# Unimodularity and invariant volume forms for Hamiltonian dynamics on
Poisson-Lie groups

I. Gutierrez-Sagredo1, D. Iglesias Ponte2, J. C. Marrero2, E. Padrón2, Z.
Ravanpak3

###### Abstract.

In this paper, we discuss several relations between the existence of invariant
volume forms for Hamiltonian systems on Poisson-Lie groups and the
unimodularity of the Poisson-Lie structure. In particular, we prove that
Hamiltonian vector fields on a Lie group endowed with a unimodular Poisson-Lie
structure preserve a multiple of any left-invariant volume on the group.
Conversely, we also prove that if there exists a Hamiltonian function such
that the identity element of the Lie group is a nondegenerate singularity and
the associated Hamiltonian vector field preserves a volume form, then the
Poisson-Lie structure is necessarily unimodular. Furthermore, we illustrate
our theory with different interesting examples, both on semisimple and
unimodular Poisson-Lie groups.

AMS Mathematics Subject Classification (2020): 37C40, 37J39, 53D17, 70G45,
70H05

Keywords: Modular vector fields, modular class, invariant volume forms,
Hamiltonian systems, Poisson-Lie groups

$\;{}^{1}$ Departamento de Matemáticas y Computación, Universidad de Burgos,
09001 Burgos, Spain

e-mail<EMAIL_ADDRESS>
$\;{}^{2}$ULL-CSIC Geometría Diferencial y Mecánica Geométrica, Departamento
de Matemáticas, Estadística e Investigación Operativa and Instituto de
Matemáticas y Aplicaciones (IMAULL)
University of La Laguna, Spain

e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
$\;{}^{3}$ Institute of Mathematics, Polish Academy of Sciences, Warsaw,
Poland

e-mail<EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
    1. 1.1 Integrability for dynamical systems and invariant volume forms
    2. 1.2 Invariant volume forms for Hamiltonian systems with respect to (almost) Poisson structures
    3. 1.3 Hamiltonian systems on Poisson-Lie groups
    4. 1.4 Aim of the paper
    5. 1.5 Organization of the paper
  2. 2 Poisson structures, Poisson-Lie groups and modular vector fields
    1. 2.1 Poisson cohomology and symplectic foliation
    2. 2.2  Poisson-Lie groups
    3. 2.3 Modular vector field of a Poisson manifold
  3. 3 Unimodularity of Poisson-Lie groups and preservation of volume forms
  4. 4 Examples
    1. 4.1 Kozlov’s unimodularity result
    2. 4.2 Contrast functions on Lie groups
    3. 4.3 A Hamiltonian system on the Poisson-Lie group $\mathrm{SL}(2,\mathbb{R})$
    4. 4.4 A Hamiltonian system on the Poisson-Lie group $S^{3}\cong\mathrm{SU}(2,\mathbb{C})$
    5. 4.5 A deformation of a conservative Lorenz system
    6. 4.6 A deformed Euler top
  5. 5 Conclusions and future work

##  1\. Introduction

###  1.1. Integrability for dynamical systems and invariant volume forms

Given a dynamical system $X\in\mathfrak{X}(M)$ on a manifold $M$ of dimension
$n$, the exact integrability is closely related to finding tensors which are
invariant for the system. It is clear that if one finds a family of
independent $C^{\infty}$-functions $\\{f_{1},\ldots,f_{n-1}\\}$ which are
first integrals for $X$, i.e., $X(f_{i})=0$, then it is possible to obtain, up
to reparametrization, the orbits for $X$.

Another approach to the integrability of the dynamical system on $M$,
following Euler and Jacobi, is to look for $n-2$ first integrals
$\\{f_{1},\ldots,f_{n-2}\\}$ which are functionally independent and an
invariant volume form $\Omega$. Under these conditions, the system can be
integrated by quadratures, that is, to determine the trajectories of $X$ by
means of a finite number of algebraic operations and quadratures of some
functions. For instance, if $dim\,M=2$, an invariant volume form $\Omega$ with
respect to $X$ implies that $i_{X}\Omega$ is a closed 1-form, so locally can
be written as $df$, $f$ being a first integral of $X$ (for more details see,
for instance, [17] and the references therein).

###  1.2. Invariant volume forms for Hamiltonian systems with respect to
(almost) Poisson structures

Among dynamical systems, those that are Hamiltonian with respect to a
symplectic structure are of particular interest. In this direction, it is
well-known that a Hamiltonian system in a symplectic manifold of dimension
$2n$ is completely integrable if it admits $n$ functionally independent first
integrals in involution with respect to the Poisson bracket. This kind of
systems can be integrated by quadratures. On the other hand, with respect to
invariant volume forms, we have Liouville’s theorem:

Given a Hamiltonian on a symplectic manifold, the flow of the Hamiltonian
vector field preserves the symplectic volume (see, for instance, [1, 2]).

In the more general case when the dynamical system is Hamiltonian with respect
to a (not necessarily symplectic) Poisson structure, there are two possible
approaches in order to find invariant volume forms: the first one is to
describe the symplectic leaves of the Poisson structure and apply Liouville’s
theorem obtaining invariant volume forms on the leaves; the second is to look
for an invariant volume form on the whole manifold. A disadvantage that shows
up for the first approach is that the symplectic leaves (and the corresponding
symplectic structures on them) for some types of Poisson manifolds are hard to
compute. Therefore, in these cases it seems more natural to use the second
approach, finding invariant forms on the whole manifold. In this direction, we
have the following result:

Given a linear Poisson structure on a vector bundle $A\to M$ and a Hamiltonian
function of mechanical type on $A$, the Hamiltonian dynamics preserves a
volume form of basic type if and only if the Poisson structure on $A$ is
unimodular. In fact, if the Hamiltonian function is of kinetic type then the
Hamiltonian dynamics preserves a volume form (not necessarily of basic type)
if and only if the Poisson structure on $A$ is unimodular  (for more details,
see [22]).

We recall that an orientable Poisson manifold $M$ is said to be unimodular if
there exists a volume form invariant for all Hamiltonian vector fields.
Equivalently, unimodularity can be expressed as the vanishing of the so-called
modular class. The modular class is a first order cohomology class in the
Poisson cohomology complex of $M$ defined by a Poisson vector field, which
measures the existence of an invariant volume form for the flows of all
Hamiltonian vector fields (see [30]). So, by Liouville’s theorem, the modular
class of a symplectic manifold is zero. Besides the importance of the modular
class in a Poisson manifold for dynamics, unimodularity has also been used to
prove the existence of a duality between Poisson homology and Poisson
cohomology [31].

Note that in the linear Poisson case, the existence of an invariant volume
form for certain kinds of Hamiltonian systems is closely related to the
unimodularity of the Poisson structure. This is still true for the more
general class of reduced symmetric nonholonomic mechanical systems subjected
to linear nonholonomic constraints. In fact, there is a geometric formulation
for these systems in terms of almost Poisson structures (see, for instance,
[19, 16]) which is used, in [14], to discuss the existence of invariant volume
forms in terms of the unimodularity of such almost Poisson structures.

###  1.3. Hamiltonian systems on Poisson-Lie groups

A Poisson-Lie group is a Lie group $G$ endowed with a Poisson structure such
that the multiplication is a Poisson map (see [12]). If the Lie group $G$ is a
vector space $V$, then $V=\mathfrak{g}^{*}$ is the dual space of a Lie algebra
$\mathfrak{g}$, endowed with the Lie-Poisson structure. Poisson-Lie groups are
in one-to-one correspondence with the so-called Lie bialgebras
$(\mathfrak{g},\mathfrak{g}^{*})$, which are compatible pairs of Lie algebras
in duality (for more details, see [11, 29] or the recent survey paper [23] and
the references therein).

We remark that some Lie bialgebras can be obtained from solutions of the
classical Yang-Baxter equation (on a Lie algebra) and these solutions are
specially relevant in the theory of integrable systems. This gives an
interesting connection between integrable systems and Poisson-Lie groups,
particularly in the case of factorisable Poisson-Lie groups as it is shown,
for instance, in [27, 28].

On the other hand, Hamiltonian systems on Poisson-Lie groups appear in the
differential equation approach to the singular value decomposition (SVD) of a
bidiagonal matrix [10]. In fact, in [25], the authors showed that the system
considered in [10] is Hamiltonian with respect to the (standard) Sklyanin
bracket $\\{\cdot,\cdot\\}$ defined on $\mathrm{SL}(n,\mathbb{R})$.

The previous facts provide a good motivation to discuss Hamiltonian systems on
Poisson-Lie groups. In particular, the existence of invariant volume forms for
such systems. We note that the description of the symplectic leaves for a
Poisson-Lie group and the symplectic structure on them may be hard. In
addition, one could consider the symplectic groupoid ${\mathcal{G}}(G)$
integrating the Poisson-Lie group $G$ (see [21]) and the invariant induced
Hamiltonian system on ${\mathcal{G}}(G)$. Then, one could apply Liouville
theorem to the symplectic Hamiltonian system. The problem is that
$\dim{\mathcal{G}}(G)=2\dim G$ and, therefore, we would need to duplicate the
number of variables. So, for these previous reasons, it seems reasonable to
discuss the existence of invariant volume forms directly in the Poisson-Lie
group. In this direction, a first result which can be found in the literature
has been obtained in [18] (see also [22]): The flow of a Hamiltonian function
of kinetic type on the dual space $\mathfrak{g}^{*}$ of a Lie algebra
$\mathfrak{g}$ (endowed with the Lie-Poisson structure) preserves a volume
form if and only if the Lie algebra $\mathfrak{g}$ is unimodular or,
equivalently, the Lie-Poisson structure on $\mathfrak{g}^{*}$ is unimodular.

###  1.4. Aim of the paper

The aim of the paper is to discuss the relation between the existence of an
invariant volume form for a Hamiltonian system on a Poisson-Lie group $G$ and
the unimodularity of the Lie-Poisson structure on $G$. In fact, the main
result of the paper is (see Corollary 3.4 and Theorem 3.7):

Given a Hamiltonian function $H$ on a Lie group $G$ endowed with a unimodular
Poisson-Lie structure $\Pi$ (that is, the dual Lie algebra of $G$ is
unimodular), then the Hamiltonian vector field of $H$, $X_{H}^{\Pi}$,
preserves a multiple of any left-invariant volume form on $G$. Conversely, if
the identity element of $G$ is a nondegenerate singularity of $H$ and
$X_{H}^{\Pi}$ preserves a volume form then the Poisson-Lie structure $\Pi$ is
unimodular.

We remark that the modular vector field of a Poisson-Lie group $G$ with
respect to a left-invariant volume form has been described in [13], using the
modular character of the dual Lie algebra of $G$.

Apart from the previous results mentioned above, we also obtain an interesting
necessary condition for a Hamiltonian system on a Poisson-Lie group to
preserve a volume form. This condition must be satisfied in the singular
points of the Hamiltonian dynamics (see Remark 3.2).

###  1.5. Organization of the paper

The paper is organized as follows. In Section 2, we recall the definition of
the Poisson cohomology associated with any Poisson structure, the basic
properties of Poisson-Lie groups and the definition of the modular vector
field associated with any Poisson structure. In Section 3, we prove the main
results of the paper: Corollary 3.4 and Theorem 3.7 (see also Remark 3.2). In
Section 4, we describe several examples which illustrate our theory:

  * •

We deduce, as a consequence, the previous result by Kozlov [18] on the
existence of invariant volume forms for Hamiltonian systems of kinetic type
with respect to Lie-Poisson structures on the dual space of a Lie algebra.

  * •

We present an interesting connection between information geometry and
Hamiltonian functions on a Poisson-Lie group with a nondegenerate singularity
at the identity element.

  * •

We discuss the existence of invariant volume forms for two Hamiltonian systems
on two semisimple Poisson-Lie groups: $\mathrm{SL}(2,\mathbb{R})$ and
$S^{3}\simeq\mathrm{SU}(2,\mathbb{C})$. The Hamiltonian system on
$\mathrm{SL}(2,\mathbb{R})$ is just the dynamical system associated with the
SDV problem for a bidiagonal matrix (see [10, 25]).

  * •

Finally, we present some interesting examples of Hamiltonian systems on
Poisson-Lie groups endowed with unimodular Poisson-Lie structures (so, all of
them admit invariant volume forms).

##  2\. Poisson structures, Poisson-Lie groups and modular vector fields

In this section, we will review some definitions and basic constructions on
Poisson geometry (for more details, see [11, 23, 29]).

###  2.1. Poisson cohomology and symplectic foliation

A $2$-vector $\Pi$ on a smooth manifold $M$ is said to be Poisson  if

(1) $[\\![\Pi,\Pi]\\!]=0,$

where $[\\![\cdot,\cdot]\\!]$ is the Schouten-Nijenhuis bracket on $M$, which
is a graded Lie bracket on the space of multivector fields extending the Lie
bracket of vector fields. In such a case, $\Pi$ is a Poisson structure on $M.$
Equivalently, a Poisson structure on $M$ is a bracket of real
$C^{\infty}$-functions on $M$,

$\\{\cdot,\cdot\\}:C^{\infty}(M)\times C^{\infty}(M)\to C^{\infty}(M),$

which is ${\mathbb{R}}$-bilinear, skew-symmetric, a derivation in each
argument with respect to the standard product on $C^{\infty}(M)$ and, in
addition, it satisfies the Jacobi identity. The relation between the two
definitions of a Poisson structure is

$\\{F,H\\}=\Pi(\mathrm{d}F,\mathrm{d}H),\mbox{ for }F,H\in C^{\infty}(M).$

The Poisson $2$-vector $\Pi$ induces, in a natural way, a morphism of vector
bundles (over the identity of $M$)

$\Pi^{\\#}:T^{*}M\to TM$

given by

$\langle\Pi^{\\#}(\alpha),\beta\rangle=\Pi(x)(\alpha,\beta),\;\;\mbox{ for
$x\in M$ and $\alpha,\beta\in T^{*}_{x}M.$}$

The Hamiltonian vector field $X_{H}^{\Pi}$ associated with a real
$C^{\infty}$-function on $M$, $H:M\to\mathbb{R},$ is just

$X_{H}^{\Pi}(x)=\Pi^{\\#}(\mathrm{d}H(x)),\mbox{ for }x\in M.$

On the other hand, using (1), one may introduce a cohomology operator
$\partial_{\Pi}$ on the space of multivector fields on $M.$ Indeed, if $P$ is
a $k$-vector on $M$ then $\partial_{\Pi}P$ is the $(k+1)$-vector defined by

$\partial_{\Pi}P=[\\![\Pi,P]\\!].$

So, $\partial_{\Pi}H$ is just the Hamiltonian vector field of $H.$

The resultant cohomology is called the Poisson cohomology of $M.$ Note that
the first cohomology group is the quotient

$H_{\Pi}^{1}(M)=\frac{P_{\Pi}(M)}{Ham_{\Pi}(M)},$

where $Ham_{\Pi}(M)$ is the space of Hamiltonian vector fields on $M$ and
$P_{\Pi}(M)$ is the space of Poisson vector fields, that is, the set of vector
fields $X$ on $M$ satisfying

(2) $[\\![X,\Pi]\\!]={\mathcal{L}}_{X}\Pi=0.$

In the particular case when the morphism $\Pi^{\\#}:T^{*}M\to TM$ is an
isomorphim, we can consider the $2$-form $\Omega$ on $M$ given by

$\Omega(x)(u,v)=\langle(\Pi^{\\#})^{-1}(u),v\rangle$

for $x\in M$ and $u,v\in T_{x}M.$ The $2$-form $\Omega$ is a symplectic
structure on $M$, that is, $\Omega$ is closed and non-degenerate.

In the general case, we have a generalized foliation ${\mathcal{F}}$ on $M,$
whose characteristic space at the point $x\in M$ is

${\mathcal{F}}(x)=\Pi^{\\#}(T_{x}^{*}M),$

and the Poisson structure induces a symplectic structure on the leaves of this
foliation. $\mathcal{F}$ is called the symplectic foliation of $M$.

###  2.2.  Poisson-Lie groups

Poisson-Lie groups are important examples of Poisson manifolds.

A Poisson structure $\Pi$ on a Lie group $G$ is said to be multiplicative  if
the multiplication $m:G\times G\to G$ is a Poisson map, when on the product
manifold $G\times G$ we consider the standard product Poisson structure, that
is,

$\Pi(gh)=(L_{g})_{\ast}(\Pi(h))+(R_{h})_{\ast}(\Pi(g))$

for $g,h\in G$. In these conditions, $\Pi$ is a Poisson-Lie structure on $G$.

In such a case, we have that $\Pi(e)=0$, with $e$ the identity element in $G$.
Moreover, if ${\mathfrak{g}}$ is the Lie algebra of $G$, the differential of
$\Pi$ at the identity $e$, $\mathrm{d}_{e}\Pi$, induces an adjoint $1$-cocycle

$\delta=\mathrm{d}_{e}\Pi\colon{\mathfrak{g}}\to\wedge^{2}{\mathfrak{g}}.$

This means that

$\delta[\xi,\eta]_{\mathfrak{g}}=[\xi,\delta\eta]_{\mathfrak{g}}-[\eta,\delta\xi]_{\mathfrak{g}},$

for all $\xi,\eta\in{\mathfrak{g}}$, where $[\cdot,\cdot]_{\mathfrak{g}}$ also
denotes the algebraic Schouten bracket, that is, the natural extension of the
Lie bracket $[\cdot,\cdot]_{\mathfrak{g}}$ on ${\mathfrak{g}}$ making on
$\oplus_{k}\wedge^{k}\mathfrak{g}$ a graded Lie algebra. In addition, (1)
implies that the dual morphism of $\delta$

$\delta^{*}=[\cdot,\cdot]_{\mathfrak{g}^{*}}:\wedge^{2}{\mathfrak{g}}^{*}\to{\mathfrak{g}}^{*}$

is a Lie bracket on ${\mathfrak{g}}^{*}.$

In other words, the pair
$(({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}}),({\mathfrak{g}}^{*},[\cdot,\cdot]_{\mathfrak{g}^{*}}))$
is a Lie bialgebra. Conversely, if
$(({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}},({\mathfrak{g}}^{*},[\cdot,\cdot]_{\mathfrak{g}^{*}}))$
is a Lie bialgebra and $G$ is a connected and simply-connected Lie group with
Lie algebra $({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}})$ then there exists
a unique Poisson-Lie structure $\Pi$ on $G$ such that the Lie bialgebra
associated with $\Pi$ is just

$(({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}}),({\mathfrak{g}}^{*},[\cdot,\cdot]_{\mathfrak{g}^{*}})).$

A first example of a non-trivial Poisson-Lie structure is the linear Poisson
structure $\Pi_{LP}$ on the dual space ${\mathfrak{g}}^{*}$ of a Lie algebra
$({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}}),$ which is characterized as
follows. If $F,G\in C^{\infty}({\mathfrak{g}}^{*})$ then

(3)
$\Pi_{LP}(\mu)(\mathrm{d}F(\mu),\mathrm{d}G(\mu))=\mu[\mathrm{d}F(\mu),\mathrm{d}G(\mu)]_{\mathfrak{g}},\mbox{
for }\mu\in{\mathfrak{g}}^{*}.$

Note that $\mathrm{d}F(\mu),\mathrm{d}G(\mu)\in
T_{\mu}^{*}{\mathfrak{g}}^{*}\cong{\mathfrak{g}}.$ The couple
$({\mathfrak{g}}^{*},\Pi_{LP})$ is an abelian Poisson-Lie group.

Other interesting examples of Poisson-Lie groups may be obtained from
solutions of the generalized Yang-Baxter equation on a Lie algebra or
$r$-matrices. More precisely, if $G$ is a Lie group with Lie algebra
$\mathfrak{g}$ and $r\in\wedge^{2}\mathfrak{g}$ is a solution of the
generalized Yang-Baxter equation, that is, $[r,r]_{\mathfrak{g}}$ is ad-
invariant, then

$\Pi=r^{l}-r^{r}$

is a Poisson-Lie structure on $G$. Here, if $P\in\wedge^{k}\mathfrak{g}$ then
$P^{l}$ (resp. $P^{r}$) is the left-invariant (resp. right-invariant)
$k$-vector field on $G$ whose value at the identity element of $G$ is $P$.
Note that the adjoint $1$-cocycle
$\delta:\mathfrak{g}\to\wedge^{2}\mathfrak{g}$ associated with a $r$-matrix is
given by

$\delta(\xi)=\mbox{ ad}_{\xi}r,\mbox{ for }\xi\in\mathfrak{g}.$

More examples of (non-abelian) Poisson-Lie groups will appear throughout the
paper.

###  2.3. Modular vector field of a Poisson manifold

Let $(M,\Pi)$ be an orientable Poisson manifold. Given a volume form $\Phi$,
we consider the vector field $\mathcal{M}_{\Phi}^{\Pi}\in{\mathfrak{X}}(M)$
given by

(4) $\mathcal{M}_{\Phi}^{\Pi}(H)=\mbox{div}_{\Phi}(X^{\Pi}_{H}),\mbox{ for
}H\in C^{\infty}(M),$

where $\mbox{div}_{\Phi}(X^{\Pi}_{H})$ is the divergence of the Hamiltonian
vector field $X^{\Pi}_{H}$ with respect to the volume form $\Phi,$ that is,

(5) $\mathcal{L}_{X^{\Pi}_{H}}\Phi=\mbox{div}_{\Phi}(X^{\Pi}_{H})\Phi.$

The vector field $\mathcal{M}_{\Phi}^{\Pi}$ is the modular vector field of
$(M,\Pi)$ with respect to $\Phi$. It is proved in [30] that
$\mathcal{M}_{\Phi}^{\Pi}$ is a Poisson vector field (see (2)). In addition,
if $\Phi^{\prime}=e^{F}\Phi$ is another positive volume form on $M$, then the
relation between $\mathcal{M}_{\Phi}^{\Pi}$ and
$\mathcal{M}_{\Phi^{\prime}}^{\Pi}$ is

(6) $\mathcal{M}_{\Phi^{\prime}}^{\Pi}=\mathcal{M}_{\Phi}^{\Pi}-X^{\Pi}_{F}.$

Therefore, the modular vector field induces a cohomology class
$[\mathcal{M}_{\Phi}^{\Pi}]$ in the first Poisson cohomology group of $M$,
which is called the modular class of $M$. The Poisson manifold is said to be
unimodular if the modular class vanishes, i.e.,
$\mathcal{M}_{\Phi}^{\Pi}=X^{\Pi}_{F}$, for $F\in C^{\infty}(M)$. From (6), it
is possible to deduce the following result.

######  Proposition 2.1.

Let $(M,\Pi)$ be a Poisson manifold. Then, $M$ is unimodular if and only if
there exists a volume form $\Phi$ which is preserved by all Hamiltonian vector
fields. In fact, if $M$ is unimodular and
$\mathcal{M}_{\Phi^{\prime}}^{\Pi}=X^{\Pi}_{F}$, for a volume form
$\Phi^{\prime}$ and a real function ${F\in C^{\infty}(M),}$ then the volume
form $\Phi=e^{F}\Phi^{\prime}$ is preserved by all Hamiltonian vector fields.

######  Example 2.2.

Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}})$ be a Lie algebra of
dimension $n$. Then, the modular character
$\mathcal{M}_{\mathfrak{g}}\in\mathfrak{g}^{*}$ of $\mathfrak{g}$ is the trace
of the adjoint action

$\mathcal{M}_{\mathfrak{g}}(\xi)=\mbox{Tr}(\mbox{ad}_{\xi}),\mbox{ for
$\xi\in{\mathfrak{g}}.$}$

So, if $\\{e_{\alpha}\\}$ is a basis of $\mathfrak{g}$ with dual basis
$\\{e^{\alpha}\\}$, we have

$\mathcal{M}_{\mathfrak{g}}=c^{\beta}_{\alpha\beta}e^{\alpha},$

where
$[e_{\alpha},e_{\beta}]_{\mathfrak{g}}=c_{\alpha\beta}^{\gamma}e_{\gamma}$.

Now, consider the Lie-Poisson structure $\Pi_{LP}$ on $\mathfrak{g}^{*}$ given
by (3). If $(x_{\gamma})$ are global coordinates on ${\mathfrak{g}}^{*}$
induced by the basis $\\{e^{\gamma}\\}$ of ${\mathfrak{g}}^{*},$ it follows
that

$\Pi_{LP}=\frac{1}{2}c_{\alpha\beta}^{\gamma}x_{\gamma}\frac{\partial}{\partial
x_{\alpha}}\wedge\frac{\partial}{\partial x_{\beta}}.$

So, the modular vector field $\mathcal{M}^{\Pi_{LP}}_{\Phi}$ with respect to
the volume form $\Phi=\mathrm{d}x_{1}\wedge\ldots\wedge\mathrm{d}x_{n}$ is

$\mathcal{M}^{\Pi_{LP}}_{\Phi}=c^{\beta}_{\alpha\beta}\frac{\partial}{\partial
x_{\alpha}}.$

In fact, following [22], one may prove that the linear Poisson structure
$\Pi_{LP}$ on ${\mathfrak{g}}^{*}$ is unimodular if and only if the Lie
algebra ${\mathfrak{g}}$ is unimodular (that is, the modular character of
${\mathfrak{g}}$ is zero).

######  Example 2.3.

Let $(G,\Pi)$ be a connected Poisson-Lie group of dimension $n$ with Lie
bialgebra $(\mathfrak{g},\mathfrak{g}^{*})$. If
$\mathcal{M}_{\mathfrak{g}}\in\mathfrak{g}^{*}$ (resp.
$\mathcal{M}_{\mathfrak{g}^{*}}\in\mathfrak{g}$) is the modular character of
$\mathfrak{g}$ (resp. $\mathfrak{g}^{*}$), then the modular vector field
associated with a left-invariant volume form $\nu^{l}$ on $G$ is

(7)
$\mathcal{M}^{\Pi}_{\nu^{l}}=\frac{1}{2}(\mathcal{M}_{\mathfrak{g}^{*}}^{l}+\mathcal{M}_{\mathfrak{g}^{*}}^{r}+\Pi^{\sharp}(\mathcal{M}_{\mathfrak{g}}^{r}))$

(see [13]). Note that if $G$ is abelian and isomorphic to a vector space $V$,
then $\Pi$ is multiplicative if it is linear. Thus, since $V^{*}$ is a Lie
algebra and the Lie algebra in $V$ is trivial, we recover Example 2.2.

Now, let $f_{0}:G\to\mathbb{R}$ be the real $C^{\infty}$-function on $G$ given
by

(8) $f_{0}(g):=\mbox{det }(Ad^{n}_{g}),\mbox{ for }g\in G,$

where $Ad^{n}:G\times\wedge^{n}\mathfrak{g}\to\wedge^{n}\mathfrak{g}$ is the
adjoint action of $G$ on $\wedge^{n}\mathfrak{g}.$ Note that $f_{0}(e)=1$ and,
thus, since $f_{0}$ is not zero at every point of $G$ and $G$ is connected, we
deduce that $f_{0}(g)>0$, for every $g\in G$. Moreover,
$\mathcal{M}_{\mathfrak{g}}^{r}=\mathrm{d}(\log f_{0})$ (see [13]) and

$[\mathcal{M}^{\Pi}_{\nu^{l}}]=[\frac{1}{2}(\mathcal{M}_{\mathfrak{g}^{*}}^{l}+\mathcal{M}_{\mathfrak{g}^{*}}^{r}+\Pi^{\sharp}(\mathrm{d}(\log
f_{0})))]=[\frac{1}{2}(\mathcal{M}_{\mathfrak{g}^{*}}^{l}+\mathcal{M}_{\mathfrak{g}^{*}}^{r})].$

So, using that $\Pi(e)=0$, it is easy to prove that the Poisson manifold
$(G,\Pi)$ is unimodular if and only if $\mathcal{M}_{\mathfrak{g}^{*}}=0$,
that is, the dual Lie algebra $\mathfrak{g}^{*}$ is unimodular. Moreover, if
$(G,\Pi)$ is unimodular it follows that the modular vector field
$\mathcal{M}^{\Pi}_{\nu^{l}}$ is just the Hamiltonian vector field of the real
function $\frac{1}{2}\log f_{0}$, that is,

$\mathcal{M}^{\Pi}_{\nu^{l}}=\frac{1}{2}X^{\Pi}_{(\log f_{0})}.$

##  3\. Unimodularity of Poisson-Lie groups and preservation of volume forms

Let $(G,\Pi)$ be a Poisson-Lie group of dimension $n$, $H:G\to\mathbb{R}$ a
Hamiltonian function and $\nu\in\wedge^{n}\mathfrak{g}^{*},$ with $\nu\not=0.$
Then, a (positive) volume form on $G$ is given by

$\Phi=e^{\sigma}\nu^{l},\mbox{ with }\sigma\in C^{\infty}(G).$

Moreover, using (4), (5) and (7), we deduce that

(9)
$\begin{array}[]{rcl}{\mathcal{L}}_{X_{H}^{\Pi}}\Phi&=&e^{\sigma}(X_{H}^{\Pi}(\sigma)+{\mathcal{M}}_{\nu^{l}}^{\Pi}(H))\nu^{l}\\\\[5.0pt]
&=&e^{\sigma}(X_{H}^{\Pi}(\sigma-
log\sqrt{f_{0}})+\frac{1}{2}({\mathcal{M}}_{\mathfrak{g}^{*}}^{l}(H)+{\mathcal{M}}^{r}_{\mathfrak{g}^{*}}(H)))\nu^{l},\end{array}$

with $f_{0}:G\to\mathbb{R}$ the real $C^{\infty}$-function on $G$ defined in
(8).

So, from (9), it follows that

######  Theorem 3.1.

Let $(G,\Pi)$ be a connected Poisson-Lie group and $H:G\to\mathbb{R}$ a
Hamiltonian function. Then, the Hamiltonian vector field $X_{H}^{\Pi}$
preserves a volume form on $G$ if and only if there exists $\sigma\in
C^{\infty}(G)$ such that

$X_{H}^{\Pi}(\sigma-
log\sqrt{f_{0}})+\frac{1}{2}({\mathcal{M}}_{\mathfrak{g}^{*}}^{l}(H)+{\mathcal{M}}^{r}_{\mathfrak{g}^{*}}(H))=0.$

Moreover, in such a case, if $\nu\in\wedge^{n}\mathfrak{g}^{*},$ with
$\nu\not=0$, then the volume form $e^{\sigma}\nu^{l}$ is preserved by
$X_{H}^{\Pi}.$

######  Remark 3.2.

If $g\in G$ is a singular point of $X_{H}^{\Pi}$ and $X_{H}^{\Pi}$ preserves a
volume form on $G$ then, from Theorem 3.1, we deduce that

${\mathcal{M}}^{l}_{\mathfrak{g}^{*}}(g)(H)+{\mathcal{M}}^{r}_{\mathfrak{g}^{*}}(g)(H)=0.$

######  Remark 3.3.

If $H\in C^{\infty}(G)$ is a first integral of vector field
${\mathcal{M}}_{\mathfrak{g}^{*}}^{l}+{\mathcal{M}}^{r}_{\mathfrak{g}^{*}}$,
then the volume form $\sqrt{f_{0}}v^{l}$ is preserved by the Hamiltonian
vector field $X_{H}^{\Pi}$.

Using Theorem 3.1, we also obtain the following result

######  Corollary 3.4.

Let $(G,\Pi)$ be a connected Poisson-Lie group of dimension $n$ such that the
dual Lie algebra $\mathfrak{g}^{*}$ is unimodular (that is, the Poisson-Lie
structure $\Pi$ is unimodular). If $\nu\in\wedge^{n}\mathfrak{g}^{*}$ with
$\nu\neq 0$, then the volume form $\sqrt{f_{0}}\nu^{l}$ is preserved by all
Hamiltonian vector fields, where $f_{0}:G\to\mathbb{R}$ is the real function
on $G$ given in (8).

Next, we will prove a converse of this result for a particular kind of
Hamiltonian functions: the Morse functions at the identity element of the
Poisson-Lie group.

######  Definition 3.5.

Let $G$ be a Lie group with identity element $e$. A function $H\in
C^{\infty}(G)$ is said to be Morse at $e$ if

  * i)

$e$ is a singular point of $H$, that is, $\mathrm{d}H(e)=0$.

  * ii)

$(\mathrm{Hess}\,H)(e)$ is nondegenerate, with
$(\mathrm{Hess}\,H)(e)\colon\mathfrak{g}\times\mathfrak{g}\to\mathbb{R}$ the
Hessian of $H$ at $e$, i.e., the symmetric bilinear form on $\mathfrak{g}$
given by

(10)
$(\mathrm{Hess}\,H)(e)(\xi,\eta)=\xi(X^{\eta}(H)),\qquad\xi,\eta\in\mathfrak{g},$

where $X^{\eta}\in\mathfrak{X}(G)$ is an arbitrary vector field such that
$X^{\eta}(e)=\eta$.

######  Remark 3.6.

  1. i)

We recall that a smooth function $H:G\to\mathbb{R}$ is Morse if the Hessian of
$H$ at each singular point of $H$ is nondegenerate. More details on
nondegenerate singular points and the definition of the Hessian can be found,
for instance, in [24, Part I]).

  2. ii)

A smooth function $H\colon G\to\mathbb{R}$ is a contrast function on $G$ if
$H(e)=0$ and $\mathrm{d}H(e)=0.$ This kind of functions plays an important
role in information geometry (see [15]). Since the Hamiltonian dynamics
induced by $H:G\to\mathbb{R}$ coincides with that induced by the function
$H-H(e)$, we can assume (without the loss of generality) that for a Morse
function at $e$, $H:G\to\mathbb{R},$ we have that $H(e)=0.$ So, a Morse
function at $e$ is a contrast function on $G$.

Now, we will prove the result announced above.

######  Theorem 3.7.

Let $(G,\Pi)$ be a connected Poisson-Lie group of dimension $n$ and $H\in
C^{\infty}(G)$ be a function on $G$ which is Morse at $e$. If $X^{\Pi}_{H}$
preserves a volume form $\Phi$ on $G$, then the dual Lie algebra
$\mathfrak{g}^{*}$ is unimodular and, therefore, the volume form
$\sqrt{f_{0}}\nu^{l}$, with $\nu\in\wedge^{n}\mathfrak{g}^{*}$ and
$\nu\not=0,$ is preserved by all Hamiltonian vector fields (in particular, for
the Hamiltonian vector field of $H$).

  * Proof. Let $0\neq\nu\in\wedge^{n}\mathfrak{g}^{*}$ and $\nu^{l}\in\Omega^{n}(G)$ be the corresponding left-invariant volume form on $G$. Suppose that there is a function $\sigma\in C^{\infty}(G)$ such that the volume form $\Phi=e^{\sigma}\nu^{l}$ is preserved by the Hamiltonian vector field $X^{\Pi}_{H}$, that is,

$\mathcal{L}_{X^{\Pi}_{H}}(e^{\sigma}\nu^{l})=0.$

Then, from (9), it follows that

$X^{\Pi}_{H}(\sigma)+\mathcal{M}^{\Pi}_{\nu^{l}}(H)=0,$

and, using the skew-symmetry of $\Pi$,

$\mathcal{M}^{\Pi}_{\nu^{l}}(H)=X^{\Pi}_{\sigma}(H)$

Thus, for any $\xi\in\mathfrak{g}$,

(11)
$(\mathrm{Hess}\,H)(e)(\xi,\mathcal{M}^{\Pi}_{\nu^{l}}(e))=\xi(\mathcal{M}^{\Pi}_{\nu^{l}}(H))=\xi(X^{\Pi}_{\sigma}(H))=(\mathrm{Hess}\,H)(e)(\xi,X^{\Pi}_{\sigma}(e))=0\mbox{
for all }\xi\in\mathfrak{g},$

because $X^{\Pi}_{F}(e)=0$ for any $F\in C^{\infty}(G)$. Since
$(\mathrm{Hess}\,H)(e)$ is nondegenerate, $\mathcal{M}^{\Pi}_{\nu^{l}}(e)=0$.
But, from (7),
$\mathcal{M}^{\Pi}_{\nu^{l}}(e)=\mathcal{M}_{\mathfrak{g}^{*}}$, the modular
character of $\mathfrak{g}^{*}$. So, the dual Lie algebra $\mathfrak{g}^{*}$
is unimodular. This, using Corollary 3.4, ends the proof of the result. $\Box$

######  Remark 3.8.

Let $(G,\Pi)$ be a connected Poisson-Lie group and $H\in C^{\infty}(G)$ be a
Hamiltonian function on $G$ such that the identity element of $G$ is a
singular point of $H$. Then, following the proof of Theorem 3.7, we deduce an
interesting result. Namely, if the Hamiltonian vector field of $H$ preserves a
volume form, we have that the modular character of the dual Lie algebra
$\mathfrak{g}^{*}$ belongs to the kernel of $(Hess\,H)(e).$

##  4\. Examples

###  4.1. Kozlov’s unimodularity result

Let $\mathfrak{g}$ be a real Lie algebra of dimension $n$ and
$\mathfrak{g}^{*}$ be the dual vector space endowed with the Lie-Poisson
structure $\Pi_{LP}$ (see Example 2.2).

If $I:\mathfrak{g}^{*}\times\mathfrak{g}^{*}\to\mathbb{R}$ is a symmetric
$\mathbb{R}$-bilinear form then we can consider the Hamiltonian function
$H_{I}:\mathfrak{\mathfrak{g}}^{*}\to\mathbb{R}$ given by

$H_{I}(\mu)=\frac{1}{2}I(\mu,\mu),\mbox{ for }\mu\in\mathfrak{g}^{*}.$

So, if $\\{e_{\gamma}\\}$ is a basis of $\mathfrak{g}$ with dual basis
$\\{e^{\gamma}\\}$ for $\mathfrak{g}^{*}$ and $\\{x_{\gamma}\\}$ the
corresponding global coordinates of $\mathfrak{g}^{*}$, we have that

$H_{I}(x)=\frac{1}{2}I^{\alpha\beta}x_{\alpha}x_{\beta}.$

It is clear that the identity element of $\mathfrak{g}^{*}$ as an abelian
Poisson-Lie group (that is, the zero vector $0$) is a singular point of
$H_{I}.$ Moreover, it is easy to prove that $H_{I}$ is Morse at $0$ if and
only if $I$ is nondegenerate. So, if $I$ is nondegenerate then, using
Corollary 3.4 and Theorem 3.7, we deduce that the Hamiltonian vector field
$X_{H_{I}}^{\Pi_{LP}}$ preserves a volume form on $\mathfrak{g}^{*}$ if only
if $\mathfrak{g}$ is unimodular. Moreover, in such a case, the function
$f_{0}:\mathfrak{g}^{*}\to\mathbb{R}$ given by (8) identically vanishes which
implies that the Hamiltonian vector field $X_{H_{I}}^{\Pi_{LP}}$ preserves the
volume form $\Phi=dx_{1}\wedge\dots\wedge dx_{n}.$

This extends a previous result by Kozlov [18] for the particular case when
$H_{I}$ is a Hamiltonian function of kinetic type, that is, the symmetric
bilinear form $I$ is positive definite.

###  4.2. Contrast functions on Lie groups

We recall that a smooth function $H\colon G\to\mathbb{R}$ on a Lie group $G$
with with identity element $e$, is a contrast function if $H(e)=0$ $and$
$\mathrm{d}H(e)=0.$ It is obvious that if $\Pi$ is a Poisson-Lie structure on
$G$ then $X^{\Pi}_{H}=X^{\Pi}_{H-H(e)}$ and so the functions $H$ and $H-H(e)$
define the same dynamics. So, we can always consider the contrast function
$H-H(e)$, when $H$ is a Morse function at $e$. This kind of functions plays an
important role in information geometry.

Now, assume that $\iota:G\to\mathbb{R}^{n}$ is an embedding of the Lie group
$G$ in $\mathbb{R}^{n}$, with $\iota(e)=0$. If $\langle\cdot,\cdot\rangle$ is
a non-degenerate bilinear symmetric form on $\mathbb{R}^{n}$, we can consider
the Hamiltonian function $H:G\to\mathbb{R}$ defined by

$H=H_{\langle\cdot,\cdot\rangle}\circ\iota:G\to\mathbb{R},$

with $H_{\langle\cdot,\cdot\rangle}:\mathbb{R}^{n}\to\mathbb{R}$ given by

$H_{\langle\cdot,\cdot\rangle}(x)=\frac{1}{2}\langle x,x\rangle,\mbox{ for
}x\in\mathbb{R}^{n}.$

Then, it is clear that the identity element $e$ is a singular point of $H$ and
$\mathrm{d}H(e)=0$, so $H$ is a contrast function. In addition, one can show
that if the restriction of the symmetric bilinear form
$\langle\cdot,\cdot\rangle$ to the subspace $\mathfrak{g}=T_{e}G\subseteq
T_{0}\mathbb{R}^{n}\cong\mathbb{R}^{n}$ is nondegenerate then $e$ is a non-
degenerate singular point for $H$. In particular, if
$\langle\cdot,\cdot\rangle$ is a definite positive symmetric bilinear form on
$\mathbb{R}^{n}$ then $H$ is Morse at $e$. Thus, if $G$ is a matrix Lie group
(a Lie subgroup of $\mathrm{GL}(n,\mathbb{R})$) then the smooth function
$H:G\to\mathbb{R}$ given by

$H(A)=\mbox{Tr}\,((Id-A)(Id-A^{t})),\qquad A\in G,$

is Morse at $Id\in G.$ Here $Id$ is the identity matrix in
$\mathrm{GL}(n,\mathbb{R}).$

So, from Corollary 3.4 and Theorem 3.7, it follows that, for a Poisson-Lie
structure $\Pi$ on $G,$ the Hamiltonian vector field of $H$ preserves a volume
form on $G$ if and only if $\Pi$ is unimodular.

######  Remark 4.1.

In the particular case when $G=\mathrm{GL}(n,\mathbb{R})$, the previous
function $H$ was used in [15] as a metric contrast function on
$\mathrm{GL}(n,\mathbb{R})$. We remark again that this kind of function plays
an important role in information geometry (see [15], for more details).

###  4.3. A Hamiltonian system on the Poisson-Lie group
$\mathrm{SL}(2,\mathbb{R})$

Let $\mathrm{SL}(2,\mathbb{R})$ be the special linear group of order $2$, that
is, the Lie subgroup of $\mathrm{GL}(2,\mathbb{R})$ defined by

(12) $\mathrm{SL(2,\mathbb{R})}=\left\\{A=\begin{pmatrix}a_{11}&&a_{12}\\\
a_{21}&&a_{22}\\\ \end{pmatrix}\in\mathrm{GL}(2,\mathbb{R})\,|\,\det
A=1\right\\}.$

The vector space $\mathfrak{gl}(2,\mathbb{R})$ of matrices of order $2$ with
real coefficients is the Lie algebra of $\mathrm{GL}(2,\mathbb{R})$ and

$\mathfrak{sl}(2,\mathbb{R})=\\{A\in\mathfrak{gl}(2,\mathbb{R})\,|\mbox{
Tr}\,A=0\\}$

is the Lie algebra of $\mathrm{SL}(2,\mathbb{R}).$

We can consider the basis
$\mathcal{B}=\\{J_{3},J_{+},J_{-},\displaystyle\frac{1}{2}Id\\}$ of
$\mathfrak{gl}(2,\mathbb{R})$, given by

$J_{3}=\begin{pmatrix}1&&0\\\ 0&&-1\\\ \end{pmatrix},\qquad
J_{+}=\begin{pmatrix}0&&1\\\ 0&&0\\\ \end{pmatrix},\qquad
J_{-}=\begin{pmatrix}0&&0\\\ 1&&0\\\ \end{pmatrix}.$

Note that $\\{J_{3},J_{+},J_{-}\\}$ is a basis of
${\mathfrak{g}}=\mathfrak{sl}(2,\mathbb{R})$ and, moreover, the non-zero
commutation relations for the basis ${\mathcal{B}}$ read

(13)
$[J_{3},J_{-}]_{\mathfrak{g}}=-2J_{-},\qquad[J_{3},J_{+}]_{\mathfrak{g}}=2J_{+},\quad[J_{+},J_{-}]_{\mathfrak{g}}=J_{3}.$

The basis of left-invariant vector fields on $\mathrm{GL}(2,\mathbb{R})$
associated with $\mathcal{B}$ is just

$\displaystyle{\frac{1}{2}Id^{l}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{(}a_{11}\frac{\partial}{\partial{a_{11}}}+a_{12}\frac{\partial}{\partial{a_{12}}}+a_{21}\frac{\partial}{\partial{a_{21}}}+a_{22}\frac{\partial}{\partial{a_{22}}}\Big{)},$
$\displaystyle J_{3}^{l}$ $\displaystyle=$ $\displaystyle
a_{11}\frac{\partial}{\partial{a_{11}}}-a_{12}\frac{\partial}{\partial{a_{12}}}+a_{21}\frac{\partial}{\partial{a_{21}}}-a_{22}\frac{\partial}{\partial{a_{22}}},$
$\displaystyle J_{+}^{l}$ $\displaystyle=$ $\displaystyle
a_{11}\frac{\partial}{\partial{a_{12}}}+a_{21}\frac{\partial}{\partial{a_{22}}},$
$\displaystyle J_{-}^{l}$ $\displaystyle=$ $\displaystyle
a_{12}\frac{\partial}{\partial{a_{11}}}+a_{22}\frac{\partial}{\partial{a_{21}}}.$

The dual basis of $\mathcal{B}$ is
$\\{(J^{3})^{l},(J^{+})^{l},(J^{-})^{l},\mathrm{d}(\det A)\\}$,
$\\{J^{3},J^{+},J^{-}\\}$ being the dual basis of $\\{J_{3},J_{+},J_{-}\\}$.

Next, consider the Poisson-Lie structure on $\mathrm{GL}(2,\mathbb{R})$ given
by

(14)
$\begin{split}&\\{a_{11},a_{12}\\}=a_{11}a_{12},\quad\\{a_{11},a_{21}\\}=a_{11}a_{21},\quad\\{a_{11},a_{22}\\}=2a_{12}a_{21},\\\
&\\{a_{12},a_{21}\\}=0,\quad\\{a_{12},a_{22}\\}=a_{12}a_{22},\quad\\{a_{21},a_{22}\\}=a_{21}a_{22}.\end{split}$

This is indeed a Poisson structure on $\mathrm{SL}(2,\mathbb{R}),$ since $\det
A=a_{11}a_{22}-a_{12}a_{21}$ is a Casimir for this structure (see, for
instance, [26]). For this Poisson-Lie group structure, we have that

(15) $A=\begin{pmatrix}a_{11}&0\\\ 0&a_{22}\end{pmatrix}\in GL(2,\mathbb{R})$

is a singular point of $\Pi$.

The previous Poisson-Lie structure is the one defined by the so-called
(standard) Drinfel’d-Jimbo $r$-matrix and, interestingly, it is the same as
the Sklyanin bracket appeared in the singular value decomposition in [25] for
Toda-SVD in [10]. With respect to the basis $\\{J_{3},J_{+},J_{-}\\}$, the
(standard) Drinfel’d-Jimbo $r$-matrix is given by

(16) $r=J_{-}\wedge J_{+}\in\wedge^{2}\mathfrak{sl}(2,\mathbb{R}).$

In order to compute the tangent Lie algebra to this Poisson-Lie structure
using that, for coboundary structures, we have

$\delta(\xi)=\mathrm{ad}_{\xi}(r),$

and, from (13) and (16),

$\delta(J_{3})=0,\qquad\delta(J_{\pm})=J_{3}\wedge J_{\pm}.$

Therefore,

(17)
$[J^{3},J^{\pm}]_{\mathfrak{g}^{*}}=J^{\pm},\qquad[J^{+},J^{-}]_{\mathfrak{g}^{*}}=0.$

So, the modular character $\mathcal{M}_{\mathfrak{g}^{*}}$ of the dual Lie
algebra can be computed from (17) and reads

$\mathcal{M}_{\mathfrak{g}^{*}}=2J_{3}\neq 0.$

This implies that the Poisson manifold
$(\mathrm{SL(2,\mathbb{R})},\\{\cdot,\cdot\\})$ is not unimodular. In fact,
the modular vector field of the Poisson manifold
$(\mathrm{SL(2,\mathbb{R})},\\{\cdot,\cdot\\})$ is

(18)
$\frac{1}{2}(\mathcal{M}_{\mathfrak{g}^{*}}^{r}\mathcal{+}{\mathcal{M}}_{\mathfrak{g}^{*}}^{l})=2a_{11}\frac{\partial}{\partial
a_{11}}-2a_{22}\frac{\partial}{\partial a_{22}},$

which once more proves the Poisson structure is not unimodular. Note that
modular character $\mathcal{M}_{\mathfrak{g}}=0$.

Now, we consider the Hamiltonian function $H\in
C^{\infty}(\mathrm{GL}(2,\mathbb{R}))$ given by

(19)
$H=\frac{1}{2}\operatorname{Tr}(A^{T}A)=\frac{1}{2}(a_{11}^{2}+a_{12}^{2}+a_{21}^{2}+a_{22}^{2})=\frac{1}{2}\sum_{i,j=1}^{2}a_{ij}^{2},$

Note that this Hamiltonian function generates the Toda-SVD flow defined by the
Poisson bracket (14) (see [25]).

Since
$\mathrm{d}H=a_{11}\mathrm{d}a_{11}+a_{12}\mathrm{d}a_{12}+a_{21}\mathrm{d}a_{21}+a_{22}\mathrm{d}a_{22}$,
it follows that

$\mathrm{d}H(Id)\not=0.$

Therefore, we cannot use Theorem 3.7 but nevertheless we can use Remark 3.2.

Indeed, the Hamiltonian vector field of $H$ is given by

$\displaystyle X_{H}^{\Pi}$ $\displaystyle=$
$\displaystyle\left(-a_{11}\left(a_{12}^{2}+a_{21}^{2}\right)-2(a_{22}-1)a_{12}a_{21}\right)\frac{\partial}{\partial
a_{11}}+\left((a_{11}-1)a_{11}a_{22}-(a_{22}-1)a_{12}a_{22}\right)\frac{\partial}{\partial
a_{12}}$
$\displaystyle+\left((a_{11}-1)a_{11}a_{12}-(a_{22}-1)a_{12}a_{22}\right)\frac{\partial}{\partial
a_{21}}+\left(2(a_{11}-1)a_{12}a_{21}+a_{22}\left(a_{12}^{2}+a_{21}^{2}\right)\right)\frac{\partial}{\partial
a_{22}}$ $\displaystyle=$ $\displaystyle X^{11}\frac{\partial}{\partial
a_{11}}+X^{12}\frac{\partial}{\partial a_{12}}+X^{21}\frac{\partial}{\partial
a_{21}}+X^{22}\frac{\partial}{\partial a_{22}},$

which is the flow of Toda-SVD in dimension two. In addition, if $a\not=0$
then, from (15),

$A=\left(\begin{array}[]{cc}a&0\\\
0&\displaystyle\frac{1}{a}\end{array}\right)\in\mathrm{SL}(2,\mathbb{R})$

is an equilibrium point for the Hamiltonian vector field $X_{H}^{\Pi}$ on
$\mathrm{SL}(2,\mathbb{R})$. Moreover, using (18) and (19), we deduce that

${\mathcal{M}}^{l}_{\mathfrak{g}^{*}}(A)(H)+{\mathcal{M}}^{r}_{\mathfrak{g}^{*}}(A)(H)=\frac{4}{a^{2}}(a^{2}-1)(a^{2}-a+1).$

In particular, if $a\not=\pm 1$ it follows that

${\mathcal{M}}^{l}_{\mathfrak{g}^{*}}(A)(H)+{\mathcal{M}}^{r}_{\mathfrak{g}^{*}}(A)(H)\not=0,$

which, from Remark 3.2, implies that $X_{H}^{\Pi}$ does not preserve a global
volume form on $\mathrm{SL}(2,\mathbb{R})$.

###  4.4. A Hamiltonian system on the Poisson-Lie group
$S^{3}\cong\mathrm{SU}(2,\mathbb{C})$

Let $S^{3}$ be the unit sphere in $\mathbb{R}^{4}-\\{(0,0,0,0)\\}$. In
$\mathbb{R}^{4}-\\{(0,0,0,0)\\}$ one can consider standard global coordinates
$(x,y,z,t)$ and the Lie group structure with multiplication

$(x,y,z,t)(x^{\prime},y^{\prime},z^{\prime},t^{\prime})=(xx^{\prime}-yy^{\prime}-zz^{\prime}-tt^{\prime},xy^{\prime}+yx^{\prime}-zt^{\prime}+tz^{\prime},zx^{\prime}-ty^{\prime}+xz^{\prime}+yt^{\prime},zy^{\prime}+tx^{\prime}+xt^{\prime}-yz^{\prime}),$

which is just the Lie group structure identifying the quaternions $\mathbb{H}$
with $\mathbb{R}^{4}-\\{(0,0,0,0)\\}$. Note that the identity element is
$(1,0,0,0)$ and that $S^{3}$ is a closed normal Lie subgroup of
$\mathbb{R}^{4}-\\{(0,0,0,0)\\}$. If $\\{e_{1},e_{2},e_{3},e_{4}\\}$ is the
canonical basis in $\mathbb{R}^{4}-\\{(0,0,0,0)\\}$, then the associated left
invariant vector fields are

$\displaystyle e_{1}^{l}$ $\displaystyle=$ $\displaystyle
x\frac{\partial}{\partial x}+y\frac{\partial}{\partial
y}+z\frac{\partial}{\partial z}+t\frac{\partial}{\partial t}$ $\displaystyle
e_{2}^{l}$ $\displaystyle=$ $\displaystyle-y\frac{\partial}{\partial
x}+x\frac{\partial}{\partial y}-t\frac{\partial}{\partial
z}+z\frac{\partial}{\partial t}$ $\displaystyle e_{3}^{l}$ $\displaystyle=$
$\displaystyle-z\frac{\partial}{\partial x}+t\frac{\partial}{\partial
y}+x\frac{\partial}{\partial z}-y\frac{\partial}{\partial t}$ $\displaystyle
e_{4}^{l}$ $\displaystyle=$ $\displaystyle-t\frac{\partial}{\partial
x}-z\frac{\partial}{\partial y}+y\frac{\partial}{\partial
z}+x\frac{\partial}{\partial t}.$

Then, the non-zero commuting relations for this basis read

$[e_{2},e_{3}]_{\mathfrak{g}}=-[e_{3},e_{2}]_{\mathfrak{g}}=-2e_{4},\qquad[e_{2},e_{4}]_{\mathfrak{g}}=-[e_{4},e_{2}]_{\mathfrak{g}}=2e_{3},\qquad[e_{3},e_{4}]_{\mathfrak{g}}=[e_{4},e_{3}]_{\mathfrak{g}}=-2e_{2}.$

In fact, $\\{e_{2},e_{3},e_{4}\\}$ is a basis of the Lie algebra of the Lie
subgroup $S^{3}\cong\mathrm{SU}(2,\mathbb{C})$. On the other hand, there is a
Poisson structure $\Pi$ on $\mathbb{R}^{4}-\\{(0,0,0,0)\\}$ whose Poisson
bracket $\\{\cdot,\cdot\\}$ is characterized by the following conditions

$\displaystyle\\{x,y\\}=-(z^{2}+t^{2}),$
$\displaystyle\\{x,z\\}=yz,\quad\\{x,t\\}=yt,$
$\displaystyle\\{y,z\\}=-xz,\quad\\{y,t\\}=-xt.$

Indeed, $\Pi$ is a Poisson-Lie structure on $\mathbb{R}^{4}-\\{(0,0,0,0)\\}$
and

$\Pi=-(z^{2}+t^{2})\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial
y}+yz\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial
z}+yt\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial
t}-xz\frac{\partial}{\partial y}\wedge\frac{\partial}{\partial
z}-xt\frac{\partial}{\partial y}\wedge\frac{\partial}{\partial t}.$

We remark that

$\|\cdot\|:\mathbb{R}^{4}-\\{(0,0,0,0)\\}\to\mathbb{R},\;\;\;(x,y,z,t)\to\|(x,y,z,t)\|^{2}=x^{2}+y^{2}+z^{2}+t^{2}$

is a Casimir function and, thus, $\Pi$ induces a Poisson-Lie structure on
$S^{3},$ which we also denote by $\Pi.$ It is the standard Poisson-Lie
structure on $S^{3}\cong\mathrm{SU}(2,\mathbb{C})$ (see [26]). Moreover, using
the fact that the adjoint 1-cocycle $\delta$ satisfies

$[\xi^{l},\Pi]=(\delta\xi)^{l},\mbox{ for }\xi\in\mathfrak{su}(2),$

we deduce that

$\delta(e_{1})=\delta(e_{2})=0,\quad\delta(e_{3})=-e_{2}\wedge
e_{3},\quad\delta(e_{4})=-e_{2}\wedge e_{4},$

and for the Lie algebra structure on $\mathfrak{g}^{*}$ the non-zero
commutation relations read

$\displaystyle\,[e^{2},e^{3}]_{\mathfrak{g}^{*}}=-[e^{3},e^{2}]_{\mathfrak{g}^{*}}=-e^{3},$
$\displaystyle\,[e^{2},e^{4}]_{\mathfrak{g}^{*}}=-[e^{4},e^{2}]_{\mathfrak{g}^{*}}=-e^{4}.$

So, the modular character of the dual of the Lie algebra,
$\mathcal{M}_{\mathfrak{g}^{*}}$, is simply $-2e_{2}$. Therefore, the modular
vector field of the Poisson manifold $(S^{3},\Pi)$ is given by

(20)
$\frac{1}{2}(\mathcal{M}_{\mathfrak{g}^{*}}^{l}+\mathcal{M}_{\mathfrak{g}^{*}}^{r})=-2(x\frac{\partial}{\partial
y}-y\frac{\partial}{\partial x}).$

Thus, it follows that $(S^{3},\\{\cdot,\cdot\\})$ is not unimodular.

Now, it is easy to show that for any smooth functions
$F\colon\mathbb{R}\to\mathbb{R}$ and $G\colon\mathbb{R}^{2}\to\mathbb{R}$, the
function

$\begin{array}[]{rccl}H\colon&\mathbb{R}^{4}-\\{(0,0,0,0)\\}&\to&\mathbb{R}\\\
&(x,y,z,t)&\mapsto&F(x^{2}+y^{2})+G(z,t)\end{array}$

is a smooth first integral of
$\frac{1}{2}(\mathcal{M}_{\mathfrak{g}^{*}}^{l}+\mathcal{M}_{\mathfrak{g}^{*}}^{r})$.
In particular, the Hamiltonian function

$H\colon S^{3}\to\mathbb{R},\quad(x,y,z,t)\mapsto P(z,t),$

where $P$ is an arbitrary smooth function on $\mathbb{R}^{2}$, is a first
integral of
$\frac{1}{2}(\mathcal{M}_{\mathfrak{g}^{*}}^{l}+\mathcal{M}_{\mathfrak{g}^{*}}^{r})_{|S^{3}}$.
Therefore, using Theorem 3.1, we deduce that the Hamiltonian vector field
$X^{\Pi}_{H}$ preserves any left-invariant volume form on $S^{3}$.

###  4.5. A deformation of a conservative Lorenz system

In [4], the authors presented a method for obtaining integrable deformations
of Lie-Poisson bi-Hamiltonian systems. The initial data is a dynamical system
$\mathcal{D}$ on $\mathbb{R}^{n}$, which is bi-Hamiltonian with respect to two
compatible linear Poisson structures. Then, they construct a bi-Hamiltonian
deformation $\mathcal{D}_{\eta}$ of the dynamical system $\mathcal{D}$, whose
bi-Hamiltonian structure is provided by a pair of Poisson-Lie structures on a
non-abelian Lie group $G_{\eta}$. Under certain conditions, the limit $\eta\to
0$ of $\mathcal{D}_{\eta}$ is just the initial dynamical system $\mathcal{D}$.
The Poisson coalgebra approach to Hamiltonian systems was discussed in [5].

In what follows, we illustrate our results with two examples coming from the
Poisson-Lie deformation theory of Lie-Poisson bi-Hamiltonian systems.

As a first example, consider the integrable deformation of a conservative
Lorenz system studied in [3], whose equations of motion read

(21) $\begin{array}[]{rcl}\dot{x}&=&y,\\\
\dot{y}&=&\displaystyle\frac{x}{2}\left(4+\displaystyle\frac{e^{-2\eta(z+w)}-1}{\eta}+\eta(2x^{2}-y^{2})\right),\\\
\dot{z}&=&xy,\\\ \dot{w}&=&0.\end{array}$

This dynamical system can be written as a Hamiltonian system on the
4-dimensional Lie group $G_{\eta}\simeq B_{2}\times B_{2}$, where $B_{2}$ is
the unique connected simply connected non-abelian Lie group of dimension $2$.
We start by introducing exponential coordinates of the second kind on
$G_{\eta}$, defined by

(22) $g(x,y,z,w)=\exp(x\,X)\exp(y\,Y)\exp(z\,Z)\exp(w\,W).$

In terms of these coordinates the group law reads

(23) $\begin{array}[]{rcl}g(x,y,z,w)\cdot
g^{\prime}(x^{\prime},y^{\prime},z^{\prime},w^{\prime})&=&g\bigg{(}x+\displaystyle\frac{e^{-\eta(z+w)}}{2}\bigg{(}2x^{\prime}\cosh\left(\displaystyle\frac{\eta
w}{\sqrt{2}}\right)-\sqrt{2}y^{\prime}\sinh\left(\displaystyle\frac{\eta
w}{\sqrt{2}}\right)\bigg{)},\\\\[8.0pt]
&&y+e^{-\eta(z+w)}\bigg{(}y^{\prime}\cosh\left(\displaystyle\frac{\eta
w}{\sqrt{2}}\right)-\sqrt{2}x^{\prime}\sinh\left(\displaystyle\frac{\eta
w}{\sqrt{2}}\right)\bigg{)},z+z^{\prime},w+w^{\prime}\bigg{)}\end{array}$

where $\eta>0$ and the Poisson bivector field is given by

(24) $\Pi_{B_{2}\times
B_{2}}=\frac{e^{-2\eta(z+w)}-1+\eta^{2}(2x^{2}-y^{2})}{4\eta}\frac{\partial}{\partial
x}\wedge\frac{\partial}{\partial y}+\frac{y}{2}\frac{\partial}{\partial
x}\wedge\frac{\partial}{\partial z}+x\frac{\partial}{\partial
y}\wedge\frac{\partial}{\partial z}.$

$\Pi_{B_{2}\times B_{2}}$ defines a Poisson-Lie structure on $G_{\eta}$ and
the dynamical system (21) can be recovered by means of the Hamiltonian
function

(25) $\mathcal{H}=2(z-w)-x^{2}.$

The Lie algebra $\mathfrak{g}_{\eta}$ associated to the Lie group $G_{\eta}$
is characterized by the commutators

(26) $[X,Z]_{\eta}=\eta X,\quad[X,W]_{\eta}=\eta(X+Y),\quad[Y,Z]_{\eta}=\eta
Y,\quad[Y,W]_{\eta}=\eta\left(\frac{1}{2}X+Y\right),\quad[X,Y]_{\eta}=[Z,W]_{\eta}=0.$

To prove that the Poisson-Lie structure (24) is unimodular and therefore the
system (21) admits invariant volume forms, we need to compute the commutator
map $\delta_{B_{2}\times B_{2}}=d_{e}\Pi_{B_{2}\times
B_{2}}:{\mathfrak{g}}\to\wedge^{2}{\mathfrak{g}}$. Explicitly, we have

(27) $\delta_{B_{2}\times B_{2}}(X)=Y\wedge Z,\qquad\delta_{B_{2}\times
B_{2}}(Y)=\frac{1}{2}X\wedge Z,\qquad\delta_{B_{2}\times
B_{2}}(Z)=-\frac{1}{2}X\wedge Y,\qquad\delta_{B_{2}\times
B_{2}}(W)=-\frac{1}{2}X\wedge Y.$

From this cocommutator, we directly obtain that the Lie algebra
$\mathfrak{g}^{*}$ of the dual Poisson-Lie group is

(28)
$[\bar{X},\bar{Y}]_{\mathfrak{g}^{*}}=-\frac{1}{2}(\bar{Z}+\bar{W}),\qquad[\bar{X},\bar{Z}]_{\mathfrak{g}^{*}}=\frac{1}{2}\bar{Y},\qquad[\bar{Y},\bar{Z}]_{\mathfrak{g}^{*}}=\bar{X},\qquad[\bar{W},\bar{X}]_{\mathfrak{g}^{*}}=[\bar{W},\bar{Y}]_{\mathfrak{g}^{*}}=[\bar{W},\bar{Z}]_{\mathfrak{g}^{*}}=0.$

Therefore,
$\mathfrak{g}^{*}\simeq\mathfrak{sl}(2,\mathbb{R})\oplus\mathbb{R}$, which is
clearly unimodular since it is a direct sum of two unimodular Lie algebras.

To have an invariant volume form, we obtain the left invariant vector fields
as

(29) $\begin{array}[]{rcl}X^{l}&=&e^{-\eta(z+w)}\cosh(\displaystyle\frac{\eta
w}{\sqrt{2}})\frac{\partial}{\partial
x}-\sqrt{2}e^{-\eta(z+w)}\sinh(\displaystyle\frac{\eta
w}{\sqrt{2}})\frac{\partial}{\partial y}\\\\[6.0pt]
Y^{l}&=&-\displaystyle\frac{\sqrt{2}}{2}e^{-\eta(z+w)}\sinh(\displaystyle\frac{\eta
w}{\sqrt{2}})\displaystyle\frac{\partial}{\partial
x}+e^{-\eta(z+w)}\cosh\displaystyle(\frac{\eta
w}{\sqrt{2}})\displaystyle\frac{\partial}{\partial y}\\\\[6.0pt]
Z^{l}&=&\displaystyle\frac{\partial}{\partial z}\\\\[6.0pt]
W^{l}&=&\displaystyle\frac{\partial}{\partial w}\\\ \end{array}$

and the right invariant vector fields as

(30) $\begin{array}[]{rcl}X^{r}&=&\displaystyle\frac{\partial}{\partial
x}\\\\[6.0pt] Y^{r}&=&\displaystyle\frac{\partial}{\partial y}\\\\[6.0pt]
Z^{r}&=&-\eta x\displaystyle\frac{\partial}{\partial x}-\eta
y\frac{\partial}{\partial y}+\displaystyle\frac{\partial}{\partial
z}\\\\[6.0pt] W^{r}&=&-(\eta
x+\displaystyle\frac{\eta}{2}y)\frac{\partial}{\partial x}-(\eta x+\eta
y)\displaystyle\frac{\partial}{\partial
y}+\displaystyle\frac{\partial}{\partial w}.\\\ \end{array}$

For invariant $1$-forms we have that

(31)
$\begin{array}[]{rcl}\bar{X}^{l}&=&e^{\eta(z+w)}\cosh(\displaystyle\frac{\eta
w}{\sqrt{2}})\mathrm{d}x+\frac{\sqrt{2}}{2}e^{\eta(z+w)}\sinh(\displaystyle\frac{\eta
w}{\sqrt{2}})\mathrm{d}y\\\\[10.0pt]
\bar{Y}^{l}&=&\sqrt{2}e^{\eta(z+w)}\sinh(\frac{\eta
w}{\sqrt{2}})\mathrm{d}x+e^{\eta(z+w)}\cosh(\displaystyle\frac{\eta
w}{\sqrt{2}})\mathrm{d}y\\\\[6.0pt] \bar{Z}^{l}&=&\mathrm{d}z\\\\[6.0pt]
\bar{W}^{l}&=&\mathrm{d}w\\\ \end{array}$

and

(32) $\begin{array}[]{rcl}\bar{X}^{r}&=&\mathrm{d}x-\eta x\mathrm{d}z+(\eta
x+\eta y)\mathrm{d}w\\\\[6.0pt] \bar{Y}^{r}&=&\mathrm{d}y-\eta
y\mathrm{d}z+(\eta x+\eta y)\mathrm{d}w\\\\[6.0pt]
\bar{Z}^{r}&=&\mathrm{d}z\\\\[6.0pt] \bar{W}^{r}&=&\mathrm{d}w,\\\
\end{array}$

The modular character of Lie algebra $\mathfrak{g}_{\eta}$ is
$\mathcal{M}_{\mathfrak{g}_{\eta}}=-2\eta(\bar{Z}+\bar{W})$ and so

$\mathcal{M}_{\mathfrak{g}_{\eta}}^{r}=-2\eta(\bar{Z}^{r}+\bar{W}^{r})=-2\eta(\mathrm{d}z+\mathrm{d}w)=\mathrm{d}(\log
f_{0}).$

Therefore, $f_{0}=e^{-2\eta(z+w)}$. For
$v=\bar{X}\wedge\bar{Y}\wedge\bar{Z}\wedge\bar{W}\in\Lambda^{4}\mathfrak{g}^{*}$,
we get the invariant volume form

(33)
$\Phi=\sqrt{f}_{0}v^{l}=e^{\eta(z+w)}\mathrm{d}x\wedge\mathrm{d}y\wedge\mathrm{d}z\wedge\mathrm{d}w.$

It is easy to see that
$\mathcal{L}_{{X_{\mathcal{H}}^{\Pi}}}\Phi=\mathrm{d}(i_{{X_{\mathcal{H}}^{\Pi}}}\Phi)=0$.

In fact, it is easy to show that $\mathcal{L}_{X_{\mathcal{F}}^{\Pi}}\Phi=0$,
for an arbitrary smooth function $\mathcal{F}:G_{\eta}\to\mathbb{R}$, in
agreement with Corollary 3.4.

######  Remark 4.2.

As stated above, (21) is an integrable deformation of a different system,
which we call a conservative Lorenz system (see [3] for details), with
equations

(34) $\dot{x}=y,\qquad\dot{y}=x(1-z),\qquad\dot{z}=xy.$

Since $B_{2}\times B_{2}$ is diffeomorphic to $\mathbb{R}^{4}$ as a smooth
manifold, this implies that we can recover the previous dynamical system (34)
starting from (21) and taking the $\eta\to 0$ limit. Moreover, we can also
take the $\eta\to 0$ limit in the group law (23) obtaining the abelian Lie
group structure on $\mathbb{R}^{4}$

(35) $g(x,y,z,w)\cdot
g^{\prime}(x^{\prime},y^{\prime},z^{\prime},w^{\prime})=g\big{(}x+x^{\prime},y+y^{\prime},z+z^{\prime},w+w^{\prime}\big{)}$

and in the Poisson-Lie bivector

(36) $\Pi_{\mathbb{R}^{4}}=-\frac{1}{2}(x+w)\frac{\partial}{\partial
x}\wedge\frac{\partial}{\partial y}+\frac{y}{2}\frac{\partial}{\partial
x}\wedge\frac{\partial}{\partial z}+x\frac{\partial}{\partial
y}\wedge\frac{\partial}{\partial z},$

if we take the $\eta\to 0$ limit in (24).

Note that this Poisson-Lie structure is nothing more than the Lie-Poisson
structure associated with $\mathfrak{g}^{*}$, i.e. the linear Poisson
structure on the vector space $(\mathfrak{g}^{*})^{*}\simeq\mathfrak{g}$
associated with the Lie algebra
$\mathfrak{g}^{*}\simeq\mathfrak{sl}(2,\mathbb{R})\oplus\mathbb{R}$.
Therefore, using the Hamiltonian (25) we obtain the system

(37) $\dot{x}=y,\qquad\dot{y}=x(2-z-w),\qquad\dot{z}=xy,\qquad\dot{w}=0.$

From this system, in order to recover (34), we simply restrict to the
submanifold $\mathcal{S}=\\{(x,y,z,w)\in\mathbb{R}^{4}|w=1\\}$, taking into
account that (36) is tangent to $\mathcal{S}$.

###  4.6. A deformed Euler top

Following a similar procedure as above we can consider the integrable
deformation of the Euler top (see [4] for more details), given by

(38) $\begin{split}\dot{x}&=e^{\eta x}(y^{2}-z^{2}),\\\ \dot{y}&=\eta e^{\eta
x}yz^{2}-\frac{1}{2}\eta e^{\eta x}y(y^{2}+z^{2})+\frac{\sinh(\eta
x)}{\eta}(2z-y),\\\ \dot{z}&=-\eta e^{\eta x}y^{2}z+\frac{1}{2}\eta e^{\eta
x}z(y^{2}+z^{2})+\frac{\sinh(\eta x)}{\eta}(z-2y).\end{split}$

This system can be written as a bi-Hamiltonian system on the so-called “book”
Lie group $G_{\eta}$, with the Lie algebra $\mathfrak{g}$ defined by

$[X,Y]=-\eta Y,\quad[X,Z]=-\eta Z,\quad[Y,Z]=0,$

where $\eta\in\mathbb{R}$ is the deformation parameter. Since $G_{\eta}$ is
diffeomorphic to $\mathbb{R}^{3}$ we can choose global coordinates such that
the group law reads

(39) $g(x,y,z)\cdot
g^{\prime}(x^{\prime},y^{\prime},z^{\prime})=g\big{(}x+x^{\prime},y+y^{\prime}e^{-\eta
x},z+z^{\prime}e^{-\eta x}\big{)}.$

The system (38) is bi-Hamiltonian since we can find two different Poisson-Lie
structures on $G_{\eta}$, namely

(40) $\Pi_{G,0}=-z\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial
y}+y\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial
z}+\frac{1}{2}\bigg{(}-\eta(y^{2}+z^{2})+\frac{e^{-2\eta
x}-1}{\eta}\bigg{)}\frac{\partial}{\partial y}\wedge\frac{\partial}{\partial
z}$

and

(41) $\Pi_{G,1}=-y\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial
y}+z\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial z}+\bigg{(}-\eta
yz+\frac{e^{-2\eta x}-1}{\eta}\bigg{)}\frac{\partial}{\partial
y}\wedge\frac{\partial}{\partial z},$

and two different functions

(42) $\mathcal{H}_{0}=yze^{\eta x}+2\left(\frac{\cosh(\eta
x)-1}{\eta^{2}}\right),$

and

(43) $\mathcal{H}_{1}=-\frac{1}{2}(y^{2}+z^{2})e^{\eta x}+\frac{\cosh(\eta
x)-1}{\eta^{2}},$

such that (38) is the dynamical system defined by the Hamiltonian vector field
$\Pi_{G,0}^{\sharp}(d\mathcal{H}_{0})=\Pi_{G,1}^{\sharp}(d\mathcal{H}_{1})$.
Note that both Poisson-Lie structures form a Poisson pencil on $G_{\eta}$,
i.e. the convex combination $\lambda\Pi_{0}+(1-\lambda)\Pi_{1}$ is also a
Poisson structure, for $\lambda\in\mathbb{R}.$

In this case, the commutator maps $\delta_{G,i}=d_{e}\Pi_{G,i}$ associated to
(40) and (41), respectively, read

(44) $\delta_{G,0}(X)=-Y\wedge Z,\qquad\delta_{G,0}(Y)=X\wedge
Z,\qquad\delta_{G,0}(Z)=-X\wedge Y,$

and

(45) $\delta_{G,1}(X)=-2Y\wedge Z,\qquad\delta_{G,1}(Y)=-X\wedge
Y,\qquad\delta_{G,1}(Z)=X\wedge Z.$

and therefore the dual Lie algebras read

(46)
$\begin{split}&[\bar{X},\bar{Y}]_{\mathfrak{g}^{*}_{0}}=-\bar{Z},\qquad[\bar{X},\bar{Z}]_{\mathfrak{g}^{*}_{0}}=\bar{Y},\qquad[\bar{Y},\bar{Z}]_{\mathfrak{g}^{*}_{0}}=-\bar{X},\\\
&[\bar{X},\bar{Y}]_{\mathfrak{g}^{*}_{1}}=-\bar{Y},\qquad[\bar{X},\bar{Z}]_{\mathfrak{g}^{*}_{1}}=\bar{Z},\qquad[\bar{Y},\bar{Z}]_{\mathfrak{g}^{*}_{1}}=-2\bar{X},\\\
\end{split}$

Since $\mathfrak{g}^{*}_{0}\simeq\mathfrak{so}(3)$ and
$\mathfrak{g}^{*}_{1}\simeq\mathfrak{sl}(2,\mathbb{R})$ are both unimodular
Lie algebras, both Poisson-Lie structures (40) and (41) are unimodular, and
therefore the dynamical system (38) admits invariant volume forms.

Taking the $\eta\to 0$ limit in the equations (39), (40) and (41) we obtain
the 3-dimensional abelian Lie group $\mathbb{R}^{3}$ together with the Poisson
bivectors

(47) $\begin{split}\Pi_{\mathbb{R}^{3},0}&=-z\frac{\partial}{\partial
x}\wedge\frac{\partial}{\partial y}+y\frac{\partial}{\partial
x}\wedge\frac{\partial}{\partial z}-x\frac{\partial}{\partial
y}\wedge\frac{\partial}{\partial z},\\\
\Pi_{\mathbb{R}^{3},1}&=-y\frac{\partial}{\partial
x}\wedge\frac{\partial}{\partial y}+z\frac{\partial}{\partial
x}\wedge\frac{\partial}{\partial z}-2x\frac{\partial}{\partial
y}\wedge\frac{\partial}{\partial z},\end{split}$

which are precisely the Lie-Poisson structures associated with the Lie
algebras $\mathfrak{g}^{*}\simeq\mathfrak{so}(3)$ and
$\mathfrak{g}^{*}\simeq\mathfrak{sl}(2,\mathbb{R})$, respectively. Also, the
Hamiltonian functions (42) and (43) go to

(48) $\mathcal{H}_{0}^{\eta=0}=x^{2}+yz,$

and

(49) $\mathcal{H}_{1}^{\eta=0}=-\frac{1}{2}(x^{2}+y^{2}+z^{2}).$

The dynamical system defined by the Hamiltonian vector field
$\Pi_{\mathbb{R}^{3},0}^{\sharp}(d\mathcal{H}_{0}^{\eta=0})=\Pi_{\mathbb{R}^{3},1}^{\sharp}(d\mathcal{H}_{1}^{\eta=0})$
reads

(50) $\dot{x}=y^{2}-z^{2},\qquad\dot{y}=x(2z-y),\qquad\dot{z}=x(z-2y),$

which is equivalent to a particular case of the Euler top.

Now, we obtain the volume form preserved by the flow of the Hamiltonian vector
field. A basis of left and right-invariant vector fields for the basis
$\\{\partial_{x},\partial_{y},\partial_{z}\\}$ on ${\mathfrak{g}}_{\eta},$ are

$X^{l}=\partial_{x},\quad Y^{l}=e^{-\eta x}\partial_{y},\quad Z^{l}=e^{-\eta
x}\partial_{z},$ $X^{r}=\partial_{x}-\eta y\partial_{y}-\eta
z\partial_{z},\quad Y^{r}=\partial_{y},\quad Z^{r}=\partial_{z},$

and a basis of the left and right invariant one forms for the dual basis
$\\{dx,dy,dz\\}$ on ${\mathfrak{g}}^{*}_{\eta}$, are

$\bar{X}^{l}=dx,\quad\bar{Y}^{l}=e^{\eta x}dy,\quad\bar{Z}^{l}=e^{\eta x}dz.$
$\bar{X}^{r}=dx,\quad\bar{Y}^{r}=dy+\eta ydx,\quad\bar{Z}^{r}=dz+\eta zdx.$

The modular character of Lie algebra $\mathfrak{g}$ is
$\mathcal{M}_{\mathfrak{g}}=-2\eta\bar{X}$ and so
$\mathcal{M}_{\mathfrak{g}}^{r}=-2\eta\mathrm{d}x=\mathrm{d}(\log f_{0})$.
Therefore, $f_{0}=e^{-2\eta x}$. Then, by the Corollary 3.4, for
$v=\bar{X}\wedge\bar{Y}\wedge\bar{Z}\in\wedge^{3}\mathfrak{g}^{*}$, the volume
form

$\Phi=\sqrt{f_{0}}\nu^{l}=e^{\eta x}dx\wedge dy\wedge dz$

is preserved by all Hamiltonian vector fields.

In fact, it is easy to show that
$\mathcal{L}_{X_{\mathcal{G}}^{\Pi_{G,0}}}\Phi=L_{X_{\mathcal{F}}^{\Pi_{G,1}}}\Phi=0$,
for arbitrary functions $\mathcal{G},\mathcal{F}:G_{\eta}\to\mathbb{R}$, in
agreement with Corollary 3.4.

##  5\. Conclusions and future work

We have discussed the existence of invariant volume forms for Hamiltonian
systems on Poisson-Lie groups. In particular, if the identity element of the
Lie group is a nondegenerate singularity of the Hamiltonian function, the
Hamiltonian system admits an invariant volume form if and only if the Poisson-
Lie structure is unimodular. Several examples which illustrate our theoretical
results are also exhibited.

We also present some examples of Hamiltonian functions on Poisson-Lie groups
with non-degenerate singularity at the identity element which appear, in a
natural way, as metric contrast functions (a special class of divergence
functions in information geometry) on matrix groups (see Section 4.2).

On the other hand, in [20] the authors discuss a connection between
information geometry and discrete geometric mechanics. More precisely, they
present the relationship between divergence functions (in information
geometry) and (regular) discrete Lagrangian functions (in discrete geometric
mechanics).

So, following the previous ideas, it would be interesting to study the
relation between Hamiltonian dynamics on Poisson-Lie groups, discrete
geometric mechanics and information geometry. This will be postponed for a
future research.

In another direction, an interesting class of Poisson manifolds closely
related to Poisson-Lie groups are Poisson homogeneous spaces, given by the
quotient $G/H$ of a Poisson-Lie group $G$ with a closed Lie subgroup $H$ of
$G$ (see [27]). As an extension of the results in this paper, it would be
interesting to discuss the existence of invariant volume forms for Hamiltonian
systems on $G/H$. In this setting we remark the following facts:

  * •

As in the case of Poisson-Lie groups, the explicit description of the
symplectic leaves and the symplectic structure on them for the Poisson
structure on $G/H$ is, in general, a hard problem.

  * •

The Poisson homogeneous space $G/H$ is integrable and an explicit symplectic
groupoid ${\mathcal{G}}(G/H)$ integrating $G/H$ may be constructed (see [21]
and also [6, 7]). But, in general, the dimension of ${\mathcal{G}}(G/H)$ is
$2\dim(G/H)$.

So, in order to study the existence of invariant volume forms, for a
Hamiltonian system on $G/H$, it seems reasonable to discuss the problem
directly in the homogeneous Poisson space $G/H$. In fact, a description of the
modular class of the Poisson structure on $G/H$ has been given in [21] (see
also [9, 13]) and, therefore, we could discuss the relation between the
unimodularity of the Poisson structure and the existence of invariant volume
forms for the Hamiltonian system.

Acknowledgments: The authors thanks the anonymous referees for their valuable
comments. D. Iglesias, JC Marrero and E. Padrón acknowledge financial support
from the Spanish Ministry of Science and Innovation under grant
PGC2018-098265-B-C32. I. Gutiérrez-Sagredo acknowledge financial support from
Agencia Estatal de Investigación (Spain) under grant
PID2019-106802GB-I00/AEI/10.13039/50110 0011033.

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# Normalization of Transliterated Words in Code-Mixed Data Using Seq2Seq Model
& Levenshtein Distance

Soumil Mandal, Karthick Nanmaran

Department of Computer Science & Engineering
SRM Institute of Science & Technology, Chennai, India
{soumil.mandal<EMAIL_ADDRESS>
###### Abstract

Building tools for code-mixed data is rapidly gaining popularity in the NLP
research community as such data is exponentially rising on social media.
Working with code-mixed data contains several challenges, especially due to
grammatical inconsistencies and spelling variations in addition to all the
previous known challenges for social media scenarios. In this article, we
present a novel architecture focusing on normalizing phonetic typing
variations, which is commonly seen in code-mixed data. One of the main
features of our architecture is that in addition to normalizing, it can also
be utilized for back-transliteration and word identification in some cases.
Our model achieved an accuracy of 90.27% on the test data.

##  1 Introduction

With rising popularity of social media, the amount of data is rising
exponentially. If mined, this data can proof to be useful for various
purposes. In countries where the number of bilinguals are high, we see that
users tend to switch back and forth between multiple languages, a phenomenon
known as code-mixing or code-switching. An interesting case is switching
between languages which share different mother scripts. On such occasions, one
of the two languages is typed in it’s phonetically transliterated form in
order to use a common script. Though there are some standard transliteration
rules, for example ITRANS 111https://en.wikipedia.org/wiki/ITRANS, ISO
222https://en.wikipedia.org/wiki/ISO_15919, but it is extremely difficult and
un-realistic for people to follow them while typing. This indeed is the case
as we see that identical words are being transliterated differently by
different people based on their own phonetic judgment influenced by dialects,
location, or sometimes even based on the informality or casualness of the
situation. Thus, for creating systems for code-mixed data, post language
tagging, normalization of transliterated text is extremely important in order
to identify the word and understand it’s semantics. This would help a lot in
systems like opinion mining, and is actually necessary for tasks like
summarization, translation, etc. A normalizing module will also be of immense
help while making word embeddings for code-mixed data.
In this paper, we present an architecture for automatic normalization of
phonetically transliterated words to their standard forms. The language pair
we have worked on is Bengali-English (Bn-En), where both are typed in Roman
script, thus the Bengali words are in their transliterated form. The canonical
or normalized form we have considered is the Indian Languages Transliteration
(ITRANS) form of the respective Bengali word. Bengali is an Indo-Aryan
language of India where 8.10%
333https://en.wikipedia.org/wiki/Languages_of_India of the total population
are $1^{st}$ language speakers and is also the official language of
Bangladesh. The mother script of Bengali is the Eastern Nagari Script
444https://www.omniglot.com/writing/bengali.htm. Our architecture utilizes
fully char based sequence to sequence learning in addition to Levenshtein
distance to give the final normalized form or as close to it as possible. Some
additional advantages of our system is that at an intermediate stage, the
back-transliterated form of the word can be fetched (i.e. word
identification), which will be very useful in several cases as original tools
(i.e. tools using mother script) can be utilized, for example emotion
lexicons. Some other important contributions of our research are the new
lexicons that have been prepared (discussed in Sec 3) which can be used for
building various other tools for studying Bengali-English code-mixed data.

##  2 Related Work

Normalization of text has been studied quite a lot Sproat et al. (1999),
especially as it acts as a pre-processing step for several text processing
systems. Using Conditional Random Fields (CRF), Zhu et al. (2007) performed
text normalization on informal emails. Dutta et al. (2015) created a system
based on noisy channel model for text normalization which handles wordplay,
contracted words and phonetic variations in code-mixed background. An
unsupervised framework was presented by Sridhar (2015) for normalizing domain-
specific and informal noisy texts using distributed representation of words.
The soundex algorithm was used in Sitaram et al. (2015) and Sitaram and Black
(2016) for spelling correction of transliterated words and normalization in a
speech to text scenario of code-mixed data respectively. Sharma et al. (2016)
build a normalization system using noisy channel framework and SILPA spell
checker in order to build a shallow parser. Sproat and Jaitly (2016) build a
system combining two models, where one essentially is a seq2seq model which
checks the possible normalizations and the other is a language model which
considers context information. Jaitly and Sproat (2017) used a seq2seq model
with attention trained at sentence level followed by error pruning using
finite-state filters to build a normalization system, mainly targeted for text
to speech purposes. A similar flow was adopted by Zare and Rohatgi (2017) as
well where seq2seq was used for normalization and a window of size 20 was
considered for context. Singh et al. (2018) exploited the fact that words and
their variations share similar context in large noisy text corpora to build
their normalizing model, using skip gram and clustering techniques. To the
best of our knowledge, the system architecture proposed by us hasn’t been
tried before, especially for code-mixed data.

##  3 Data Sets

On a whole, three data sets or lexicons were created. The first data set was a
parallel lexicon (PL) where the 1st column had phonetically transliterated Bn
words in Roman taken from code-mixed data prepared in Mandal et al. (2018b).
The 2nd column consisted of the standard Roman transliterations (ITRANS) of
the respective words. To get this, we first manually back-transliterated
PLcol_1 to the original word in Eastern Nagari script, and then converted it
into standardized ITRANS format. The final size of the PL was 6000. The second
lexicon we created was a transliteration dictionary (BN_TRANS) where the first
column had Bengali words in Eastern Nagari script taken from samsad
555http://dsal.uchicago.edu/dictionaries/biswas-bengali/, while the second
column had the standard transliterations (ITRANS). The number of entries in
the dictionary was 21850. For testing, we took the data used in Mandal and Das
(2018), language tagged it using the system described in Mandal et al.
(2018a), and then collected Bn tagged tokens. Post manual checking and
discarding of misclassified tokens, the size of the list was 905. Finally,
each of the words were tagged with their ITRANS using the same approach used
for making PL. For PLcol_1 and test data, some initial rule based
normalization techniques were used. If the input string contains a digit, it
was replaced by the respective phone (e.g. ek for 1, dui for 2, etc), and if
there are n consecutive identical characters where n $>$ 2 (elongation), it
was trimmed down to 2 consecutive characters (e.g. baaaad will become baad),
as no word in it’s standard form has more than two consecutive identical
characters.

##  4 Proposed Method

Our method is a two step modular approach comprising of two degrees of
normalization. The 1∘ normalization module does an initial normalization and
tries to convert the input string closest to the standard transliteration. The
2∘ normalization module takes the output from the first module and tries to
match with the standard transliterations present in the dictionary (BN_TRANS).
The candidate with the closest match is returned as the final normalized
string.

##  5 1∘ Normalization Module

The purpose of this module is to phonetically normalize the word as close to
the standard transliteration as possible, to make the work of the matching
module easier. To achieve this, our idea was to train a sequence to sequence
model where the input sequences are user transliterated words and the target
sequences are the respective ITRANS transliterations. We had specifically
chosen this architecture as it has performed amazingly well in complex
sequence mapping tasks like neural machine translation and summarization.

###  5.1 Seq2Seq Model

The sequence to sequence model Sutskever et al. (2014) is a relatively new
idea for sequence learning using neural networks. It has been especially
popular since it achieved state of the art results in machine translation
task. Essentially, the model takes as input a sequence X = {x1, x2, …, xn} and
tries to generate another sequence Y = {y1, y2, …, ym}, where xi and yi are
the input and target symbols respectively. The architecture of seq2seq model
comprises of two parts, the encoder and decoder. As the input and target
vectors were quite small (words), attention Vaswani et al. (2017) mechanism
was not incorporated.

####  5.1.1 Encoder

Encoder essentially takes a variable length sequence as input and encodes it
into a fixed length vector, which is suppose to summarize it’s meaning taking
into context as well. A recurrent neural network (RNN) cell is used to achieve
this. The directional encoder reads the sequence from one end to the other
(left to right in our case).

$\vec{h}\textsubscript{t}=\vec{f}\textsubscript{enc}(E\textsubscript{x}(x\textsubscript{t}),\vec{h}\textsubscript{t-1})$

Here, Ex is the input embedding lookup table (dictionary), $\vec{f}$enc are
the transfer function for the recurrent unit e.g. Vanilla, LSTM or GRU. A
contiguous sequence of encodings C = {h1, h2, …, hn} is constructed which is
then passed on to the decoder.

####  5.1.2 Decoder

Decoder takes input context vector C from the encoder, and computes the hidden
state at time t as,

$s\textsubscript{t}=f\textsubscript{dec}(E\textsubscript{y}(y\textsubscript{t-1}),s\textsubscript{t-1},c\textsubscript{t})$

Subsequently, a parametric function outk returns the conditional probability
using the next target symbol being k. Here, the concept of teacher forcing is
utilized, the strategy of feeding output of the model from a prior time-step
as input.

$\begin{multlined}p(y\textsubscript{t}=k\mid y<{t},X)=\\\
\frac{1}{Z}exp(out\textsubscript{k}(E\textsubscript{y}(y\textsubscript{t}-1),s\textsubscript{t},c\textsubscript{t}))\end{multlined}p(y\textsubscript{t}=k\mid
y<{t},X)=\\\
\frac{1}{Z}exp(out\textsubscript{k}(E\textsubscript{y}(y\textsubscript{t}-1),s\textsubscript{t},c\textsubscript{t}))$

Z is the normalizing constant

$\sum\textsubscript{j}exp(out\textsubscript{j}(E\textsubscript{y}(y\textsubscript{t}-1),s\textsubscript{t},c\textsubscript{t}))$

###  5.2 Training

The model is trained by minimizing the negative log-likelihood. For training,
we used the fully character based seq2seq model Lee et al. (2016) with stacked
LSTM cells. The input units were user typed phonetic transliterations
(PLcol_1) while the target units were respective standard transliterations
(PLcol_2). Thus, the model learns to map user transliterations to standard
transliterations, effectively learning to normalize phonetic variations. The
lookup table Ex we used for character encoding was a dictionary where the keys
were the 26 English alphabets and the values were the respective index.
Encodings at character level were then padded to the length of the maximum
existing word in the dataset, which was 14, and was converted to one-hot
encodings prior to feeding the to the seq2seq model. We created our seq2seq
model using the Keras Chollet et al. (2015) library. The batch size was set to
64, and number of epochs was set to 100. The size of the latent dimension was
kept at 128. Optimizer we chose was rmsprop, learning rate set at 0.001, loss
was categorical crossentropy and transfer function used was softmax. Accuracy
and loss graphs during training with respect to epochs are shown in Fig 1.

Figure 1: Training accuracy and loss.

As we can see from Fig 1, the accuracy reached at the end of training was not
too high (around 41.2%) and the slope became asymptotic. This is quite
understandable as the amount of training data was relatively quite low for the
task, and the phonetic variations were quite high. On running this module on
our testing data, an accuracy of 51.04% was achieved. It should be noted that
even a single character predicted wrongly by the softmax function reduces the
accuracy.

##  6 2∘ Normalization Module

This module basically comprises of the string matching algorithm. For this, we
have used Levenshtein distance (LD) Levenshtein (1966), which is a string
metric for measuring difference between two sequences. It does so by
calculating the minimum number of insertions, deletions and substitutions
required for converting one sequence to the other. Here, the output from the
previous module is compared with all the standard ITRANS entries present in
BN_TRANS and the string with the least Levenshtein distance is given as
output, which is the final normalized form. If there are ties, the instance
which has higher matches traversing from left to right is given more priority.
Also, observing the errors from 1∘ normalizer, we noticed that in a lot of
cases, the character pairs {a,o} and {b,v} are used interchangeably quite
often, both in phonetic transliterations alone, as well as when compared with
ITRANS. Thus, along with the standard approach, we tried a modified version as
well where the cost of the above mentioned character pairs are same, i.e. they
are treated as identical characters. This was simply done by assigning special
symbols to those pairs, and replacing them in the input parameters. For
example, post replacement, distance(chalo, chala) will become distance(ch$l$,
ch$l$).

##  7 Evaluation

Our system was evaluated in two ways, one at word level and another at task
level.

###  7.1 Word Level

Here, the basic idea was compare the normalized words with the respective
standard transliterations. For this, the testing data discussed in Sec 3 was
used. For comparison purposes, three other setups other than our proposed
model (setup_4) were tested, all of which are described in Table 1.

Model |  1∘ Norm | LD | Acc
---|---|---|---
setup_1 | no | standard | 58.78
setup_2 | no | modified | 61.10
setup_3 | yes | standard | 89.72
setup_4 | yes | modified | 90.27
Table 1: Comparison of different setups.

From Table 1, we can see that the jump in accuracy from setup_1 to setup_3 is
quite significant (by 30.94%). This proves that instead of simple distance
comparison with lexicon entries, a prior seq2seq normalization can have great
impact on the performance. Additionally, we can also see that when modified
input is given to the Levenshtein distance (LD), the accuracies achieved are
slightly better. On analyzing the errors, we found out that majority (92%) of
them is due to the fact that the standard from was not present in BN_TRANS,
i.e. was out of vocabulary. These words were mostly slangs, expressions, or
two words joined into a single one. The other 8% was due to the 1∘ module
casuing substantial deviation from normal form. For deeper analysis, we
collected the ITRANS of errors due out of vocab, and on comparison with the 1∘
normalizations, the mean LD was calculated to be 1.89, which is suggesting
that if they were present in BN_TRANS, the normalizer would have given the
correct output.

###  7.2 Task Level

For task level evaluation, we decided to go with sentiment analysis using the
exact setup and data described in Mandal et al. (2018b), on Bengali-English
code-mixed data. All the training and testing data were normalized using our
system along with the lexicons that are mentioned. Finally, the same steps
were followed and the different metrics were calculated. The comparison of the
systems prior (noisy) and post normalization (normalized) is shown in Table 2.

Model | Acc | Prec | Rec | F1
---|---|---|---|---
noisy | 80.97 | 81.03 | 80.97 | 81.20
normalized | 82.47 | 82.52 | 82.47 | 82.61
Table 2: Prior and post normalization results.

We can see an improvement in the accuracy (by 1.5%). On further investigation,
we saw that the unigram, bigram and trigram matches with the bag of n-grams
and testing data increased by 1.6%, 0.4% and 0.1% respectively. The accuracy
can be improved further more if back-transliteration is done and Bengali
sentiment lexicons are used but that is beyond the scope of this paper.

##  8 Discussion

Though our proposed model achieved high accuracy, some drawbacks are there.
Firstly is the requirement for the parallel corpus (PL) for training a seq2seq
model, as manual checking and back-transliteration is quite tedious. Speed of
processing in terms of words/second is not very high due to the fact that both
seq2seq and Levenshthein distance calculation is computationally heavy, plus
the O(n) search time. For string matching, simpler and faster methods can be
tested and search area reduction algorithms (e.g. specifying search domains
depending on starting character) can be tried to improve the processing speed.
A simple lexical checker can be added as well before using seq2seq and/or
matching module to see if the word is already in it’s transliterated form.

##  9 Conclusion & Future Work

In this article, we have presented a novel architecture for normalization of
transliterated words in code-mixed scenario. We have employed the seq2seq
model with LSTM cells for initial normalization followed by evaluating
Levenshthein distance to retrieve the standard transliteration from a lexicon.
Our approach got an accuracy of 90.27% on testing data, and improved the
accuracy of a pre-existing sentiment analysis system by 1.5%. In future, we
would like to collect more transliterated words and increase the data size in
order to improve both PL and BN_TRANS. Also, combining this module with a
context capturing system and expanding to other Indic languages like Hindi,
Tamil will be one of the goals as well.

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