partial_{x}\wedge\partial_{y}-z\partial_{x}\wedge\partial_{z}$

$((\@slowromancap ii@,0),(\@slowromancap v@,bX_{1}))$

$E=-b\partial_{x}$

$\Lambda=-(1+b)y\partial_{x}\wedge\partial_{y}-(1+b)z\partial_{x}\wedge\partial_{z}$

$((\@slowromancap iii@,b\tilde{X}^{1}),(\@slowromancap iii@.i,-X_{2}+X_{3}))$

$E=\partial_{y}-\partial_{z}$

\begin{array}[]{l}\Lambda=-\frac{1}{b}(1-e^{-bx})\partial_{x}\wedge\partial_{y}+\frac{1}{b}(1-e^{-bx})\partial_{x}\wedge\partial_{z}\\ \qquad-\frac{2}{b}(y+z)e^{-bx}\partial_{y}\wedge\partial_{z}\end{array}

Table 2: (Continued.)

$((\mathfrak{g},\phi_{0}),(\mathfrak{g^{*}},X_{0}))$ vector field $E$ and 2-vector $\Lambda$

$((\@slowromancap iii@,-b\tilde{X}^{2}+b\tilde{X}^{3}),(\@slowromancap iii@.iv,bX_{1}))$ $E=-b\partial_{x}$

\begin{array}[]{l}\Lambda=\frac{1}{2}(1-e^{b(y-z)})\partial_{x}\wedge\partial_{y}-\frac{1}{2}(1-e^{b(y-z)})\partial_{x}\wedge\partial_{z}\\ \qquad+(y+z)e^{b(y-z)}\partial_{y}\wedge\partial_{z}\end{array}

$((\@slowromancap iii@,0),(\@slowromancap iii@.v,\frac{1}{2}X_{2}-\frac{1}{2}X_{3}))$ $E=-\frac{1}{2}\partial_{y}+\frac{1}{2}\partial_{z}$

\begin{array}[]{l}\Lambda=\dfrac{e^{2x}-1}{4}\partial_{x}\wedge\partial_{y}+\dfrac{e^{2x}-1}{4}\partial_{x}\wedge\partial_{z}\\ \qquad-\frac{1}{2}(y+z+1-e^{2x})\partial_{y}\wedge\partial_{z}\end{array}

$((\@slowromancap iii@,0),(\@slowromancap iv@.iv,X_{2}-X_{3}))$ $E=-\partial_{y}+\partial_{z}$

$\Lambda=\frac{1}{2}(e^{2x}-1)\partial_{y}\wedge\partial_{z}$

$((\@slowromancap iii@,-2\tilde{X}^{1}),(\@slowromancap v@.i,-X_{2}-X_{3}))$ $E=e^{2x}\partial_{y}+e^{2x}\partial_{z}$

$\Lambda=0$

$((\@slowromancap iii@,0),(\@slowromancap vi@_{0}.iv,X_{2}-X_{3}))$ $E=-\partial_{y}+\partial_{z}$

$\Lambda=-2(y+z)\partial_{y}\wedge\partial_{z}$

$((\@slowromancap iii@,0),(\@slowromancap vi@_{a}.vii,-X_{2}+X_{3}))$ $E=\partial_{y}-\partial_{z}$

$\Lambda=\dfrac{-2}{a-1}(y+z)\partial_{y}\wedge\partial_{z}$

$((\@slowromancap iii@,0),(\@slowromancap vi@_{a}.viii,-X_{2}+X_{3}))$ $E=\partial_{y}-\partial_{z}$

$\Lambda=\dfrac{2}{a+1}(y+z)\partial_{y}\wedge\partial_{z}$

$((\@slowromancap iv@,-\tilde{X}^{1}),(\@slowromancap iii@.vi,-X_{2}-X_{3}))$ $E=e^{x}\partial_{y}+e^{x}(1-x)\partial_{z}$

$\Lambda=-xe^{x}\partial_{x}\wedge\partial_{z}+e^{x}(z-y-1+e^{x})\partial_{y}\wedge\partial_{z}$

Table 2: (Continued.)

$((\mathfrak{g},\phi_{0}),(\mathfrak{g^{*}},X_{0}))$

vector field $E$ and 2-vector $\Lambda$

$((\@slowromancap iv@,-\tilde{X}^{1}),(\@slowromancap iv@.i,-bX_{3}))$

$E=be^{x}\partial_{z}$

$\Lambda=e^{x}(e^{x}-1)\partial_{y}\wedge\partial_{z}$

$((\@slowromancap iv@,-\tilde{X}^{1}),(\@slowromancap iv@.ii,-bX_{3}))$

$E=be^{x}\partial_{z}$

$\Lambda=-e^{x}(e^{x}-1)\partial_{y}\wedge\partial_{z}$

$((\@slowromancap iv@,-\tilde{X}^{1}),(\@slowromancap vi@_{0}.i,-X_{3}))$

$E=e^{x}\partial_{z}$

$\Lambda=-2ye^{x}\partial_{y}\wedge\partial_{z}$

$((\@slowromancap iv@,-\tilde{X}^{1}),(\@slowromancap vi@_{a}.i,-X_{3}))$

$E=e^{x}\partial_{z}$

$\Lambda=\dfrac{2}{a-1}ye^{x}\partial_{y}\wedge\partial_{z}$

$((\@slowromancap iv@,-\tilde{X}^{1}),(\@slowromancap vi@_{a}.ii,-X_{3}))$

$E=e^{x}\partial_{z}$

$\Lambda=-\dfrac{2}{a+1}ye^{x}\partial_{y}\wedge\partial_{z}$

$((\@slowromancap v@,-2\tilde{X}^{1}),(\@slowromancap v@.i,-2X_{2}-2X_{3}))$

$E=2e^{x}\partial_{y}+2e^{x}\partial_{z}$

\begin{array}[]{l}\Lambda=e^{x}(1-e^{x})\partial_{x}\wedge\partial_{y}+e^{x}(1-e^{x})\partial_{x}\wedge\partial_{z}\\ \qquad+e^{2x}(z-y)\partial_{y}\wedge\partial_{z}\end{array}

$((\@slowromancap v@,-\frac{2a}{a-1}\tilde{X}^{1}),(\@slowromancap vi@_{a}.i,-\frac{2a}{a-1}X_{3}))$

$E=\frac{2a}{a-1}e^{x}\partial_{z}$

$\Lambda=(e^{x}-e^{\frac{2a}{a-1}x})\partial_{x}\wedge\partial_{z}-ye^{\frac{2a}{a-1}x}\partial_{y}\wedge\partial_{z}$

$((\@slowromancap v@,-\frac{2a}{a+1}\tilde{X}^{1}),(\@slowromancap vi@_{a}.ii,-\frac{2a}{a+1}X_{3}))$

$E=\frac{2a}{a+1}e^{x}\partial_{z}$

$\Lambda=(e^{x}-e^{\frac{2a}{a+1}x})\partial_{x}\wedge\partial_{z}-ye^{\frac{2a}{a+1}x}\partial_{y}\wedge\partial_{z}$

Table 2: (Continued.)

$((\mathfrak{g},\phi_{0}),(\mathfrak{g^{*}},X_{0}))$ vector field $E$ and 2-vector $\Lambda$

$((\@slowromancap vi@_{0},\tilde{X}^{3}),(\@slowromancap iii@.vii,-X_{1}-X_{2}))$ $E=\partial_{x}+\partial_{y}$

\begin{array}[]{l}\Lambda=(y-x)\partial_{x}\wedge\partial_{y}+(1-e^{-2z})\partial_{x}\wedge\partial_{z}\\ \qquad+(1-e^{-2z})\partial_{y}\wedge\partial_{z}\end{array}

$((\@slowromancap vi@_{0},\tilde{X}^{3}),(\@slowromancap iii@.ix,-X_{1}))$ $E=\partial_{x}$

\begin{array}[]{l}\Lambda=(1+y-e^{-z})\partial_{x}\wedge\partial_{y}+e^{-z}\sinh(z)\partial_{x}\wedge\partial_{z}\\ \qquad+(1-e^{-z}\cosh(z))\partial_{y}\wedge\partial_{z}\end{array}

$((\@slowromancap vi@_{0},\tilde{X}^{3}),(\@slowromancap vi@_{0}.ii,-X_{1}+X_{2}))$ $E=\partial_{x}-\partial_{y}$

$\Lambda=2(x+y)\partial_{x}\wedge\partial_{y}$

$((\@slowromancap vi@_{0},-2\tilde{X}^{3}),(\@slowromancap vi@_{0}.ii,2X_{1}-2X_{2}))$ $E=-2\partial_{x}+2\partial_{y}$

\begin{array}[]{l}\Lambda=-(x+y)\partial_{x}\wedge\partial_{y}+(1-e^{3z})\partial_{x}\wedge\partial_{z}\\ \qquad-(1-e^{3z})\partial_{y}\wedge\partial_{z}\end{array}

$((\@slowromancap vi@_{0},\tilde{X}^{3}),(\@slowromancap vi@_{a}.iii,-X_{1}+X_{2}))$ $E=\partial_{x}-\partial_{y}$

$\Lambda=-\frac{2}{a-1}(x+y)\partial_{x}\wedge\partial_{y}$

$((\@slowromancap vi@_{0},\tilde{X}^{3}),(\@slowromancap vi@_{a}.iv,-X_{1}+X_{2}))$ $E=\partial_{x}-\partial_{y}$

$\Lambda=\frac{2}{a+1}(x+y)\partial_{x}\wedge\partial_{y}$

$((\@slowromancap vi@_{0},\frac{2}{a-1}\tilde{X}^{3}),(\@slowromancap vi@_{a}.iii,-\frac{2}{a-1}(X_{1}-X_{2})))$ $E=\frac{2}{a-1}\partial_{x}-\frac{2}{a-1}\partial_{y}$

\begin{array}[]{l}\Lambda=-(x+y)\partial_{x}\wedge\partial_{y}+(1-e^{\frac{a-3}{a-1}z})\partial_{x}\wedge\partial_{z}\\ \qquad-(1-e^{\frac{a-3}{a-1}z})\partial_{y}\wedge\partial_{z}\end{array}

$((\@slowromancap vi@_{0},-\frac{2}{a+1}\tilde{X}^{3}),(\@slowromancap vi@_{a}.iv,\frac{2}{a+1}(X_{1}-X_{2})))$ $E=-\frac{2}{a+1}\partial_{x}+\frac{2}{a+1}\partial_{y}$

\begin{array}[]{l}\Lambda=-(x+y)\partial_{x}\wedge\partial_{y}+(1-e^{\frac{a+3}{a+1}z})\partial_{x}\wedge\partial_{z}\\ \qquad-(1-e^{\frac{a+3}{a+1}z})\partial_{y}\wedge\partial_{z}\end{array}

Table 2: (Continued.)

$((\mathfrak{g},\phi_{0}),(\mathfrak{g^{*}},X_{0}))$ vector field $E$ and 2-vector $\Lambda$

$((\@slowromancap vi@_{a},-(a-1)\tilde{X}^{1}),(\@slowromancap iii@.ii,-\frac{a-1}{a+1}(X_{2}+X_{3})))$ $E=\frac{a-1}{a+1}e^{(a+1)x}\partial_{y}+\frac{a-1}{a+1}e^{(a+1)x}\partial_{z}$

\begin{array}[]{l}\Lambda=\frac{1}{a+1}(e^{(a+1)x}-e^{(a-1)x})\partial_{x}\wedge\partial_{y}\\ \qquad+\frac{1}{a+1}(e^{(a+1)x}-e^{(a-1)x})\partial_{x}\wedge\partial_{z}\\ \qquad+\frac{1}{a+1}e^{(a-1)x}(1-a)(y-z)\partial_{y}\wedge\partial_{z}\end{array}

$((\@slowromancap vi@_{a},-(a+1)\tilde{X}^{1}),(\@slowromancap iii@.v,\frac{1}{a-1}(X_{2}-aX_{3})))$ \begin{array}[]{l}E=\frac{1}{a-1}e^{ax}(-\cosh(x)+a\sinh(x))\partial_{y}\\ \qquad+\frac{1}{a-1}e^{ax}(-\sinh(x)+a\cosh(x))\partial_{z}\end{array}

\begin{array}[]{l}\Lambda=\frac{1}{a-1}e^{ax}\sinh(x)\partial_{x}\wedge\partial_{y}\\ \qquad+\frac{1}{a-1}(e^{ax}\cosh(x)-e^{(a+1)x})\partial_{x}\wedge\partial_{z}\\ \qquad+\frac{1}{a-1}(e^{2ax}-e^{(a+1)x}(1+ay+z))\partial_{y}\wedge\partial_{z}\end{array}

$((\@slowromancap vi@_{a},-(a-1)\tilde{X}^{1}),(\@slowromancap iii@.v,\frac{1}{a+1}(X_{2}-aX_{3})))$ \begin{array}[]{l}E=\frac{1}{a+1}e^{ax}(-\cosh(x)+a\sinh(x))\partial_{y}\\ \qquad+\frac{1}{a+1}e^{ax}(-\sinh(x)+a\cosh(x))\partial_{z}\end{array}

\begin{array}[]{l}\Lambda=\frac{1}{a+1}e^{ax}\sinh(x)\partial_{x}\wedge\partial_{y}\\ \qquad+\frac{1}{a+1}(e^{ax}\cosh(x)-e^{(a-1)x})\partial_{x}\wedge\partial_{z}\\ \qquad+\frac{1}{a+1}(e^{2ax}-e^{(a-1)x}(1+ay+z))\partial_{y}\wedge\partial_{z}\end{array}

$((\@slowromancap vi@_{a},-(a+1)\tilde{X}^{1}),(\@slowromancap iii@.x,-\frac{a+1}{a-1}(X_{2}-X_{3})))$ $E=\frac{a+1}{a-1}e^{(a-1)x}\partial_{y}-\frac{a+1}{a-1}e^{(a-1)x}\partial_{z}$

\begin{array}[]{l}\Lambda=\frac{1}{a-1}(e^{(a-1)x}-e^{(a+1)x})\partial_{x}\wedge\partial_{y}\\ \qquad+\frac{1}{a-1}(e^{(a+1)x}-e^{(a-1)x})\partial_{x}\wedge\partial_{z}\\ \qquad+\frac{1}{a-1}e^{(a+1)x}(1+a)(y+z)\partial_{y}\wedge\partial_{z}\end{array}

$((\@slowromancap vi@_{a},-(a+1)\tilde{X}^{1}),(\@slowromancap vi@_{b}.v,-X_{2}-X_{3}))$ $E=e^{(a+1)x}\partial_{y}+e^{(a+1)x}\partial_{z}$

$\Lambda=\frac{2}{(b-1)}e^{(a+1)x}(y-z)\partial_{y}\wedge\partial_{z}$

$((\@slowromancap vi@_{a},-(a+1)\tilde{X}^{1}),(\@slowromancap vi@_{b}.vi,-X_{2}+X_{3}))$ $E=e^{(a+1)x}\partial_{y}+e^{(a+1)x}\partial_{z}$

$\Lambda=-\frac{2}{(b+1)}e^{(a+1)x}(y-z)\partial_{y}\wedge\partial_{z}$

Table 2: (Continued.)

$((\mathfrak{g},\phi_{0}),(\mathfrak{g^{*}},X_{0}))$ vector field $E$ and 2-vector $\Lambda$

$((\@slowromancap vi@_{a},-(a-1)\tilde{X}^{1}),(\@slowromancap vi@_{b}.vii,-X_{2}+X_{3}))$ $E=e^{(a-1)x}\partial_{y}-e^{(a-1)x}\partial_{z}$

$\Lambda=-\frac{2}{(b-1)}e^{(a-1)x}(y+z)\partial_{y}\wedge\partial_{z}$

$((\@slowromancap vi@_{a},-(a-1)\tilde{X}^{1}),(\@slowromancap vi@_{b}.viii,-X_{2}+X_{3}))$ $E=e^{(a-1)x}\partial_{y}-e^{(a-1)x}\partial_{z}$

$\Lambda=\frac{2}{(b+1)}e^{(a-1)x}(y+z)\partial_{y}\wedge\partial_{z}$

$((\@slowromancap vi@_{a},-\frac{2(ab+1)}{b-1}\tilde{X}^{1}),(\@slowromancap vi@_{b}.v,-\frac{2(ab+1)}{(a+1)(b-1)}(X_{2}+X_{3})))$ \begin{array}[]{l}E=\frac{2(ab+1)}{(a+1)(b-1)}e^{(a+1)x}\partial_{y}\\ \qquad+\frac{2(ab+1)}{(a+1)(b-1)}e^{(a+1)x}\partial_{z}\end{array}

\begin{array}[]{l}\Lambda=\frac{1}{a+1}(e^{(a+1)x}-e^{\frac{2(ab+1)}{b-1}x})\partial_{x}\wedge\partial_{y}\\ \qquad+\frac{1}{a+1}(e^{(a+1)x}-e^{\frac{2(ab+1)}{b-1}x})\partial_{x}\wedge\partial_{z}\\ \qquad+\frac{1}{a+1}e^{\frac{2(ab+1)}{b-1}x}(1-a)(y-z)\partial_{y}\wedge\partial_{z}\end{array}

$((\@slowromancap vi@_{a},-\frac{2(ab-1)}{b+1}\tilde{X}^{1}),(\@slowromancap vi@_{b}.vi,-\frac{2(ab-1)}{(a+1)(b+1)}(X_{2}+X_{3})))$ \begin{array}[]{l}E=\frac{2(ab-1)}{(a+1)(b+1)}e^{(a+1)x}\partial_{y}\\ \qquad+\frac{2(ab-1)}{(a+1)(b+1)}e^{(a+1)x}\partial_{z}\end{array}

\begin{array}[]{l}\Lambda=\frac{1}{a+1}(e^{(a+1)x}-e^{\frac{2(ab-1)}{b+1}x})\partial_{x}\wedge\partial_{y}\\ \qquad+\frac{1}{a+1}(e^{(a+1)x}-e^{\frac{2(ab-1)}{b+1}x})\partial_{x}\wedge\partial_{z}\\ \qquad+\frac{1}{a+1}e^{\frac{2(ab-1)}{b+1}x}(1-a)(y-z)\partial_{y}\wedge\partial_{z}\end{array}

$((\@slowromancap vi@_{a},-\frac{2(ab-1)}{b-1}\tilde{X}^{1}),(\@slowromancap vi@_{b}.vii,-\frac{2(ab-1)}{(a1)(b-1)}(X_{2}-X_{3})))$ \begin{array}[]{l}E=\frac{2(ab-1)}{(a-1)(b-1)}e^{(a-1)x}\partial_{y}\\ \qquad-\frac{2(ab-1)}{(a-1)(b-1)}e^{(a-1)x}\partial_{z}\end{array}

\begin{array}[]{l}\Lambda=\frac{1}{a-1}(e^{(a-1)x}-e^{\frac{2(ab-1)}{b-1}x})\partial_{x}\wedge\partial_{y}\\ \qquad+\frac{1}{a-1}(-e^{(a-1)x}+e^{\frac{2(ab-1)}{b-1}x})\partial_{x}\wedge\partial_{z}\\ \qquad+\frac{1}{a-1}e^{\frac{2(ab-1)}{b-1}x}(1+a)(y+z)\partial_{y}\wedge\partial_{z}\end{array}

$((\@slowromancap vi@_{a},-\frac{2(ab+1)}{b+1}\tilde{X}^{1}),(\@slowromancap vi@_{b}.viii,-\frac{2(ab+1)}{(a-1)(b+1)}(X_{2}-X_{3})))$ \begin{array}[]{l}E=\frac{2(ab+1)}{(a-1)(b+1)}e^{(a-1)x}\partial_{y}\\ \qquad-\frac{2(ab+1)}{(a-1)(b+1)}e^{(a-1)x}\partial_{z}\end{array}

\begin{array}[]{l}\Lambda=\frac{1}{a-1}(e^{(a-1)x}-e^{\frac{2(ab+1)}{b+1}x})\partial_{x}\wedge\partial_{y}\\ \qquad+\frac{1}{a-1}(-e^{(a-1)x}+e^{\frac{2(ab+1)}{b+1}x})\partial_{x}\wedge\partial_{z}\\ \qquad+\frac{1}{a-1}e^{\frac{2(ab+1)}{b+1}x}(1+a)(y+z)\partial_{y}\wedge\partial_{z}\end{array}

Table 3: vector field $E$ and 2-vector $\Lambda$ related to real three dimensional coboundary Jacobi-Lie bialgebras.

$((\mathfrak{g},\phi_{0}),(\mathfrak{g^{*}},X_{0}))$

vector field $E$ and 2-vector $\Lambda$

$((\@slowromancap i@,0),(\@slowromancap v@,-X_{1}))$

$E=\partial_{x}$

$\Lambda=0$

$((\@slowromancap ii@,-\tilde{X}^{2}+\tilde{X}^{3}),(\@slowromancap iii@,-2X_{1}))$

$E=2\partial_{x}$

$\Lambda=(-1+e^{y-z})\partial_{x}\wedge\partial_{y}+(1-e^{y-z})\partial_{x}\wedge\partial_{z}$

$((\@slowromancap iii@,-\tilde{X}^{2}+\tilde{X}^{3}),(\@slowromancap iii@.iii,X_{2}+X_{3}))$

$E=-e^{2x}\partial_{y}-e^{2x}\partial_{z}$

$\Lambda=(-e^{2x}+e^{y-z})\partial_{y}\wedge\partial_{z}$

$((\@slowromancap iii@,0),(\@slowromancap iii@.x,-X_{2}+X_{3}))$

$E=\partial_{y}-\partial_{z}$

$\Lambda=(y+z)\partial_{y}\wedge\partial_{z}$

$((\@slowromancap iv@,-\tilde{X}^{1}),(\@slowromancap iii@.v,-X_{3}))$

$E=e^{x}\partial_{z}$

$\Lambda=-(y+1-e^{x})e^{x}\partial_{y}\wedge\partial_{z}$

$((\@slowromancap iv@,-2\tilde{X}^{1}),(\@slowromancap v@.ii,-2X_{3}))$

$E=2e^{x}\partial_{z}$

$\Lambda=(e^{x}-e^{2x})\partial_{x}\wedge\partial_{z}-ye^{2x}\partial_{y}\wedge\partial_{z}$

$((\@slowromancap iv@,-\frac{2a}{a-1}\tilde{X}^{1}),(\@slowromancap vi@_{a}.i,-\frac{2a}{a-1}X_{3}))$

$E=\frac{2a}{a-1}e^{x}\partial_{z}$

$\Lambda=(e^{x}-e^{\frac{2a}{a-1}x})\partial_{x}\wedge\partial_{z}-ye^{\frac{2a}{a-1}x}\partial_{y}\wedge\partial_{z}$

$((\@slowromancap iv@,-\frac{2a}{a+1}\tilde{X}^{1}),(\@slowromancap vi@_{a}.ii,-\frac{2a}{a+1}X_{3}))$

$E=\frac{2a}{a+1}e^{x}\partial_{z}$

$\Lambda=(e^{x}-e^{\frac{2a}{a+1}x})\partial_{x}\wedge\partial_{z}-ye^{\frac{2a}{a+1}x}\partial_{y}\wedge\partial_{z}$

Table 3: (Continued.)

$((\mathfrak{g},\phi_{0}),(\mathfrak{g^{*}},X_{0}))$

vector field $E$ and 2-vector $\Lambda$

$((\@slowromancap v@,-\tilde{X}^{1}),(\@slowromancap vi@_{0}.i,-X_{3}))$

$E=e^{x}\partial_{z}$

$\Lambda=-2ye^{x}\partial_{y}\wedge\partial_{z}$

$((\@slowromancap v@,-\tilde{X}^{1}),(\@slowromancap vi@_{a}.i,-X_{3}))$

$E=e^{x}\partial_{z}$

$\Lambda=\frac{2}{a-1}ye^{x}\partial_{y}\wedge\partial_{z}$

$((\@slowromancap v@,-\tilde{X}^{1}),(\@slowromancap vi@_{a}.ii,-X_{3}))$

$E=e^{x}\partial_{z}$

$\Lambda=-\frac{2}{a+1}ye^{x}\partial_{y}\wedge\partial_{z}$

$((\@slowromancap vi@_{0},\tilde{X}^{3}),(\@slowromancap iii@.viii,-X_{1}+X_{2}))$

$E=\partial_{x}-\partial_{y}$

$\Lambda=(x+y)\partial_{x}\wedge\partial_{y}$

$((\@slowromancap vi@_{a},-(a+1)\tilde{X}^{1}),(\@slowromancap iii@.ii,-X_{2}-X_{3}))$

$E=e^{(a+1)x}\partial_{y}+e^{(a+1)x}\partial_{z}$

$\Lambda=-(y-z)e^{(a+1)x}\partial_{y}\wedge\partial_{z}$

$((\@slowromancap vi@_{a},-(a-1)\tilde{X}^{1}),(\@slowromancap iii@.x,-X_{2}+X_{3}))$

$E=e^{(a-1)x}\partial_{y}+e^{(a-1)x}\partial_{z}$

$\Lambda=(y+z)e^{(a-1)x}\partial_{y}\wedge\partial_{z}$