# Quantum groups, discrete Magnus expansion,
pre-Lie $\&$ tridendriform algebras

Anastasia Doikou  Department of Mathematics, Heriot-Watt University, Edinburgh
EH14 4AS $\&$ Maxwell Institute for Mathematical Sciences, Edinburgh EH8 9BT,
UK<EMAIL_ADDRESS>
###### Abstract.

We review the discrete evolution problem and the corresponding solution as a
discrete Dyson series in order to rigorously derive a discrete version of
Magnus expansion. We also systematically derive the discrete analogue of the
pre-Lie Magnus expansion and show that the elements of the discrete Dyson
series are expressed in terms of a tridendriform algebra action. Key links
between quantum algebras, tridendriform and pre-Lie algebras are then
established. This is achieved by examining tensor realizations of quantum
groups, such as the Yangian. We show that these realizations can be expressed
in terms of tridendriform and pre-Lie algebras actions. The continuous limit
as expected provides the corresponding non-local charges of the Yangian as
members of the pre-Lie Magnus expansion.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Rota Baxter, pre-Lie $\&$ tridendriform algebras
    2. 2.2 From pre-Lie algebras to braces
  3. 3 Discrete Magnus expansion $\&$ Pre-Lie algebras
    1. 3.1 An alternative discrete expansion
    2. 3.2 Gauge fields $\&$ open boundary systems
  4. 4 Quantum groups as tridendriform $\&$ pre-Lie algebras
    1. 4.1 The Yangian ${\cal Y}(\mathfrak{gl}_{\cal N})$
    2. 4.2 Tensor realizations of the Yangian $\&$ Pre-Lie algebras

##  1\. Introduction

In the present study we identify interesting links between quantum groups [25,
19, 27], and tridendriform [30, 31] and pre-Lie algebras (also studied under
the name chronological algebras) [2, 26, 47] (see also [5, 35] for a recent
reviews). Specifically, we systematically derive the discrete analogues of
Dyson series [20] and Magnus expansion [34] as solutions of a discrete
evolution problem. We then show that the Dyson series members are expressed in
terms of a tridendriform algebra action, whereas the discrete Magnus expansion
members are derived in relation to a pre-Lie algebra action in analogy to the
continuous case (see also relevant findings in [22]). The use of Rota-Baxter
operators [6, 39] has been essential in expressing the discrete series in
connection with tridendriform and pre-Lie algebra actions. On the other hand,
tensor realizations of quantum groups [25, 19], such as the Yangian [19], are
also solutions of a discrete evolution problem. Hence, we deduce that the
coproducts of the elements of the Yangian can be re-expressed in terms of
suitable tridendriform and pre-Lie algebras actions.

Before we describe in detail what is achieved in each section we first recall
the general set up and some necessary preliminaries on the Magnus expansion as
a solution of a linear evolution problem, whereas in the subsequent section we
briefly recall basic notions on Rota-Baxter, pre-Lie and tridendriform
algebras. We note that interesting relations between pre-Lie algebras, rooted
tree graphs, (tri)dendriform and Rota-Baxter algebras have been reported (see
for instance [30, 22, 21, 23]), whereas links between tridendriform, Rota-
Baxter and (quasi)shuffle algebras have been also revealed in [32]. The recent
findings on the relationships between pre-Lie algebras and braces (nipotent
rings) [41, 46], have generated increased interest on these distinct algebraic
structures opening up unexplored research avenues. It is worth pointing out
that the notion of infinitesimal Hopf algebras and their connections to pre-
Lie and dendriform algebras have been explored in [3]. In this study however
we establish links with typical Hopf algebras, such as the Yangian that also
appear in quantum integrable systems. Some of these profound emerging
relations in the frame of classical and quantum integrability lie in the
epicenter of our analysis, while others will be further examined and extended
in future works.

We start our discussion by recalling the initial value problem associated to a
linear differential equation. Indeed, let $A,T$ be in general some linear
operators e.g. ($T,\ A\in\mbox{End}({\mathbb{C}}^{\mathcal{N}})$) depending on
two parameters, $\xi\in{\mathbb{R}},\ \alpha\in{\mathbb{C}}$, such that

$\partial_{\xi}T(\xi,\alpha)=\alpha A(\xi)T(\xi,\alpha),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
T(x_{0},a)=T_{0}.$ (1.1)

The formal solution of the evolution equation above can be given as (we
consider simple initial conditions $T(x_{0},\alpha)=1$)

$T(x,\alpha)=\overset{\curvearrowleft}{\mathrm{P}}\text{exp}\big{(}\alpha\int_{x_{0}}^{x}A(\xi)d\xi\big{)},\quad
x>x_{0}.$ (1.2)

The latter solution is a path ordered exponential (monodromy), which is
formally expressed in terms of Dyson series [20]

$\displaystyle\overset{\curvearrowleft}{\mathrm{P}}\mbox{exp}\Big{(}\alpha\int_{x_{0}}^{x}A(\xi)d\xi\
\Big{)}=\sum_{n=0}^{\infty}\alpha^{n}\int_{x_{0}}^{x}dx_{n}A(x_{n})\int_{x_{0}}^{x_{n}}dx_{n-1}A(x_{n-1})\ldots\int_{x_{0}}^{x_{2}}dx_{1}A(x_{1}).$

Magnus [34] suggested that the solution $T(x,\alpha)$ of the linear evolution
problem can be expressed as a real exponential, i.e.
$T(x,\alpha)=e^{{\mathbb{Q}}(x,\alpha)}$ such that $\
e^{{\mathbb{Q}}(x,\alpha)}:=1+\sum_{n=1}^{\infty}{{\mathbb{Q}}^{n}(x,a)\over
n!},$ where the following formal series are considered
$T(x,\alpha)=1+\sum_{n=1}^{\infty}\alpha^{n}T^{(n)}(x),$ $\leavevmode\nobreak\
{\mathbb{Q}}(x,\alpha)=\sum_{n=1}^{\infty}\alpha^{n}{\mathbb{Q}}^{(n)}(x)$ and

$T^{(n)}(x)=\int_{x_{0}}^{x}dx_{n}A(x_{n})\int_{x_{0}}^{x_{n}}dx_{n-1}A(x_{n-1})\ldots\int_{x_{0}}^{x_{2}}dx_{1}A(x_{1}).$
(1.3)

Comparing the $\alpha$ series expansion of $T(x,\alpha)$ and
$e^{{\mathbb{Q}}(x,\alpha)}$ we obtain the coefficients ${\mathbb{Q}}^{(n)}$
as symmetric polynomials of $T^{(n)}$ ($T^{(n)},\ {\mathbb{Q}}^{(n)}$ below
depend on $x$):

$\displaystyle{\mathbb{Q}}^{(1)}$ $\displaystyle=$ $\displaystyle
T^{(1)},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
{\mathbb{Q}}^{(2)}=T^{(2)}-{1\over 2}(T^{(1)})^{2},$ (1.4)
$\displaystyle{\mathbb{Q}}^{(3)}$ $\displaystyle=$ $\displaystyle
T^{(3)}-{1\over 2}\left(T^{(1)}T^{(2)}+T^{(2)}T^{(1)}\right)+{1\over
3}(T^{(1)})^{3},\leavevmode\nobreak\ \leavevmode\nobreak\ \ldots$ (1.5)

and vise versa all $T^{(m)}$’s can be expressed in terms of
${\mathbb{Q}}^{(m)}$’s,

$\displaystyle T^{(1)}$ $\displaystyle=$
$\displaystyle{\mathbb{Q}}^{(1)},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ T^{(2)}={\mathbb{Q}}^{(2)}+{1\over
2}({\mathbb{Q}}^{(1)})^{2},$ (1.6) $\displaystyle T^{(3)}$ $\displaystyle=$
$\displaystyle{\mathbb{Q}}^{(3)}+{1\over
2}\left({\mathbb{Q}}^{(1)}{\mathbb{Q}}^{(2)}+{\mathbb{Q}}^{(2)}{\mathbb{Q}}^{(1)}\right)+{1\over
3!}({\mathbb{Q}}^{(1)})^{3},\leavevmode\nobreak\ \leavevmode\nobreak\ \ldots$
(1.7)

Every element ${\mathbb{Q}}^{(m)}$ of the series expansion can be obtained by
means of the generic recursive formula (see for instance [8] and references
therein):

$\displaystyle{\mathbb{Q}}^{(m)}=T^{(m)}-\sum_{k=2}^{m}(-1)^{m}{\Pi^{(m)}_{k}\over
k},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ n>1,$ (1.8)

where $\Pi^{(n)}_{k}=\sum T^{(j_{1})}\ldots T^{(j_{k})}:\leavevmode\nobreak\
\leavevmode\nobreak\ j_{1}+\ldots+j_{k}=n$ and satisfy the recursion formula:

$\Pi^{(n)}_{k}=\sum_{m=1}^{n-k+1}\Pi^{(m)}_{1}\Pi^{(n-m)}_{k-1}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{and}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\Pi^{(n)}_{1}=T^{(n)},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \Pi^{(n)}_{n}=(T^{(1)})^{n}.$ (1.9)

Hence, we obtain, after recalling (4.3), the explicit expressions for the
first three terms of the series (Magnus expansion):

$\displaystyle{\mathbb{Q}}^{(1)}(x)$ $\displaystyle=$
$\displaystyle\int_{x_{0}}^{x}dx_{1}A(x_{1}),$ (1.10)
$\displaystyle{\mathbb{Q}}^{(2)}(x)$ $\displaystyle=$ $\displaystyle{1\over
2}\int_{x_{0}}^{x}dx_{2}\int_{x_{0}}^{x_{2}}dx_{1}\big{[}A(x_{2}),A(x_{1})\big{]},$
$\displaystyle{\mathbb{Q}}^{(3)}(x)$ $\displaystyle=$ $\displaystyle{1\over
6}\int_{x_{0}}^{x}dx_{3}\int_{x_{0}}^{x_{3}}dx_{2}\int_{x_{0}}^{x_{2}}dx_{1}\big{(}\big{[}A(x_{3}),\big{[}A(x_{2}),A(x_{1})\big{]}+\big{[}\big{[}A(x_{3}),A(x_{2})\big{]},A(x_{1})\big{]}\big{)}.$

######  Remark 1.1.

Magnus (Theorem [34]) obtained the general expression for ${\mathbb{Q}}(x)$ as
an infinite series involving Bernoulli’s numbers (see also [8] and references
therein). We first introduce some useful notation: Let $A,B$ be linear
operators, and recall the binary operation $\big{[}\ ,\ \big{]}:(A,B)\mapsto
AB-BA,$ i.e. the familiar Lie commutator. We also define

$\mbox{ad}_{A}B=\big{[}A,\ B\big{]},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{ad}^{n}_{A}=\big{[}A,\
\mbox{ad}^{n-1}_{A}B\big{]},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{ad}^{0}_{A}B=B,$ (1.11)

and recall the Bernoulli numbers $B_{n}$ defined as
$\sum_{n=o}^{\infty}{B_{n}\over n!}z^{n}={z\over e^{z}-1}.$ Then
${\mathbb{Q}}(x)$ can be expressed in a compact form as

${\mathbb{Q}}(x)=\int_{x_{0}}^{x}ds\sum_{n=0}^{\infty}{B_{n}\over
n!}\mbox{ad}^{n}_{{\mathbb{Q}}(x)}A(s).$ (1.12)

Expressions (1.10) can be then obtained from (1.12) by iteration. For a
detailed discussion on Magnus expansion, convergence issues, expansion
generators and applications the interested reader is referred for instance to
[8] and references therein.

After the brief review on Magnus expansion we recall in the subsequent section
some of the fundamental notions necessary of our analysis in Sections 3 and 4.
More precisely, we describe below what is achieved in each section.

  * $\circ$

In Subsection 2.1 we recall the definitions of Rota-Baxter, pre-Lie and
tridendriform algebras and we then discuss the connections among these
algebras. To illustrate these relations we use two simple examples, which will
be exploited anyway in this study. In subsection 2.2 in order to further
motivate the study of deep interconnections among seemingly distinct algebraic
structures we recall the passage from pre-Lie algebras to braces [41, 46].

  * $\circ$

In section 3 the rigorous derivation of the discrete analogue of Magnus
expansion is exhibited, after having first derived the discrete version of
Dyson series. These derivations are realized by means of the discrete analogue
of the evolution problem (1.1). Linearization of the discrete evolution
problem leads naturally to the continuous equation (1.1). Furthermore, we show
that the discrete Dyson series are expressed in terms of a tridendriform
algebra actions, whereas the Magnus expansion is expressed in terms of a pre-
Lie algebra action. Construction of a brace multiplication from this pre-Lie
algebra immediately follows. Explicit expressions of the first few elements of
both expansions are provided. Taking the continuum limits of these expressions
we recover the continuous Magnus expression. and the pre-Lie Magnus formula.
In subsection 3.1 we consider alternative discrete Dyson and Magnus
expansions, which again can be expressed using tridendriform and pre-Lie
algebra actions. The various discrete expansions are associated to distinct
quantum algebras as will be transparent in section 4. In subsection 3.2 we
present two basic examples/applications related to the “backward” and
“forward” Dyson and Magnus expansions. The first example is related to gauge
transformations of matrix valued fields, whereas the second one describes the
discrete and continuous evolution problem of certain classes of open boundary
systems.

  * $\circ$

In section 4 we investigate the strong ties between quantum algebras,
specifically the Yangian and tridedriform and pre-Lie algebras. We first
recall the derivation of the Yangian via the Faddev-Reshetikhin-Takhtajan
(FRT) construction. We extract an alternative set of generators using the Lie
exponential of the solutions of the FRT relation and derive the defining
algebraic relations of the alternative set of generators. We then move on to
study tensor realizations of the Yangian for both sets of generators and
express co-products of the Yangian generators using actions of pre-Lie and
tridendriform algebras. The classical Yangian is briefly discussed after we
introduce the notions of the classical $r$-matrix and Sklyanin’s bracket [24].
We conclude that the non-local charges of the classical Yangian are naturally
expressed in terms of (tri)dendriform algebra actions.

##  2\. Preliminaries

###  2.1. Rota Baxter, pre-Lie $\&$ tridendriform algebras

We shall now briefly recall the notions of pre-Lie algebras [2, 26] and the
pre-Lie Magnus expansion, which will be essential to our analysis here. Before
we comment on this interesting connection we first introduce the definitions
of Rota-Baxter and pre-Lie algebras [6, 39] (see also [5, 22, 23, 35] and
references therein).

######  Definition 2.1.

A Rota-Baxter algebra is a $k$-vector space ${\cal A}$ equipped with a linear
map $R:{\cal A}\to{\cal A},$ such that $\forall a,b\in{\cal A}$

$R(a)R(b)=R(R(a)b+aR(b)+\theta ab),$ (2.1)

where $\theta\in k$ is a fixed parameter.

The map $R$ is called a Rota–Baxter operator of weight $\theta$.

######  Definition 2.2.

A pre-Lie algebra is a $k$-vector space ${\mathrm{A}}$ with a binary operation
$\triangleright:(a,b)\mapsto a\triangleright b,$ such that it satisfies the
pre-Lie identity $\forall a,b,c\in A,$

$(a\triangleright b)\triangleright c-a\triangleright(b\triangleright
c)=(b\triangleright a)\triangleright c-b\triangleright(a\triangleright c).$
(2.2)

Analogously, a right pre-Lie algebra with a binary operation $\triangleleft$
can be defined with a pre-Lie identity:

$(a\triangleleft b)\triangleleft c-a\triangleleft(b\triangleleft
c)=(a\triangleleft c)\triangleleft b-a\triangleleft(c\triangleleft b).$ (2.3)

In the following Proposition it is shown that Rota-Baxter operators of weight
$\theta$ can be used to construct a pre-Lie algebra action (see also for
instance [5, 22] and references therein).

######  Proposition 2.3.

Let $R:{\cal A}\to{\cal A}$ be a Rota-Baxter operator of weight $\theta$ and
let $\triangleright$ be a binary operation: $(a,b)\mapsto a\triangleright b,$
$\forall a,b\in{\cal A}$ such that

$a\triangleright b:=\big{[}R(a),\ b\big{]}+\theta ab.$ (2.4)

Then the binary operation $\triangleright$ satisfies the pre-Lie identity
(2.2).

###### Proof.

Using definition (2.4) and assuming (in accordance to the cases examined here)
that $R(a+b)=R(a)+R(b)$ we first show that $R(-x)=R(0)-R(x),$ indeed

$\displaystyle R(x-x)=R(0)\Rightarrow R(x)+R(-x)=R(0)\Rightarrow
R(-x)=R(0)-R(x).$ (2.5)

We then explicitly compute:

$\displaystyle(x\triangleright y)\triangleright z$ $\displaystyle=$
$\displaystyle R(R(x)y-yR(x)+\theta xy)z-zR(R(x)y-yR(x)+\theta xy)$ (2.6)
$\displaystyle+$ $\displaystyle\theta\big{(}R(x)yz-yR(x)z+\theta xyz\big{)}.$
(2.7)

Similarly, using (2.1):

$\displaystyle x\triangleright(y\triangleright z)$ $\displaystyle=$
$\displaystyle R(R(x)y+xR(y)+\theta xy)z+zR(R(y)x+yR(x)+\theta yx)$
$\displaystyle+$ $\displaystyle\theta\Big{(}R(x)yz-yzR(x)+xR(y)z-xzR(y)+\theta
xyz\big{)}-R(x)zR(y)-R(y)zR(x)$

And from the two last expressions we conclude that (recall $R(a+b)=R(a)+R(b)$
and $R(-x)=R(0)-R(x)$):

$\displaystyle(x\triangleright y)\triangleright
z-x\triangleright(y\triangleright z)=$ $\displaystyle-$ $\displaystyle
R(R(x)y+xR(y))z-zR(R(x)y+xR(y))$ (2.9) $\displaystyle-$ $\displaystyle\theta
z\big{(}R(xy)+R(yx)\big{)}-\theta\big{(}yR(x)+xR(y)\big{)}$ (2.10)
$\displaystyle+$
$\displaystyle\theta\big{(}yzR(x)+xzR(y)\big{)}+R(x)zR(y)+R(y)zR(x).$ (2.11)

The latter expression is symmetric in $x$ and $y$ and leads to the pre-Lie
algebra identity. ∎

We present two simple examples of Rota-Baxter operators, which will be used in
our analysis.

######  Example 2.4.

A simple example of Rota–Baxter operator is given by the ordinary Riemann
integral, which is a weight zero Rota–Baxter map (other examples can be found
for instance in [22] and references therein). Indeed, let
$S(f)_{x}:=\int_{0}^{x}f(\zeta)d\zeta,$ then

$\displaystyle S(f)_{x}S(g)_{x}$ $\displaystyle=$
$\displaystyle\int_{0}^{x}f(\zeta)d\zeta\int_{0}^{x}g(\xi)d\xi=\int_{0}^{x}d\zeta\int_{0}^{\zeta}d\xi
f(\zeta)g(\xi)+\int_{0}^{x}d\zeta\int_{0}^{\zeta}d\xi f(\xi)g(\zeta)$ (2.12)
$\displaystyle=$ $\displaystyle
S(S(f)_{\zeta}g_{\zeta}+f_{\zeta}S(g)_{\zeta})_{x}$ (2.13)

i.e. $S$ is a Rota-Baxter operator of zero weight.

Let $A,B$ be linear operators that depend on a continuous parameter $x$, we
define the following action

$(A\triangleright B)(x):=\big{[}\int_{0}^{x}A(s)ds,\ B(x)\big{]},$ (2.14)

which provides a non-commutative binary operation, e.g. $A$ and $B$ can be
matrix valued functions of $x$. It turns out according to Proposition 2.3 that
the binary operation defined in (2.14) satisfies the pre-Lie identity (2.2).

The elements of the Magnus expansion can be re-expressed in terms of a pre-Lie
algebra action –pre-Lie Magnus expansion– (see also [8, 22] and references
therein). It is a matter of some tedious computation and use of the form of
expressions (1.10) and (2.14) to show that

$\displaystyle{\mathbb{Q}}^{(1)}(x)$ $\displaystyle=$
$\displaystyle\int_{x_{0}}^{x}dx_{1}A(x_{1}),$ (2.15)
$\displaystyle{\mathbb{Q}}^{(2)}(x)$ $\displaystyle=$ $\displaystyle-{1\over
2}\int_{x_{0}}^{x}dx_{2}\big{(}A\triangleright A\big{)}(x_{2}),$ (2.16)
$\displaystyle{\mathbb{Q}}^{(3)}(x)$ $\displaystyle=$
$\displaystyle\int_{x_{0}}^{x}dx_{3}\Big{(}{1\over 4}\big{(}(A\triangleright
A)\triangleright A\big{)}(x_{3})+{1\over
12}\big{(}A\triangleright(A\triangleright A)\big{)}(x_{3})\Big{)}.$ (2.17)

######  Example 2.5.

Let $f,g$ be linear operators and let the map $\Sigma:$
$\Sigma(f)_{n}=\sum_{m=1}^{n-1}f_{m}.$ Then

$\Sigma(f)\Sigma(g)=\Sigma(\Sigma(f)g+f\Sigma(g)+fg),$ (2.18)

i.e. $\Sigma$ is a Rota-Baxter operator of weight one.

Indeed, we rewrite equation (2.18) with the suitable discrete variables:

$\displaystyle\Sigma(f)_{n}\Sigma(g)_{n}$ $\displaystyle=$
$\displaystyle\sum_{m=1}^{n-1}f_{m}\sum_{k=1}^{n-1}g_{k}=\sum_{m>k}f_{m}g_{k}+\sum_{m<k}f_{k}g_{m}+\sum_{m=1}^{n-1}f_{n}g_{n}$
$\displaystyle=$
$\displaystyle\sum_{m=1}^{n-1}\big{(}f_{m}\Sigma(g)_{m}+\Sigma(f)_{m}g_{m}+f_{m}g_{m}\big{)}$
$\displaystyle=$
$\displaystyle\Sigma(f_{m}\Sigma(g)_{m}+\Sigma(f)_{m}g_{m}+f_{m}g_{m})_{n},$

and this concludes the proof.

As mentioned in the Introduction interesting relationships between pre-Lie
algebras, rooted tree graphs, dendriform and Rota-Baxter algebras already
exist (see for instance [30, 31]), whereas links between tridendriform, Rota-
Baxter and (quasi)shuffle algebras have been also studied in [32]. We shall
briefly recall now the relation between pre-Lie algebras and (tri)-dendriform
algebras.

######  Definition 2.6.

A tridendriform algebra ${\cal D}$ is a $k$-vector space equipped with three
binary operations $\prec,$ $\succ,$ $\cdot$and the following axioms [30, 31] :

  1. (1)

$(a\prec b)\prec c=a\prec(b\prec c+b\succ c+b\cdot c)$

  2. (2)

$(a\prec b)\succ c=a\succ(b\prec c)$

  3. (3)

$a\succ(b\succ c)=(a\prec b+a\succ b+a\cdot b)\succ c$

  4. (4)

$a\cdot(b\cdot c)=(a\cdot b)\cdot c$

  5. (5)

$(a\succ b)\cdot c=a\succ(b\cdot c)$

  6. (6)

$(a\prec b)\cdot c=a\cdot(b\succ c)$

  7. (7)

$(a\cdot b)\prec c=a\cdot(b\prec c).$

A dendriform algebra is defined by setting the product $\cdot$ to zero in the
above axioms, consequently the rules of a dendriform algebra are given in
terms of axioms (1)-(3) without the $\cdot$ term.

######  Remark 2.7.

We note that the product $x*y=x\prec y+x\succ y+x\cdot y$ is associative [31],
whereas the product $x\triangleright y=x\succ y-y\prec x+x\cdot y$
($x\triangleleft y=x\prec y-y\succ x+x\cdot y$) defines a pre-Lie algebra
${\mathrm{A}},$ $\forall x,y\in{\cal D}.$ In the case of a dendriform algebra
the term $x\cdot y$ is not present in both products, i.e. $x*y=x\prec y+x\succ
y$ and $x\triangleright y=x\succ y-y\prec x$ ($x\triangleleft y=x\prec
y-y\succ x$).

The binary actions of the (tri)dendriform algebra can be defined $\forall
x,y\in{\cal A}$ in terms of Rota-Baxter operators as

$x\succ y:=R(x)y,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ x\prec y:=xR(y),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ a\cdot b=\theta
ab,$ (2.19)

which leads to the findings of Proposition 2.3.

###  2.2. From pre-Lie algebras to braces

We report in this subsection the recent findings on the relations between pre-
Lie algebras and braces [41, 46], and in particular the passage from pre-Lie
algebras to braces. Before we discuss this passage we recall the definition of
a brace [40, 9] and we note that braces were essentially introduced in order
to derive set-theoretic solutions of the Yang-Baxter equation [40]. The
already known relationships between braces, the Yang-Baxter equation and
quantum integrability [13, 14], as well as the passage described below are
expected to lead to even deeper associations and the study of possibly novel
algebraic structures.

######  Definition 2.8.

A left brace is a set $B$ together with two group operations $+,\circ:B\times
B\to B$, the first is called addition and the second is called multiplication,
such that $\forall a,b,c\in B$,

$a\circ(b+c)=a\circ b-a+a\circ c.$ (2.20)

The additive identity of a left brace $B$ will be denoted by $0$ and the
multiplicative identity by $1,$ and in every left brace $0=1.$ Also, let
$(N,+,\cdot)$ be an associative ring which is a nilpotent. For $a,b\in N$
define $a\circ b=a\cdot b+a+b,$ then $(N,+,\circ)$ is a brace [40].

The group of formal flows constructed from a pre-Lie algebra was introduced in
[2] (see also [35]). We summarize below the passage from pre-Lie algebras to
[41, 46] after recalling the notion of the group of formal flows [2, 35]. We
also assume as in [46] that ${\mathrm{A}}$ is a nilpotent pre-Lie algebra.

  1. (1)

Let $a\in{\mathrm{A}}$, and let $L_{a}:{\mathrm{A}}\rightarrow{\mathrm{A}}$
denote the left multiplication by $a$, so $L_{a}(b):=a\triangleright b$. We
define $L_{c}\triangleright
L_{b}(a):=L_{c}(L_{b}(a))=c\triangleright(b\triangleright a),$ and

$e^{L_{a}}(b)=b+a\triangleright
b+{\frac{1}{2!}}a\triangleright(a\triangleright
b)+{\frac{1}{3!}}a\triangleright(a\triangleright(a\triangleright b))+\cdots$

  2. (2)

We formally consider the element ${\bf 1},$ such that ${\bf 1}\triangleright
a=a\triangleright{\bf 1}=a$ in the pre-Lie algebra (as in [35]) and define

$W(a):=e^{L_{a}}({\bf 1})-{\bf 1}=a+{\frac{1}{2!}}a\triangleright
a+{\frac{1}{3!}}a\triangleright(a\triangleright a)+\cdots$

$W(a):{\mathrm{A}}\rightarrow{\mathrm{A}}$ is a bijective function, provided
that ${\mathrm{A}}$ is a nilpotent pre-Lie algebra.

  3. (3)

Let $\Omega(a):{\mathrm{A}}\rightarrow{\mathrm{A}}$ be the inverse function to
the function $W(a)$, i.e. $\Omega(W(a))=W(\Omega(a))=a$. Following [35] the
first terms of $\Omega$ are

$\Omega(a)=a-{\frac{1}{2}}a\triangleright a+{\frac{1}{4}}(a\triangleright
a)\triangleright a+{\frac{1}{12}}a\triangleright(a\triangleright a)+\ldots$

  4. (4)

We define the multiplication,

$a\circ b:=a+e^{L_{\Omega(a)}}(b).$

The addition is the same as in the pre-Lie algebra ${\mathrm{A}}$. It was
shown in [2] that $({\mathrm{A}},\circ)$ is a group. It is then
straightforward to show that $({\mathrm{A}},\circ,+)$ is a left brace, indeed

$a\circ(b+c)+a=a+e^{L_{\Omega(a)}}(b+c)+a=(a+e^{L_{\Omega(a)}}(b))+(a+e^{L_{\Omega(a)}}(c))=a\circ
b+a\circ c.$

The above formula can also be written using the Baker-Campbell-Hausdorff (BCH)
formula, (see [2, 35]). We first recall that the Lie algebra $L({\mathrm{A}})$
is obtained from a pre-Lie algebra $A$ by defining $[a,b]=a\triangleright
b-b\triangleright a$ (with the same addition as ${\mathrm{A}}$). By means of
the BCH formula, $e^{L_{a}}(e^{L_{b}}({\bf 1}))=e^{L_{C(a,b)}}({\bf 1}),$ the
element $C(a,b)$ can be represented in the form of a series as
$C(a,b)=a+b+{\frac{1}{2}}[a,b]+{\frac{1}{12}}([a,[a,b]]+[b,[b,a]])+\ldots$

######  Lemma 2.9.

The following formula for the multiplication $\circ$ defined above holds (see
e.g. [35]):

$W(a)\circ W(b)=W(C(a,b)),$

where $C(a,b)$ is obtained using the BCH series in the Lie algebra
$L({\mathrm{A}})$.

###### Proof.

The proof is immediate from the definition of the brace multiplication:

$\displaystyle W(a)\circ W(b)$ $\displaystyle=$ $\displaystyle
W(a)+e^{L_{\Omega(W(a))}}(W(b))=e^{L_{a}}({\bf 1})-{\bf
1}+e^{L_{a}}(e^{L_{b}}({\bf 1})-{\bf 1})=$ $\displaystyle=$ $\displaystyle
e^{L_{a}}(e^{L_{b}}({\bf 1}))-{\bf 1}=e^{L_{C(a,b)}}({\bf 1})-{\bf
1}=W(C(a,b)).\hfill\qed$

######  Example 2.10.

We recall the expression from Magnus expansion
${\mathbb{Q}}(x,\alpha)=\sum_{m>0}{\mathbb{Q}}^{(m)}(x)\alpha^{m}$ (2.17). Let
${\mathbb{Q}}(x,\alpha)=\int_{x_{0}}^{x}\Omega(\xi,\alpha;A)d\xi,$ then from
(2.17) we read off:

$\Omega(\xi,\alpha;A)=\alpha A(\xi)-{1\over 2}\alpha^{2}(A\triangleright
A)(\xi)+{1\over 4}\alpha^{3}\big{(}(A\triangleright A)\triangleright
A\big{)}(\xi)+{1\over 12}\alpha^{3}\big{(}A\triangleright(A\triangleright
A)\big{)}(\xi)+\ldots,$

where for any linear operators $A,B:$ $(L_{A}(B))(x):=(A\triangleright
B)(x)=\big{[}\int_{x_{0}}^{x}A(s)ds,\ B(x)\big{]},$ and

$\big{(}e^{L_{A}}(B)\big{)}(x)=B(x)+(A\triangleright
B)(x)+{\frac{1}{2!}}\big{(}A\triangleright(A\triangleright
B)\big{)}(x)+{\frac{1}{3!}}\big{(}A\triangleright(A\triangleright(A\triangleright
B))\big{)}(x)+\ldots$

And in this case the brace multiplication in defined as: $(A\circ
B)(x):=A(x)+\big{(}e^{L_{\Omega(\alpha,A)}}(B)\big{)}(x)$ (see also [2]).

##  3\. Discrete Magnus expansion $\&$ Pre-Lie algebras

In Section 1 we described the solution of the linear evolution problem
expressed as Dyson and Magnus series and we recalled the pre-Lie Magnus
expansion. We are now focusing on the discrete evolution problem and the
derivation of the discrete analogue of the Magnus expansion. Our starting
point will be the discrete evolution problem; indeed let ${\mathbb{L}},T$ be
some linear operators depending on some discrete index $n\in{\mathbb{Z}}^{+}$
and an extra parameter $\alpha\in{\mathbb{C}}$ (e.g.
${\mathbb{L}},T\in\mbox{End}({\mathbb{C}}^{\cal N})$):

$T_{n+1}(\alpha)={\mathbb{L}}_{n}(\alpha)T_{n}(\alpha).$ (3.1)

The solution of the difference equation above is found by iteration and is
given by
$T_{N+1}(\alpha)={\mathbb{L}}_{N}(\alpha)\ldots{\mathbb{L}}_{1}(\alpha)$ (let
us choose for simplicity $T_{1}(\alpha)=1$ as initial condition), and as in
the continuous case we express the monodromy matrix as
$T_{N+1}(\alpha)=1+\sum_{n>1}\alpha^{n}T^{(n)}(N+1)$.

We consider the general scenario, where the ${\mathbb{L}}$-operator is
formally expressed as ${\mathbb{L}}(\alpha)=1+\sum_{k>1}\alpha^{k}L^{(k)},$
where in general $L^{(m)}$ are linear operators (or algebraic objects), for
instance for our purposes here we will be considering
$L^{(m)}\in\mbox{End}({\mathbb{C}}^{\cal N})\otimes{\mathfrak{A}},$ where
${\mathfrak{A}}$ is some quantum algebra to be identified in the subsequent
sections, or $L^{(m)}\in\mbox{End}({\mathbb{C}}^{\cal N}).$ Notice that
${\mathbb{L}}$ may be re-expressed as
${\mathbb{L}}(\alpha)=e^{{\mathrm{Q}}(\alpha)}$ (see next section for a
detailed exposition and relation to the Yangian algebra). Considering the
generic form of ${\mathbb{L}}$ we obtain the form of the coefficients of the
monodromy:

$\displaystyle
T^{(m)}(N+1)=\sum_{n=1}^{N}\sum_{k=1}^{m}\Big{(}L_{n}^{(m_{k})}\sum_{n_{k-1}=1}^{n-1}L^{(m_{k-1})}_{n_{k-1}}\ldots\sum_{n_{1}=1}^{n_{2}-1}L^{(m_{1})}_{n_{1}}\Big{)}_{\sum_{j=1}^{k}m_{j}=m}.$
(3.2)

In the following Proposition we express the elements of the discrete analogue
of Dyson’s series (3.2) in terms of the tridendriform algebra.

######  Proposition 3.1.

The elements of the discrete analogue of Dyson series (3.2) are expressed in
terms of the tridendriform action $\prec$ as

$\displaystyle
T^{(m)}(N+1)=\sum_{n=1}^{N}\sum_{k=1}^{m}\Big{(}L^{(m_{k})}\prec(L^{(m_{k-1})}\prec(\ldots\prec(L^{(m_{2})}\prec
L^{(m_{1})}))\ldots)\Big{)}_{n}\big{|}_{\sum_{j=1}^{k}m_{j}=m}.$ (3.3)

###### Proof.

We first recall (2.19) and the fact that in this case the Rota-Baxter operator
of weight one is the summation, i.e.
$R(x):=\Sigma(x)_{n}=\sum_{m=1}^{n-1}x_{m},$ then $(x\prec
y)_{n}=x_{n}\Sigma(y)_{n},$ where $x,y$ are in general some linear operators.
Using the definitions of $\Sigma$ and the explicit expressions of $T^{(m)},$
$m\in\\{1,2,\ldots,N\\}$ (3.2), we write

$T^{(m)}(N+1)=\sum_{n=1}^{N}L^{(m_{k})}_{n}\Sigma(L^{(m_{k-1})}\Sigma(L^{(m_{k-2})}\Sigma(\ldots\Sigma(L^{(m_{1})}))\ldots))_{n}\big{|}_{\sum_{j=1}^{k}m_{j}=m}.$
(3.4)

Via $(x\prec y)_{n}=x_{n}\Sigma(y)_{n},$ expression (3.4) leads to (3.3).

For instance, for $n=1,$ $T^{(1)}(N+1)=\sum_{n=1}^{N}L^{(1)}_{n},$ for $n=2$

$\displaystyle T^{(2)}(N+1)$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{N}\Big{(}L^{(1)}_{n}\sum_{m=1}^{n-1}L^{(1)}_{m}+L^{(2)}_{n}\Big{)}=\sum_{n=1}^{N}\Big{(}L^{(1)}_{n}\Sigma(L^{(1)})_{n}+L^{(2)}_{n}\Big{)}$
$\displaystyle=$ $\displaystyle\sum_{n=1}^{N}\Big{(}(L^{(1)}\prec
L^{(1)})_{n}+L_{n}^{(2)}\Big{)}$

for $n=3,$

$\displaystyle T^{(3)}(N+1)$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{N}\Big{(}L^{(1)}_{n}\sum_{m=1}^{n-1}L^{(1)}_{m}\sum_{k=1}^{m-1}L^{(1)}_{k}+L^{(1)}_{n}\sum_{m=1}^{n-1}L^{(2)}_{m}+L^{(2)}_{n}\sum_{m=1}^{n-1}L^{(1)}_{m}+L_{n}^{(3)}\Big{)}$
(3.6) $\displaystyle=$
$\displaystyle\sum_{n=1}^{N}\Big{(}L^{(1)}_{n}\Sigma(L^{(1)}\Sigma(L^{(1)}))_{n}+L^{(1)}_{n}\Sigma(L^{(2)})_{n}+L_{n}^{(2)}\Sigma(L^{(1)})_{n}+L_{n}^{(3)}\Big{)}$
$\displaystyle=$
$\displaystyle\sum_{n=1}^{N}\Big{(}\big{(}L^{(1)}\prec(L^{(1)}\prec
L^{(1)})\big{)}_{n}+\big{(}L^{(1)}\prec L^{(2)}\big{)}_{n}+\big{(}L^{(2)}\prec
L^{(1)}\big{)}_{n}+L_{n}^{(3)}\Big{)}.\hfill\qed$

We come now to the precise derivation of the discrete Magnus expansion. We
consider the Lie exponential $T_{N+1}(\alpha)=e^{{\mathbb{Q}}_{N+1}(\alpha)},$
${\mathbb{Q}}_{N+1}(\alpha)=\sum_{m=1}^{\infty}\alpha^{m}{\mathbb{Q}}^{(m)}(N+1),$
which will lead to the discrete analogue of Magnus expansion; expressions
(1.4), (1.5), (1.8) and (1.9) hold.

######  Lemma 3.2.

The quantities ${\mathbb{Q}}^{(k)}(N+1):=\sum_{n=1}^{N}\Omega^{(k)}_{n},$
where $\Omega^{(k)}_{n}={\mathbb{Q}}^{(k)}(n+1)-{\mathbb{Q}}^{(k)}(n),$ are
expressed explicitly as:

$\displaystyle{\mathbb{Q}}^{(1)}(N+1)$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{N}L^{(1)}_{n},$ (3.8)
$\displaystyle{\mathbb{Q}}^{(2)}(N+1)$ $\displaystyle=$ $\displaystyle{1\over
2}\sum_{n>n_{1}=1}^{N}\big{[}L^{(1)}_{n},\ L^{(1)}_{n_{1}}\big{]}-{1\over
2}\sum_{n=1}^{N}(L^{(1)}_{n})^{2}+\sum_{n=1}^{N}L_{n}^{(2)},$ (3.9)
$\displaystyle{\mathbb{Q}}^{(3)}(N+1)$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{N}\Big{(}{1\over
6}\sum_{n_{2}>n_{1}=1}^{n-1}\big{(}\big{[}L^{(1)}_{n},\big{[}L^{(1)}_{n_{2}},\
L^{(1)}_{n_{1}}\big{]}\big{]}+\big{[}\big{[}L^{(1)}_{n},\
L^{(1)}_{n_{2}}\big{]},L^{(1)}_{n_{1}}\big{]}\big{)}$ (3.10) $\displaystyle+$
$\displaystyle{1\over
6}\sum_{n_{1}=1}^{n-1}\big{(}L^{(1)}_{n_{1}}\big{[}L^{(1)}_{n_{1}},\
L^{(1)}_{n}\big{]}+\big{[}L^{(1)}_{n_{1}},\
L^{(1)}_{n}\big{]}L^{(1)}_{n}\big{)}$ (3.11) $\displaystyle+$
$\displaystyle{1\over 6}\sum_{n_{1}=1}^{n-1}\big{(}\big{[}L^{(1)}_{n_{1}},\
(L^{(1)}_{n})^{2}\big{]}+\big{[}(L^{(1)}_{n_{1}})^{2},\
L^{(1)}_{n}\big{]}\big{)}-{1\over
2}\sum_{n=1}^{N}\big{(}L_{n}^{(1)}L_{n}^{(2)}+L_{n}^{(2)}L_{n}^{(1)}\big{)}$
(3.12) $\displaystyle+$ $\displaystyle{1\over 3}\ (L^{(1)}_{n})^{3}-{1\over
2}\sum_{m=1}^{n-1}\big{(}\big{[}L^{(1)}_{m},L_{n}^{(2)}\big{]}+\big{[}L^{(2)}_{m},L^{(1)}_{n}\big{]}\big{)}+L_{n}^{(3)}\Big{)},\
\ldots$ (3.13)

###### Proof.

Recursive expressions (1.8), (1.9) apparently still hold, but now $T^{(m)}$
are given by (3.2). Notice that in the discrete case both $T_{N+1}(\lambda)$
and $T^{(m)}(N+1)$ depend on a discrete parameter $N,$ which replaces the
continuum parameter $x$ of the continuous analogue discussed in Section 1. The
quantities $\Omega^{(k)}$ can be immediately read from (3.13), and are the
discrete analogues of the derivatives $\dot{\mathbb{Q}}^{(k)}(x)$, of Magnus
expansion in Section 1. ∎

######  Corollary 3.3.

Let $A,B$ be linear operators (or generic algebraic objects) that depend on a
discrete parameter $n\in{\mathbb{Z}}^{+},$ and define the binary operation
$\triangleright$: $(A,B)\mapsto A\triangleright B,$ such that

$(A\triangleright B)_{n}:=\big{[}\sum_{m=1}^{n-1}A_{m},\
B_{n}\big{]}+A_{n}B_{n}.$ (3.14)

Then the pre-Lie identity is satisfied, i.e.

$\big{(}(A\triangleright B)\triangleright
C\big{)}_{n}-\big{(}A\triangleright(B\triangleright
C)\big{)}_{n}=\big{(}(B\triangleright A)\triangleright
C\big{)}_{n}-\big{(}B\triangleright(A\triangleright C)\big{)}_{n}.$ (3.15)

###### Proof.

The proof is immediate by means of Proposition 2.3 and Example 2.5. ∎

######  Proposition 3.4.

The elements of the discrete Magnus expansion (3.13) can be re-expressed in
terms of the pre-Lie action (3.14).

###### Proof.

We recall the definition of the pre-Lie binary operation (3.14), then

$(x\triangleright x)_{n}=\big{[}\sum_{m=1}^{n-1}x_{m},\
x_{n}\big{]}+x_{n}^{2},$ (3.16)

also, from expressions (2.7) and (2.1), for $R\to\Sigma,$ and by setting
$x=y=z$ we obtain:

$\displaystyle{1\over 4}\big{(}(x\triangleright x)\triangleright
x\big{)}_{n}+{1\over 12}\big{(}x\triangleright(x\triangleright x)\big{)}_{n}$
$\displaystyle=$ $\displaystyle{1\over
6}\sum_{n_{2}>n_{1}=1}^{n-1}\Big{(}\big{[}x_{n},\big{[}x_{n_{2}},\
x_{n_{1}}\big{]}\big{]}+\big{[}\big{[}x_{n},\
x_{n_{2}}\big{]},x_{n_{1}}\big{]}\big{)}$ (3.17) $\displaystyle+$
$\displaystyle{1\over 6}\sum_{n_{1}=1}^{n-1}\Big{(}x_{n_{1}}\big{[}x_{n_{1}},\
x_{n}\big{]}+\big{[}x_{n_{1}},\ x_{n}\big{]}x_{n}\Big{)}$ (3.18)
$\displaystyle+$ $\displaystyle{1\over
6}\sum_{n_{1}=1}^{n-1}\Big{(}\big{[}x_{n_{1}},\
x^{2}_{n}\big{]}+\big{[}x^{2}_{n_{1}},\ x_{n}\big{]}\big{)}+{1\over
3}x^{3}_{n}.$ (3.19)

Comparing expressions (3.13) with (3.16) and (3.19) we conclude

$\displaystyle{\mathbb{Q}}^{(1)}(N+1)=\sum_{n=1}^{N}L^{(1)}_{n},$ (3.20)
$\displaystyle{\mathbb{Q}}^{(2)}(N+1)=-{1\over
2}\sum_{n=1}^{N}(L^{(1)}\triangleright
L^{(1)})_{n}+\sum_{n=1}^{N}L_{n}^{(2)},$
$\displaystyle{\mathbb{Q}}^{(3)}(N+1)=\sum_{n=1}^{N}\Big{(}{1\over
4}\big{(}(L^{(1)}\triangleright L^{(1)})\triangleright
L^{(1)}\big{)}_{n}+{1\over
12}\big{(}L^{(1)}\triangleright(L^{(1)}\triangleright
L^{(1)})\big{)}_{n}\Big{)}$ (3.21) $\displaystyle\qquad\qquad\qquad-{1\over
2}\sum_{n=1}^{N}\Big{(}(L^{(2)}\triangleright
L^{(1)})_{n}+(L^{(1)}\triangleright
L^{(2)})_{n}\Big{)}+\sum_{n=1}^{N}L_{n}^{(3)}.$ (3.22)

The above expressions provide elegant expressions of the discrete analogues of
the pre-Lie Magnus expansion. Higher order terms are computed by iteration via
(1.8), (1.9). ∎

######  Remark 3.5.

It is worth focusing on the simple linear case, where
${\mathbb{L}}(\alpha)=1+\alpha{\mathbb{P}},$ i.e. $L^{(1)}={\mathbb{P}}$ and
$L^{(m)}=0$ $\forall m>1.$ Then we obtain the following explicit expressions,
which provide a simpler discrete analogue of Dyson’s series:

$\displaystyle
T_{N+1}(\alpha)=1+\sum_{m=1}^{N}\alpha^{m}T^{(m)}(N+1),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
T^{(m)}(N+1)=\sum_{n_{m}>\ldots>n_{1}=1}^{N}{\mathbb{P}}_{n_{m}}\ldots{\mathbb{P}}_{n_{1}}.$
(3.23)

According to Proposition 3.1 the elements of the discrete Dyson series (3.23)
are expressed in terms of the tridendriform action $\prec$ as

$\displaystyle
T^{(m)}(N+1)=\sum_{n=1}^{N}\Big{(}{\mathbb{P}}\prec({\mathbb{P}}\prec({\mathbb{P}}\prec(\ldots\prec({\mathbb{P}}\prec{\mathbb{P}}))\ldots))\Big{)}_{n},$
(3.24)

with $m$ ${\mathbb{P}}$-terms. We also recall that $T:=e^{{\mathbb{Q}}},$ then
via (1.4) and (1.5) we immediately obtain the formulas for
${\mathbb{Q}}^{(m)}(N+1)$, given the findings of the generic case, and the
pre-Lie discrete Magnus expansion in this case takes the simple form
resembling structurally the continuum case,

$\displaystyle{\mathbb{Q}}^{(1)}(N+1)=\sum_{n=1}^{N}{\mathbb{P}}_{n},$ (3.25)
$\displaystyle{\mathbb{Q}}^{(2)}(N+1)=-{1\over
2}\sum_{n=1}^{N}({\mathbb{P}}\triangleright{\mathbb{P}})_{n}$
$\displaystyle{\mathbb{Q}}^{(3)}(N+1)=\sum_{n=1}^{N}\Big{(}{1\over
4}\big{(}({\mathbb{P}}\triangleright{\mathbb{P}})\triangleright{\mathbb{P}}\big{)}_{n}+{1\over
12}\big{(}{\mathbb{P}}\triangleright({\mathbb{P}}\triangleright{\mathbb{P}})\big{)}_{n}\Big{)}.$
(3.26)

Higher order terms are obtained via (1.8), (1.9). Expressions (3.26) are much
more concise compared to the general ones (3.22), and apparently similar to
the corresponding continuous formulas, given that certain “boundary terms” are
missing in the linear scenario.

######  Remark 3.6.

We briefly discuss now the associated brace structure emerging from the
discrete pre-Lie Magnus expansion, which provides one more example of brace
construction in accordance to Subsection 2.2. We recall expressions
${\mathbb{Q}}_{N+1}(\alpha)=\sum_{m>0}{\mathbb{Q}}^{(m)}(N+1)\alpha^{m}$
(2.17). Let
${\mathbb{Q}}_{N+1}(\alpha)=\sum_{n=1}^{N}\Omega_{n}(\alpha;{\mathbb{P}}),$
then from (3.26) we read off:

$\Omega_{n}(\alpha;{\mathbb{P}})=\alpha{\mathbb{P}}_{n}-{1\over
2}\alpha^{2}({\mathbb{P}}\triangleright{\mathbb{P}})_{n}+{1\over
4}\alpha^{3}\big{(}({\mathbb{P}}\triangleright{\mathbb{P}})\triangleright{\mathbb{P}}\big{)}_{n}+{1\over
12}\alpha^{3}\big{(}{\mathbb{P}}\triangleright({\mathbb{P}}\triangleright{\mathbb{P}})\big{)}_{n}+\ldots,$
(3.27)

where we recall that for ${\mathbb{P}},B$ being linear operators
$(L_{{\mathbb{P}}}(B))_{n}:=({\mathbb{P}}\triangleright
B)_{n}=\big{[}\sum_{m=1}^{n-1}{\mathbb{P}}_{m},\
B_{n}\big{]}+{\mathbb{P}}_{n}B_{n},$ and

$\big{(}e^{L_{{\mathbb{P}}}}(B)\big{)}_{n}=B_{n}+({\mathbb{P}}\triangleright
B)_{n}+{\frac{1}{2!}}\big{(}{\mathbb{P}}\triangleright({\mathbb{P}}\triangleright
B)\big{)}_{n}+{\frac{1}{3!}}\big{(}{\mathbb{P}}\triangleright({\mathbb{P}}\triangleright({\mathbb{P}}\triangleright
B))\big{)}_{n}+\ldots$

And in this case the brace multiplication in defined as: $({\mathbb{P}}\circ
B)_{n}={\mathbb{P}}_{n}+\big{(}e^{L_{\Omega(\alpha,{\mathbb{P}})}}(B)\big{)}_{n}.$

######  Remark 3.7.

(The continuum limit). Recall the discrete evolution problem (3.1), then
rescale $\alpha\to\delta\alpha$ ($\delta<<1$) and consider the general form
${\mathbb{L}}(\lambda)=1+\sum_{m>1}\alpha^{m}\delta^{n}L^{(m)}.$ Expression
(3.1) can be rewritten as (keeping only linear terms in $\delta$)

$T_{n+1}(\alpha)=(1+\alpha\delta L^{(1)}_{n+1})T_{n}(\alpha).$ (3.28)

By considering the following “dictionary” as $\delta\to 0$: $L^{(1)}_{n+1}\to
A(x),$ and ${\Psi_{n+1}-\Psi_{n}\over\delta}\to\partial_{\xi}\Psi(\xi)$ we
arrive at the continuum limit of (3.1), which is the linear evolution problem

$\partial_{\xi}T(\xi,\alpha)=\alpha A(\xi)T(\xi,\alpha).$ (3.29)

Detailed proof on the continuum limit of the discrete monodromy that lead to
continuum monodromy, based on a “power counting rule” is given in [4]. The
counting rule relies on the fact
$\delta\sum_{n=1}^{N}f_{n}\to\int_{0}^{x}f(\xi)d\xi,$ and terms of the form:
$\delta^{\sum_{j=1}^{m}n_{j}}\sum_{k_{1},k_{2},...,k_{l}}L_{k_{1}}^{(n_{1})}\ldots
L_{k_{m}}^{(n_{m})}\to 0$ in the continuum limit for $\sum_{j=1}^{m}n_{j}>m.$

In the continuum limit the monodromy matrix $T$ is expressed as a Dyson series
with terms that are written, via (3.23), (3.24) in terms of a dendriform
action $(A\prec B)(x):=A(x)\int_{0}^{x}B(\xi)d\xi,$ as

$\displaystyle
T^{(m)}(x)=\int_{0}^{x}d\zeta\Big{(}A\prec(A\prec(A\prec(\ldots\prec(A\prec
A))\ldots))\Big{)}(\zeta).$ (3.30)

###  3.1. An alternative discrete expansion

Let us now consider a slightly different scenario, where the
${\mathbb{L}}$-operator in (3.1) is of the form
${\mathbb{L}}(\alpha)={\mathrm{M}}+\alpha{\mathrm{L}},$ ${\mathrm{M}}\neq 1$
and ${\mathrm{M}},$ ${\mathrm{L}}$ are some linear operators. Before we state
the main findings in the next Proposition it is useful to introduce some
notation.We set:

  1. (1)

${\mathbb{M}}_{N+1,n-1}:={\mathrm{M}}_{N}\ldots{\mathrm{M}}_{n}{\mathrm{M}}_{n-1}$

  2. (2)

$\hat{\mathrm{M}}:={\mathrm{M}}^{-1}\ $ and $\
\hat{\mathbb{M}}_{n-1,N+1}=\hat{\mathrm{M}}_{n-1}\hat{\mathrm{M}}_{n}\ldots\hat{\mathrm{M}}_{N}$

  3. (3)

${\mathbb{M}}_{N+1,1}:={\mathbb{M}}_{N+1}\ $ and $\
\hat{\mathbb{M}}_{1,N+1}:=\hat{\mathbb{M}}_{N+1}.$

######  Proposition 3.8.

Let ${\mathbb{L}}$ in (3.1) be of the form
${\mathbb{L}}(\alpha)={\mathrm{M}}+\alpha{\mathrm{L}},$ and the monodromy be
$T_{N+1}(\alpha)=\sum_{n=0}^{N}\alpha^{n}T^{(n)}(N+1),$ where in general
${\mathrm{M}}\neq 1$ and ${\mathrm{L}}$ are linear operators. Then the
monodromy matrix $T_{N+1}(\alpha)$ can be expressed as

$T_{N+1}(\alpha)=e^{{\mathbb{Q}}_{N+1}(\alpha)}{\mathbb{M}}_{N+1},$ (3.31)

where
${\mathbb{Q}}_{N+1}(\alpha)=\sum_{m=1}^{N+1}\alpha^{m}{\mathbb{Q}}^{(m)}(N+1)$
is the discrete Magnus expansion as in Remark 3.5 and the elements
${\mathbb{Q}}^{(m)}(N+1)$ are given in (3.26) as the pre-Lie Magnus expansion.

###### Proof.

We start with the standard ordered expansion of the monodromy matrix, given
the choice ${\mathbb{L}}(\alpha)=M+\alpha L$ we obtain

$\displaystyle T_{N+1}(\alpha)$ $\displaystyle=$
$\displaystyle{\mathbb{M}}_{N+1}+\alpha\sum_{n=1}^{N}{\mathrm{M}}_{N+1}\ldots{\mathrm{M}}_{n+1}{\mathrm{L}}_{n}{\mathrm{M}}_{n-1}\ldots{\mathrm{M}}_{1}$
(3.32) $\displaystyle+$
$\displaystyle\alpha^{2}\sum_{n>m=1}^{N}{\mathrm{M}}_{N}\ldots{\mathrm{M}}_{n+1}{\mathrm{L}}_{n}{\mathrm{M}}_{n-1}\ldots{\mathrm{M}}_{m+1}{\mathrm{L}}_{m}{\mathrm{M}}_{m-1}\ldots{\mathrm{M}}_{1}+\ldots$

after some lengthy, but straightforward computation and after defining:

${\bf
P}_{n}:={\mathbb{M}}_{N,n-1}{\mathrm{L}}_{n}\hat{\mathrm{M}}_{n}\hat{\mathbb{M}}_{n-1,N},$
(3.33)

we arrive at

$\displaystyle
T_{N+1}(\alpha)=\Big{(}1+\sum_{m=1}^{N}\alpha^{m}\sum_{n_{m}>...>n_{1}=1}^{N}{\bf
P}_{n_{m}}{\bf P}_{n_{m-1}}\ldots{\bf P}_{n_{1}}\Big{)}{\mathbb{M}}_{N+1}.$
(3.34)

The bracket in the expression above is just the discrete ordered expansion
(3.24), which as shown in the previous subsection can be expressed as the
discrete analogue of Magnus expansion, and also in terms of a suitable pre-Lie
action. Notice however that contrary to the previous analysis of Section 3,
the objects ${\bf P}_{n}$ (3.33) are not local anymore, but are “dressed”
quantities via a gauge transformation ${\mathbb{M}}_{N,n-1}$. ∎

###  3.2. Gauge fields $\&$ open boundary systems

We consider in this subsection two fundamental problems/applications very much
associated to the derivations of the previous subsection. The first problem is
the transformation of the field ${\mathbb{L}}$ ($A$ in the continuous case)
via a gauge transformation, and the second one is the construction of the
evolution problem and solution for certain classes of systems with open
boundary conditions.

Gauge fields. Let $T_{n}$ be a solution of the difference equation (3.1) and
$\hat{T}_{n}$ a solution of

$\hat{T}_{n+1}(\alpha)=\hat{\mathbb{L}}_{n}(\alpha)\hat{T}_{n}(\alpha).$
(3.35)

Let also ${\mathrm{G}}(\alpha)$ be a linear operator, such that
$\hat{T}_{n}(\alpha)={\mathrm{G}}_{n}(\alpha)T_{n}(\alpha),$ then via (3.1),
(3.35), we conclude for the transformed field,
$\hat{\mathbb{L}}_{n}(\alpha)={\mathrm{G}}_{n+1}(\alpha){\mathbb{L}}_{n}(\alpha){\mathrm{G}}_{n}^{-1}(\alpha).$
Assuming that the operators ${\mathbb{L}_{n}},\hat{\mathbb{L}}_{n}$ are given
we solve the following difference equation to identify the gauge
transformation

${\mathrm{G}}_{n+1}(\alpha)=\hat{\mathbb{L}}_{n}(\alpha){\mathrm{G}}_{n}(\alpha){\mathbb{L}}^{-1}_{n}(\alpha).$
(3.36)

######  Lemma 3.9.

We recall that
$\hat{T}_{N+1}(\alpha):=\hat{\mathbb{L}}_{N}(\alpha)\ldots\hat{\mathbb{L}}_{1}(\alpha)$
and
$T^{-1}_{N+1}(\alpha):={\mathbb{L}}^{-1}_{1}(\alpha)\ldots{\mathbb{L}}^{-1}_{N}(\alpha).$
The solution of the difference equation (3.36) is given by
${\mathrm{G}}_{N+1}(\alpha)=\hat{T}_{N+1}(\alpha){\mathrm{G}}_{1}(\alpha)T^{-1}_{N+1}(\alpha),$
${\mathrm{G}}_{1}(\alpha)$ is some generic initial value.

###### Proof.

The solution of (3.36) is obtained directly by iteration. ∎

Similarly, in the continuous case (see also Remark 3.7) we can either consider
the continuous limit of the above expressions or directly apply the gauge
transformation: $\hat{T}(x,\alpha)={\mathrm{G}}(x,\alpha)T(x,\alpha)$, where
$T$ satisfies (1.1) and $\hat{T}$ satisfies
$\partial_{\xi}\hat{T}(\xi,\alpha)=\alpha\hat{A}(\xi)\hat{T}(\xi,\alpha).$
Hence, we conclude that the transformed field is given as
$\hat{A}(\xi)={\mathrm{G}}(\xi,\alpha)A{\mathrm{G}}^{-1}(\xi,\alpha)+\alpha^{-1}\partial_{\xi}{\mathrm{G}}(\xi,\alpha){\mathrm{G}}^{-1}(\xi,\alpha),$
which is the familiar transformation of a gauge field $A.$ Suppose that
$A,\hat{A}$ are given, then ${\mathrm{G}}(\xi,\alpha)$ satisfies the evolution
problem

$\partial_{\xi}{\mathrm{G}}(\xi,\alpha)=\alpha\hat{A}(\xi){\mathrm{G}}(\xi,\alpha)-\alpha{\mathrm{G}}(\xi,\alpha)A(\xi).$
(3.37)

The solution of the latter equation is given by
${\mathrm{G}}(x,\alpha)=\hat{T}(x,\alpha){\mathrm{G}}_{0}(\alpha)T^{-1}(x,\alpha),$
where we recall that,
$T(x,\alpha)=\overset{\curvearrowleft}{\mathrm{P}}\mbox{exp}\Big{(}\alpha\int_{0}^{x}A(\xi)d\xi\
\Big{)}$ (similarly for $\hat{T}(x,\alpha)$) and ${\mathrm{G}}_{0}(\alpha)$ is
some initial value at $x=0.$

This type of problems systematically appear in the context of integrable
systems, where a Lax pair and strong compatibility conditions (zero curvature
condition, i.e. equations of motion) exist [29, 1, 24]. In this frame dressing
schemes [49, 17, 36] are used in order to obtain solutions of the associated
integrable non-linear ODEs and PDEs that emerge from the zero curvature
condition.

Typically, in integrable systems due to the existence of a Lax pair or the
existence of a classical or quantum $R$-matrix [24, 25] (see also [11] and
references therein) it turns out that $trT(\alpha)$ provides a hierarchy of
conserved quantities; for instance the Hamiltonian of the system with periodic
boundary conditions is a member of this hierarchy (a more detailed discussion
will follow later in the text). We are regarding in the next section both
quantum and classical integrable systems, and we establish fundamental
connections with the findings of the present section. A relevant construction,
which leads to systems with open boundary conditions is presented below.

Open boundary conditions. We now focus on the evolution problem associated to
integrable systems with open boundary conditions, although we are not going to
discuss here about the notion of integrability for such systems (the
interested reader is referred to [45] for a detailed exposition). In order to
describe discrete systems with open boundary conditions within integrability
[45], we also need to consider the following difference equation

$\hat{T}_{n+1}(\alpha)=\hat{T}_{n}(\alpha)\hat{\mathbb{L}}_{n}(\alpha).$
(3.38)

The solution of equation (3.38) is found by iteration and is given by
$\hat{T}_{N+1}(\alpha)=\hat{\mathbb{L}}_{1}(\alpha)\ldots\hat{\mathbb{L}}_{N}(\alpha)$
(we choose for simplicity $\hat{T}_{1}(\alpha)=1$ as initial condition). We
express the monodromy as
$\hat{T}_{N+1}(\alpha)=1+\sum_{n>1}\alpha^{n}\hat{T}^{(n)}(N+1)$ and recall
the exponential map
$\hat{T}_{N+1}(\alpha)=e^{\hat{\mathbb{Q}}_{N+1}(\alpha)};$ expressions
analogous to (4.5), (1.5), (1.8) and (1.9) naturally hold.

We consider the general scenario, where
$\hat{\mathbb{L}}(\alpha)=1+\sum_{k>1}\alpha^{k}\hat{L}^{(k)},$ and in general
$\hat{L}^{(m)}$ are linear operators, then the form of the coefficients of the
monodromy $\hat{T}$ are given as

$\displaystyle\hat{T}^{(m)}(N+1)=\sum_{n_{1}=1}^{N}\sum_{k=1}^{m}\Big{(}\sum_{n_{k}=1}^{n_{k-1}-1}\hat{L}_{n_{k}}^{(m_{k})}\sum_{n_{k-1}=1}^{n_{k-2}-1}\hat{L}^{(m_{k-1})}_{n_{k-1}}\ldots\sum_{n_{2}=1}^{n_{1}-1}\hat{L}^{(m_{2})}_{n_{1}}\hat{L}^{(m_{1})}_{n_{1}}\Big{)}_{\sum_{j=1}^{k}m_{j}=m}.$
(3.39)

We may than express the elements of the discrete analogue of Dyson’s series
(3.39) in terms of the tridendriform algebra in analogy to Proposition 3.1.

######  Lemma 3.10.

The elements of the generalized discrete Dyson series (3.39) are expressed in
terms of the tridendriform action $\succ$ as

$\displaystyle\hat{T}^{(m)}(N+1)=\sum_{n=1}^{N}\sum_{k=1}^{m}\Big{(}(\ldots(\hat{L}^{(m_{k})}\succ\hat{L}^{(m_{k-1})})\succ\ldots)\succ\hat{L}^{(m_{1})}\Big{)}_{n}\big{|}_{\sum_{j=1}^{k}m_{j}=m}.$
(3.40)

For instance, for $n=1,$ $\hat{T}^{(1)}(N+1)=\sum_{n=1}^{N}\hat{L}^{(1)}_{n},$
for $n=2$ and $n=3$

$\displaystyle\hat{T}^{(2)}(N+1)$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{N}\Big{(}(\hat{L}^{(1)}\succ\hat{L}^{(1)})_{n}+\hat{L}_{n}^{(2)}\Big{)}$
(3.41) $\displaystyle\hat{T}^{(3)}(N+1)$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{N}\Big{(}\big{(}(\hat{L}^{(1)}\succ\hat{L}^{(1)})\succ\hat{L}^{(1)}\big{)}_{n}+\big{(}\hat{L}^{(1)}\succ
L^{(2)}\big{)}_{n}+\big{(}\hat{L}^{(2)}\succ\hat{L}^{(1)}\big{)}_{n}+\hat{L}_{n}^{(3)}\Big{)}.$

###### Proof.

The proof goes along the same lines as in Proposition 3.1. ∎

We recall the exponential map $\hat{T}(\alpha)=e^{\hat{\mathbb{Q}}(\alpha)},$
$\hat{\mathbb{Q}}(\alpha)=\sum_{m=1}^{\infty}\alpha^{m}\hat{\mathbb{Q}}^{(m)},$
which will lead to the discrete analogue of Magnus expansion.

######  Lemma 3.11.

The quantities
$\hat{\mathbb{Q}}^{(k)}(N+1):=\sum_{n=1}^{N}\hat{\Omega}^{(k)}_{n},$ where
$\hat{\Omega}^{(k)}_{n}=\hat{\mathbb{Q}}^{(k)}(n+1)-\hat{\mathbb{Q}}^{(k)}(n),$
are expressed explicitly as:

$\displaystyle\hat{\mathbb{Q}}^{(1)}(N+1)$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{N}\hat{L}^{(1)}_{n},$ (3.42)
$\displaystyle\hat{\mathbb{Q}}^{(2)}(N+1)$ $\displaystyle=$
$\displaystyle{1\over 2}\sum_{n>n_{1}=1}^{N}\big{[}\hat{L}^{(1)}_{n_{1}},\
L^{(1)}_{n}\big{]}-{1\over
2}\sum_{n=1}^{N}(\hat{L}^{(1)}_{n})^{2}+\sum_{n=1}^{N}\hat{L}_{n}^{(2)},$
(3.43) $\displaystyle\hat{\mathbb{Q}}^{(3)}(N+1)$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{N}\Big{(}{1\over
6}\sum_{m>k=1}^{n-1}\big{(}\big{[}\hat{L}^{(1)}_{k},\big{[}\hat{L}^{(1)}_{m},\
\hat{L}^{(1)}_{n}\big{]}\big{]}+\big{[}\big{[}\hat{L}^{(1)}_{k},\
\hat{L}^{(1)}_{m}\big{]},\hat{L}^{(1)}_{n}\big{]}\big{)}$ (3.44)
$\displaystyle+$ $\displaystyle{1\over
6}\sum_{m=1}^{n-1}\big{(}\hat{L}^{(1)}_{n}\big{[}\hat{L}^{(1)}_{n},\
\hat{L}^{(1)}_{m}\big{]}+\big{[}\hat{L}^{(1)}_{n},\
\hat{L}^{(1)}_{m}\big{]}\hat{L}^{(1)}_{m}\big{)}$ (3.45) $\displaystyle+$
$\displaystyle{1\over 6}\sum_{m=1}^{n-1}\big{(}\big{[}\hat{L}^{(1)}_{n},\
(\hat{L}^{(1)}_{m})^{2}\big{]}+\big{[}(\hat{L}^{(1)}_{n})^{2},\
\hat{L}^{(1)}_{m}\big{]}\big{)}-{1\over
2}\sum_{n=1}^{N}\big{(}\hat{L}_{n}^{(1)}\hat{L}_{n}^{(2)}+\hat{L}_{n}^{(2)}\hat{L}_{n}^{(1)}\big{)}$
(3.46) $\displaystyle+$ $\displaystyle{1\over 3}\
(\hat{L}^{(1)}_{n})^{3}-{1\over
2}\sum_{m=1}^{n-1}\big{(}\big{[}\hat{L}^{(1)}_{n},\hat{L}_{m}^{(2)}\big{]}+\big{[}\hat{L}^{(2)}_{n}\hat{L}^{(1)}_{m}\big{]}\big{)}+\hat{L}_{n}^{(3)}\Big{)},\
\ldots$ (3.47)

###### Proof.

The proof is along the lines of Lemma 3.2. ∎

We are now in the position to express the discrete Magnus expansion elements
is terms of the Pre-Lie action $\triangleleft$. Let $A,B$ be linear operators
that depend on a discrete parameter $n\in{\mathbb{N}}$ and define the binary
operation $\triangleleft$ : $(A,B)\mapsto A\triangleleft B,$ such that

$(A\triangleleft B)_{n}:=\big{[}A_{n},\
\sum_{m=1}^{n-1}B_{m}\big{]}+A_{n}B_{n}.$ (3.48)

Then the pre-Lie identity is satisfied, i.e.

$\big{(}(A\triangleleft B)\triangleleft
B\big{)}_{n}-\big{(}A\triangleleft(B\triangleleft
C)\big{)}_{n}=\big{(}(A\triangleleft C)\triangleleft
B\big{)}_{n}-\big{(}A\triangleleft(C\triangleleft B)\big{)}_{n}.$ (3.49)

######  Proposition 3.12.

The elements of the discrete Magnus expansion (3.47) can be re-expressed in
terms of the pre-Lie action (3.48).

###### Proof.

Recalling that $(a\triangleleft b)_{n}=\big{[}a_{n},\
\sum_{m=1}^{n-1}b_{m}\big{]}+a_{n}b_{n},$ we conclude

$\displaystyle\hat{\mathbb{Q}}^{(1)}(N+1)=\sum_{n=1}^{N}\hat{L}^{(1)}_{n},$
(3.50) $\displaystyle\hat{\mathbb{Q}}^{(2)}(N+1)=-{1\over
2}\sum_{n=1}^{N}(\hat{L}^{(1)}\triangleleft\hat{L}^{(1)})_{n}+\sum_{n=1}^{N}\hat{L}_{n}^{(2)},$
$\displaystyle\hat{\mathbb{Q}}^{(3)}(N+1)=\sum_{n=1}^{N}\Big{(}{1\over
12}\big{(}(\hat{L}^{(1)}\triangleleft\hat{L}^{(1)})\triangleleft\hat{L}^{(1)}\big{)}_{n}+{1\over
4}\big{(}\hat{L}^{(1)}\triangleleft(\hat{L}^{(1)}\triangleleft\hat{L}^{(1)})\big{)}_{n}\Big{)}$
(3.51) $\displaystyle\quad\quad\qquad\quad\quad-{1\over
2}\sum_{n=1}^{N}\Big{(}(\hat{L}^{(2)}\triangleleft
L^{(1)})_{n}+(\hat{L}^{(1)}\triangleleft\hat{L}^{(2)})_{n}\Big{)}+\sum_{n=1}^{N}\hat{L}_{n}^{(3)}.\hfill\qed$

A remark similar to Remark 3.5 applies here too by setting
$\hat{L}^{(1)}=\hat{\mathbb{P}},$ $\hat{L}^{(j)}=0,$ $\forall j>1$.

Now that we have the discrete Magnus expansions for both the “forward” and
“backward” evolution problems we may identify the solution of the open
boundary problem [45].

######  Lemma 3.13.

Let $T_{N+1}(\alpha)={\mathbb{L}}_{N}(\alpha)\ldots{\mathbb{L}}_{1}(\alpha)$
and
$\hat{T}_{N+1}(\alpha)=\hat{\mathbb{L}}_{1}(\alpha)\ldots\hat{\mathbb{L}}_{N}(\alpha)$
be solutions of (3.1) and (3.38) respectively. Let also $K(\alpha)$ be a
linear operator, then the quantity
${\mathbb{T}}_{N+1}(\alpha)=T_{N+1}(\alpha)K(\alpha)\hat{T}_{N+1}(\alpha)$
[45], is a solution of the difference equation

${\mathbb{T}}_{n+1}(\alpha)={\mathbb{L}}_{n}(\alpha){\mathbb{T}}_{n}(\alpha)\hat{\mathbb{L}}_{n}(\alpha).$
(3.52)

###### Proof.

The proof is straightforward by means of (3.1) and (3.38). ∎

######  Remark 3.14.

We consider $\hat{\mathbb{L}}(\alpha)={\mathbb{L}}^{-1}(-\alpha)$ and
consequently $\hat{T}_{N+1}(\alpha)=T_{N+1}^{-1}(-\alpha).$ This case is
associated to the so-called reflection algebra in integrable systems [10, 45].
In the continuum limit, after recalling
${\mathbb{T}}_{n+1}\to{\mathbb{T}}(x+\delta),$ ${\mathbb{L}}_{n}\to
1+\delta\alpha A(x)$ and keeping linear terms we obtain from (3.52)

$\partial_{\xi}{\mathbb{T}}(\xi;\alpha)=\alpha
A(\xi){\mathbb{T}}(\xi;\alpha)+\alpha{\mathbb{T}}(\xi;\alpha)A(\xi),$ (3.53)

which is the evolution equation for a system with open boundary conditions.
The solution of (3.53) is given by
${\mathbb{T}}(\xi,\alpha)=T(\xi,\alpha)K(\alpha)\hat{T}(\xi,\alpha),$ where
$T(\xi,\alpha)=e^{{\mathbb{Q}}(\xi,\alpha)}$ is given in (1.2), and
$\hat{T}(\xi,\alpha)=T^{-1}(\xi,-\alpha).$ Notice that although (3.37) and
(3.53) are mathematically similar, they have distinct physical interpretations
as already pointed out.

##  4\. Quantum groups as tridendriform $\&$ pre-Lie algebras

After the derivation of the discrete analogue of the Magnus expansion and its
relation to pre-Lie algebras we are ready to study some key relationships
between quantum groups, tridendriform and pre-Lie algebras. We employ the
Yangian ${\cal Y}(\mathfrak{gl}_{\cal N})$ [19, 37] as our key paradigm and we
show that the $N$-coproducts of the Yangian elements can be re-expressed in
terms of suitable tridendriform and pre-Lie algebra actions.

We first recall the derivation of quantum groups associated to solutions
$R:{\mathbb{C}}^{\cal N}\otimes{\mathbb{C}}^{\cal N}\to{\mathbb{C}}^{\cal
N}\otimes{\mathbb{C}}^{\cal N}$ of the Yang-Baxter equation (YBE) [7, 48]

$R_{12}(u_{1},u_{2})R_{13}(u_{1},u_{3})R_{23}(u_{2},u_{3})=R_{23}(u_{2},u_{3})R_{13}(u_{1},u_{3})R_{12}(u_{1},u_{2}),$
(4.1)

where $u_{1},u_{2}\in{\mathbb{C}}$. Let $R=\sum_{x}a_{x}\otimes b_{x}$ then in
the “index notation”: $R_{12}=\sum_{x}a_{x}\otimes b_{x}\otimes 1_{V}$,
$R_{23}=1_{V}\otimes\sum_{x}a_{x}\otimes b_{x},$ and
$R_{13}=\sum_{x}a_{x}\otimes 1_{V}\otimes b_{x}.$ For the derivation of a
quantum algebra associated to an $R$-matrix we employ the FRT construction.
Given a solution of the YBE the associated quantum algebra is a quotient of
the free algebra, generated by $\Big{\\{}L^{(m)}_{x,y}|x,y\in\\{1,\ldots,{\cal
N}\\}\Big{\\}}$ and relations:

$R_{12}(u_{1},u_{2})\ {\mathbb{L}}_{1}(u_{1})\
{\mathbb{L}}_{2}(u_{2})={\mathbb{L}}_{2}(u_{2})\ {\mathbb{L}}_{1}(u_{1})\
R_{12}(u_{1},u_{2}),$ (4.2)

where $\ {\mathbb{L}}(\lambda)\in\mbox{End}({\mathbb{C}}^{\cal
N})\otimes{\mathfrak{A}}$ and ${\mathfrak{A}}$111Notice that in ${\mathbb{L}}$
in addition to the indices 1 and 2 in (4.2) there is also an implicit “quantum
index” $n$ associated to ${\mathfrak{A}},$ which for now is omitted, i.e. one
writes ${\mathbb{L}}_{1n},\ {\mathbb{L}}_{2n}$. is the quantum algebra defined
by (4.2). Let ${\mathbb{L}}(u)=\sum_{m=0}^{\infty}u^{m}L^{(m)},$ and $e_{x,y}$
be ${\cal N}\times{\cal N}$ matrices with entries
$(e_{x,y})_{z,w}=\delta_{x,z}\delta_{y,w}.$ Then in the index notation,
$R_{12}=R\otimes 1_{\mathfrak{A}}$ and

$L_{1}^{(m)}=\sum_{x,y\in 1}^{\cal N}e_{x,y}\otimes 1_{V}\otimes
L_{x,y}^{(m)},\qquad L_{2}^{(m)}=\sum_{x,y\in 1}^{\cal N}1_{V}\otimes
e_{x,y}\otimes L^{(m)}_{x,y}.$

###  4.1. The Yangian ${\cal Y}(\mathfrak{gl}_{\cal N})$

We focus now on the Yangian ${\cal Y}(\mathfrak{gl}_{\cal N});$ in this case
$\alpha^{-1}=u:=\lambda$ (additive parameter) and
$R(u_{1},u_{2})=R(\lambda_{1}-\lambda_{2}).$ Specifically,
$R(\lambda_{1},\lambda_{2})=(\lambda_{1}-\lambda_{2})1_{V\otimes V}+{\cal P},$
where ${\cal P}=\sum_{i,j=1}^{\cal N}e_{i,j}\otimes e_{j,i}$ is the
permutation operator. Via the fundamental relation (4.2) we obtain the
algebraic relations among the generators [37]:

$\Big{[}L_{i,j}^{(n+1)},\ L_{k,l}^{(m)}\Big{]}-\Big{[}L_{i,j}^{(n)},\
L_{k,l}^{(m+1)}\Big{]}=L_{k,j}^{(m)}L_{i,l}^{(n)}-L_{k,j}^{(n)}L_{i,l}^{(m)},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
i,j,k,l\in\\{1,\ldots{\cal N}\\}.$ (4.3)

We recall the Lie exponential is written as
$e^{\mathrm{Q}}:=1+\sum_{n=1}^{\infty}{{\mathrm{Q}}^{n}\over n!}.$ We focus
here on the case were ${\mathbb{L}},\
{\mathrm{Q}}\in\mbox{End}({\mathbb{C}}^{\cal N})\otimes{\cal
Y}(\mathfrak{gl}_{\cal N}),$ and ${\mathbb{L}}$ satisfies relation (4.2). We
express the generic solution of (4.2) as
${\mathbb{L}}(\lambda)=e^{{\mathrm{Q}}(\lambda)}$ and consider the formal
$\lambda$ series expansions:

${\mathbb{L}}(\lambda)=1_{V\otimes{\cal
Y}}+\sum_{m=1}^{\infty}\lambda^{-m}L^{(m)},\quad{\mathrm{Q}}(\lambda)=\sum_{m=1}^{\infty}\lambda^{-m}{\mathrm{Q}}^{(m)}.$
(4.4)

Then comparing the series expansion ${\mathbb{L}}(\lambda)$ and
$e^{{\mathrm{Q}}(\lambda)},$ using also (4.4), we obtain expressions of
${\mathrm{Q}}^{(m)}$ in terms of symmetric polynomials of $L^{(m)}$ (see
(1.4), (1.5), ${\mathbb{Q}}^{(n)}\to{\mathrm{Q}}^{(n)}$, $T^{(n)}\to
L^{(n)}$).

$\displaystyle{\mathrm{Q}}^{(1)}=L^{(1)},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ {\mathrm{Q}}^{(2)}=L^{(2)}-{1\over
2}(L^{(1)})^{2}$ (4.5) $\displaystyle{\mathrm{Q}}^{(3)}=L^{(3)}-{1\over
2}\left(L^{(1)}L^{(2)}+L^{(2)}L^{(1)}\right)+{1\over
3}(L^{(1)})^{3},\leavevmode\nobreak\ \leavevmode\nobreak\ \ldots$ (4.6)

and vise versa as in (1.6), (1.7), i.e. similarly, the formal logarithm can be
defined such that $ln({\mathbb{L}}(\lambda))={\mathrm{Q}}(\lambda)$.

Our aim now is to derive an alternative set of generators of the Yangian based
on expressions (4.5), (4.6). Indeed, let us focus in the first few explicit
exchange relations from (4.3)

  1. (1)

$n=0,$ $m=1$ ($L_{i,j}^{(0)}=\delta_{i,j}$):

$\big{[}L_{i,j}^{(1)},\
L_{k,l}^{(1)}\big{]}=\delta_{i,l}L_{k,j}^{(1)}-\delta_{k,j}L_{i,l}^{(1)}$

the latter are the familiar ${\mathfrak{g}l}_{\cal N}$ exchange relations.

  2. (2)

$n=2,$ $m=0$:

$\big{[}L_{i,j}^{(2)},\
L_{k,l}^{(1)}\big{]}=\delta_{i,l}L_{k,j}^{(2)}-\delta_{k,j}L_{i,l}^{(2)}$

  3. (3)

$n=2,$ $m=1$:

$\big{[}L_{i,j}^{(3)},\ L_{k,l}^{(1)}\big{]}-\big{[}L_{i,j}^{(2)},\
L_{k,l}^{(2)}\big{]}=L_{k,j}^{(1)}L_{i,l}^{(2)}-L_{k,j}^{(2)}L_{i,l}^{(1)}$

  4. (4)

$n=3,$ $m=0$

$\big{[}L_{i,j}^{(3)},\
L_{k,l}^{(1)}\big{]}=\delta_{i,l}L_{k,j}^{(3)}-\delta_{k,j}L_{i,l}^{(3)}$

######  Lemma 4.1.

The algebraic relations for the alternative set of generators of the Yangian,
${\mathrm{Q}}_{i,j}^{(m)},$ $i,j\in\\{1,\ldots{\cal N}\\},$ $\
m\in\\{1,2\ldots\\}$ are given:

$\displaystyle\big{[}{\mathrm{Q}}_{i,j}^{(1)},\
{\mathrm{Q}}_{k,l}^{(1)}\big{]}$ $\displaystyle=$
$\displaystyle\delta_{i,l}{\mathrm{Q}}^{(1)}_{k,j}-\delta_{k,j}{\mathrm{Q}}^{(1)}_{i,l}$
(4.7) $\displaystyle\big{[}{\mathrm{Q}}_{i,j}^{(1)},\
{\mathrm{Q}}_{k,l}^{(2)}\big{]}$ $\displaystyle=$
$\displaystyle\delta_{i,l}{\mathrm{Q}}^{(2)}_{k,j}-\delta_{k,j}{\mathrm{Q}}^{(2)}_{i,l}$
(4.8) $\displaystyle\big{[}{\mathrm{Q}}_{i,j}^{(2)},\
{\mathrm{Q}}_{k,l}^{(2)}\big{]}$ $\displaystyle=$
$\displaystyle\delta_{i,l}{\mathrm{Q}}^{(3)}_{k,j}-\delta_{k,j}{\mathrm{Q}}^{(3)}_{i,l}-{1\over
4}{\mathrm{Q}}_{k,j}^{(1)}\sum_{x=1}^{\cal
N}{\mathrm{Q}}_{i,x}^{(1)}{\mathrm{Q}}_{x,l}^{(1)}+{1\over 4}\sum_{x=1}^{\cal
N}{\mathrm{Q}}_{k,x}^{(1)}{\mathrm{Q}}_{x,j}^{(1)}{\mathrm{Q}}_{i,l}^{(1)}$
(4.9) $\displaystyle+$ $\displaystyle{1\over
12}\Big{(}\delta_{k,j}\sum_{x,y=1}^{\cal
N}{\mathrm{Q}}_{i,x}^{(1)}{\mathrm{Q}}_{x,y}^{(1)}){\mathrm{Q}}_{y,l}^{(1)}-\delta_{i,l}\sum_{x,y=1}^{\cal
N}{\mathrm{Q}}_{k,x}^{(1)}{\mathrm{Q}}_{x,y}^{(1)}){\mathrm{Q}}_{y,j}^{(1)}\Big{)},\
\ldots$ (4.10)

###### Proof.

The proof is based on (4.6) and the exchange relations (1)-(4). ∎

######  Remark 4.2.

The Yangian is a quasi-triangular Hopf algebra on ${\mathbb{C}}$ [19] equipped
with:

  1. (1)

A co-product $\Delta:{\cal Y}(\mathfrak{gl}_{\cal N})\to{\cal
Y}(\mathfrak{gl}_{\cal N})\otimes{\cal Y}(\mathfrak{gl}_{\cal N})$ such that

$(\mbox{id}\otimes\Delta){\mathbb{L}}(\lambda)={\mathbb{L}}_{13}(\lambda){\mathbb{L}}_{12}(\lambda),\quad(\Delta\otimes\mbox{id}){\mathbb{L}}(\lambda)={\mathbb{L}}_{13}(\lambda){\mathbb{L}}_{23}(\lambda).\
$ (4.11)

We define the $N$-coproduct as
$(\mbox{id}\otimes\Delta^{(N)}){\mathbb{L}}(\lambda)={\mathbb{L}}_{0N}\ldots{\mathbb{L}}_{01}.$

  2. (2)

A counit $\epsilon:{\cal Y}(\mathfrak{gl}_{\cal N})\to{\mathbb{C}},$ such that
$(\epsilon\otimes\mbox{id}){\mathbb{L}}(\lambda)=1_{\mathfrak{A}},$
$(\mbox{id}\otimes\epsilon){\mathbb{L}}(\lambda)=1_{V}.$

  3. (3)

An antipode $S:{\cal Y}(\mathfrak{gl}_{\cal N})\to{\cal Y}(\mathfrak{gl}_{\cal
N}):$ $(S\otimes\mbox{id}){\mathbb{L}}(\lambda)={\mathbb{L}}^{-1}(\lambda),$
$\ (\mbox{id}\otimes S){\mathbb{L}}^{-1}(\lambda)={\mathbb{L}}(\lambda).$

Specifically, for the generators of the Yangian algebra (Hopf algebra):

$\displaystyle\Delta({\mathrm{Q}}_{ab}^{(1)})$ $\displaystyle=$
$\displaystyle{\mathrm{Q}}_{ab}^{(1)}\otimes
1+1\otimes{\mathrm{Q}}_{ab}^{(1)}$ (4.12)
$\displaystyle\Delta({\mathrm{Q}}_{ab}^{(2)})$ $\displaystyle=$
$\displaystyle{\mathrm{Q}}_{ab}^{(2)}\otimes
1+1\otimes{\mathrm{Q}}_{ab}^{(2)}+{1\over
2}\sum_{d=1}^{n}({\mathrm{Q}}_{ad}^{(1)}\otimes{\mathrm{Q}}_{db}^{(1)}-{\mathrm{Q}}_{db}^{(1)}\otimes{\mathrm{Q}}_{ad}^{(1)}).$
(4.13)

Also, $\epsilon({\mathrm{Q}}^{(1)}_{ab})=\epsilon({\mathrm{Q}}^{(2)}_{ab})=0,$
and $S({\mathrm{Q}}^{(1)}_{ab})=-{\mathrm{Q}}_{ab}^{(1)},\quad
S({\mathrm{Q}}^{(2)}_{ab})=-{\mathrm{Q}}^{(2)}_{ab}+{1\over
2}{\mathrm{Q}}^{(1)}_{ab}.$

The Yangian as a Hopf algebra is associative, and the $n$-coproducts can be
derived by iteration via
$\Delta^{(n+1)}=(\mbox{id}\otimes\Delta^{(n)})\Delta=(\Delta^{(n)}\otimes\mbox{id})\Delta.$

In the next subsection we examine tensor realizations of the Yangian and
express the $N$-coproducts of the algebra generators as elements of the
discrete Magnus expansion. We also express coproduts of the algebra in terms
of suitable tridendriform and pre-Lie algebra actions.

###  4.2. Tensor realizations of the Yangian $\&$ Pre-Lie algebras

In order to demonstrate the links between the Yangian, tridendriform and pre-
Lie algebras we consider tensor realizations of the Yangian. Let the solution
of relation (4.2) be of the general form
${\mathbb{L}}(\lambda)=1_{V\otimes{\cal
Y}}+\sum_{m>1}{L^{(m)}\over\lambda^{m}}\in\mbox{End}({\mathbb{C}}^{\cal
N})\otimes{\cal Y}(\mathfrak{gl}_{\cal N}),$ with the $R$-matrix being the
Yangian $R(\lambda)=1_{V\otimes V}+\lambda^{-1}{\cal P}$ and
$L^{(m)}=\sum_{x,y=1}^{{\mathcal{N}}}e_{x,y}\otimes L^{(m)}_{x,y},$ where
$L^{(m)}_{x,y}$ are the generators of the Yangian $\mathfrak{gl}_{\cal N}$. We
introduce the monodromy matrix $T\in\mbox{End}({\mathbb{C}}^{\cal
N})\otimes{\cal Y}(\mathfrak{gl}_{\cal N})^{\otimes N}$:

$T_{0,12..N}(\lambda):=(\mbox{id}\otimes\Delta^{(N)}){\mathbb{L}}(\lambda)={\mathbb{L}}_{0N}(\lambda)\ldots{\mathbb{L}}_{01}(\lambda),$
(4.14)

which also satisfies the algebraic relation (4.2) and is a solution of the
discrete evolution problem (3.1). Historically, the index $0$ is called
“auxiliary”, whereas the indices $1,2,\ldots,N$ are called “quantum”, and they
are usually suppressed for simplicity, i.e. we simply write $T_{0,N+1}$ (or
$T_{N+1}$). We also note that a standard simple solution of the fundamental
relation (4.2) for the Yangian is
${\mathbb{L}}(\lambda)=1+\lambda^{-1}{\mathbb{P}},$ where
${\mathbb{P}}=\sum_{i,j=1}^{\cal N}e_{i,j}\otimes{\mathbb{P}}_{i,j}$ and
${\mathbb{P}}_{i,j}\in\mathfrak{gl}_{\cal N}:$ $\big{[}{\mathbb{P}}_{i,j},\
{\mathbb{P}}_{k,l}\big{]}=\delta_{i,l}{\mathbb{P}}_{k,j}-\delta_{k,j}{\mathbb{P}}_{i,l}.$

######  Remark 4.3.

We define the transfer matrix
${\mathfrak{t}}_{N+1}(\lambda)=tr_{0}\big{(}T_{0,N+1}(\lambda)\big{)}\in\mbox{End}\big{(}({\mathbb{C}}^{{\mathcal{N}}})^{\otimes
N}\big{)}$. The monodromy matrix $T$ satisfies (4.2) (it is a tensor
realization of the algebra defined by (4.2)), and hence it is shown that the
transfer matrix provides a family of mutually commuting quantities [25]
(${\mathfrak{t}}_{N+1}(\lambda)=\lambda^{N}\sum_{k=1}^{N}{{\mathfrak{t}}_{N+1}^{(k)}\over\lambda^{k}}$):

$\Big{[}{\mathfrak{t}}_{N+1}(\lambda),\ {\mathfrak{t}}_{N+1}(\mu)\Big{]}=0\
\Rightarrow\ \Big{[}{\mathfrak{t}}_{N+1}^{(k)},\
{\mathfrak{t}}_{N+1}^{(l)}\Big{]}=0.$ (4.15)

These commutation relations guarantee the “quantum integrability” of a spin-
chain like system with periodic boundary conditions. For instance the
Hamiltonian and momentum of the system belong to the family of the mutual
commuting quantities.

######  Proposition 4.4.

Let $L_{a,b}^{(m)},$ $a,b\in\\{1,2,\ldots,{\cal N}\\},$ $m=\\{1,2,\ldots\\}$
be the generators of the Yangian $\mathfrak{gl}_{\cal N}$ (4.3). The
coproducts of the algebra generators are expressed in terms of tridendriform
algebras actions $\prec$.

###### Proof.

Before we carry on with the proof it is useful to recall the tensor index
notation: let $A,B$ be elements of some algebra ${\mathfrak{A}},$ then
$A_{n}B_{m}=B_{m}A_{n}=B_{m}\otimes A_{n}$ $n>m,$ where the indices $n,m$
denote the position of the objects $A,B$ on an $N$-tensor product, i.e.

$A_{n}:=1\otimes\ldots 1\otimes\underbrace{A}_{\mbox{n${}^{th}$
position}}\otimes 1\ldots\otimes 1.$ (4.16)

We recall that the quantum monodromy is expressed as
$T_{N+1}(\alpha)=1+\sum_{m>1}\alpha^{m}T^{(m)}(N+1)$
($\alpha={1\over\lambda}$), with coefficients given in (3.2) and in terms of a
tridendriform action in (3.3). We also recall that
$T_{N+1}(\lambda)=(\mbox{id}\otimes\Delta^{(N)}){\mathbb{L}}(\lambda),$ hence
$T^{(m)}_{a,b}(N+1)=\Delta^{(N)}(L^{(m)}_{a,b})$ ($\Delta$ is an algebra
homomorphism), which leads to

$\displaystyle\Delta^{(N)}(L^{(m)}_{a,b})$ $\displaystyle=$
$\displaystyle\sum_{n>...>n_{1}=1}^{N}\sum_{k=1}^{m}\Big{(}(L^{(m_{k})}_{a,b_{k}})_{n}(L^{(m_{k-1})}_{b_{k},b_{k-1}})_{n_{k-1}}\ldots(L^{(m_{1})}_{b_{1},b})_{n_{1}}\Big{)}_{\sum_{j=1}^{k}m_{j}=m}$
$\displaystyle=$
$\displaystyle\sum_{n=1}^{N}\sum_{k=1}^{m}\Big{(}L^{(m_{k})}_{a,b_{k}}\prec(L^{(m_{k-1})}_{b_{k},b_{k-1}}\prec(\ldots\prec(L^{(m_{2})}_{b_{2},b_{1}}\prec
L^{(m_{1})}_{b_{1},b})\ldots))\Big{)}_{n}\Big{|}_{\sum_{j=1}^{k}m_{j}=m}.$

We read off the coproducts for the first three orders $m=1,2,3:$

  1. (1)

$\Delta^{(N)}(L^{(1)}_{a,b})=\sum_{n=1}^{N}(L^{(1)}_{a,b})_{n},$

  2. (2)

$\Delta^{(N)}(L^{(2)}_{a,b})=\sum_{n=1}^{N}(L^{(2)}_{a,b})_{n}+\sum_{n=1}^{N}\big{(}L^{(1)}_{a,b_{1}}\prec
L^{(1)}_{b_{1},b}\big{)}_{n}.$

  3. (3)

$\Delta^{(N)}(L^{(3)}_{a,b})=\sum_{n=1}^{N}(L^{(3)}_{a,b})_{n}+\sum_{n=1}^{N}\Big{(}\big{(}L^{(2)}_{a,b_{1}}\prec
L^{(1)}_{b_{1},b}\big{)}_{n}\ +\big{(}L^{(1)}_{a,b_{1}}\prec
L^{(2)}_{b_{1},b}\big{)}_{n}\Big{)}\\\
+\sum_{n=1}^{N}\big{(}L^{(1)}_{a,b_{2}}\prec(L^{(1)}_{b_{2},b_{1}}\prec
L^{(1)}_{b_{1},b})\big{)}_{n}.$ ∎

We note that a presentation of the Yangian is given via the evaluation
homomorphism, $\mbox{ev}:{\cal Y}(\mathfrak{gl}_{\cal
N})\to\mathfrak{gl}_{\cal N}$, such that
$L_{a,b}^{(m)}\mapsto\theta^{m}{\mathbb{P}}_{a,b},$ $\theta\in{\mathbb{C}},$
and ${\mathbb{P}}_{a,b}$ are the $\mathfrak{gl}_{\cal N}$ generators. We also
recall that ${\mathbb{L}}(\lambda)=\sum_{m>0}\lambda^{-m}{\mathbb{P}}^{m}$ is
a solution of (4.2), hence the following map also exists, $\sigma:{\cal
Y}(\mathfrak{gl}_{\cal N})\to\mathfrak{gl}_{\cal N}$, such that
$L_{a,b}^{(m)}\mapsto({\mathbb{P}}^{m})_{a,b}.$ Before we come to the second
key link between the coproducts of the alternative Yangian generators
${\mathrm{Q}}_{a,b}$ and pre-Lie algebras, we first introduce a useful Lemma
and some handy notation.

######  Lemma 4.5.

Let $x,y$ be general linear operators and $R(x),R(y)$ be Rota-Baxter
operators, such that $[R(x),y]=[R(y),x]=0.$ Assume also that the actions of a
tridendriform algebra are defined in (2.19), then $x\prec y=y\succ x.$

###### Proof.

From the definitions in (2.19) we immediately conclude that $x\prec y=y\succ
x.$ ∎

######  Remark 4.6.

We introduce some notation that will be used in the following Proposition. Let
${\cal O}\in\mbox{End}({\mathbb{C}}^{\cal N})\otimes{\mathfrak{A}},$ then
${\cal O}_{n}=\sum_{a,b}^{\cal N}e_{a,b}\otimes({\cal O}_{a,b})_{n},$ where
$({\cal O}_{a,b})_{n}$ is defined as in (4.16). We assume that
$A,B\in\mbox{End}({\mathbb{C}}^{\cal N})\otimes{\mathfrak{A}},$ and we define

$\displaystyle\big{(}(A\triangleright B)_{n}\big{)}_{a,b}$ $\displaystyle:=$
$\displaystyle\Big{(}\big{[}\sum_{m=1}^{n}A_{m},B_{n}\big{]}+A_{n}B_{n}\Big{)}_{a,b}$
$\displaystyle=$ $\displaystyle\sum_{c=1}^{\cal
N}\Big{(}\sum_{m=1}^{n-1}(A_{a,c})_{m}(B_{c,b})_{n}-(B_{a,c})_{n}\sum_{m=1}^{n-1}(A_{c,b})_{m}+(A_{a,c})_{n}(B_{c,b})_{n}\Big{)}.$

Recalling the definitions of the tridendriform actions (2.19), we conclude for
the above expression

$\big{(}(A\triangleright B)_{n}\big{)}_{a,b}=\sum_{c=1}^{\cal
N}\Big{(}(A_{a,c}\succ B_{c,b})_{n}-(B_{a,c}\prec
A_{c,b})_{n}+(A_{a,c}B_{c,b})_{n}\Big{)}.$ (4.19)

Due to the fact that $\big{[}(A_{a,c})_{m},\ (B_{c,b})_{n}\big{]}=0,$ $n\neq
m,$ Lemma 4.5 applies.

######  Proposition 4.7.

Let ${\mathrm{Q}}_{a,b}^{(m)},$ $a,b\in\\{1,2,\ldots,{\cal N}\\}$ and
$m=\\{1,2,\ldots\\}$ be the alternative generators of the Yangian
$\mathfrak{gl}_{\cal N}$ (Lemma 4.1). The coproducts of these generators are
emerging from a pre-Lie algebra and are explicitly expressed in terms of
tridendriform algebras actions.

###### Proof.

First we use the fact that
$(\mbox{id}\otimes\Delta^{(N)}){\mathrm{Q}}^{(m)}={\mathbb{Q}}^{(m)}(N+1)$
(4.14), we also recall the expressions ${\mathbb{Q}}^{(m)}(N+1)$ (3.22), then
via (4.5), (4.6) we conclude

$\displaystyle(\mbox{id}\otimes\Delta^{(N)}){\mathrm{Q}}^{(1)}=\sum_{n=1}^{N}{\mathrm{Q}}^{(1)}_{n},$
(4.20) $\displaystyle(\mbox{id}\otimes\Delta^{(N)}){\mathrm{Q}}^{(2)}=-{1\over
2}\sum_{n=1}^{N}({\mathrm{Q}}^{(1)}\triangleright{\mathrm{Q}}^{(1)})_{n}+\sum_{n=1}^{N}\big{(}{\mathrm{Q}}_{n}^{(2)}+{1\over
2}({\mathrm{Q}}^{(1)}_{n})^{2}\big{)},$
$\displaystyle(\mbox{id}\otimes\Delta^{(N)}){\mathrm{Q}}^{(3)}=\sum_{n=1}^{N}\Big{(}{1\over
4}\big{(}({\mathrm{Q}}^{(1)}\triangleright{\mathrm{Q}}^{(1)})\triangleright{\mathrm{Q}}^{(1)}\big{)}_{n}+{1\over
12}\big{(}{\mathrm{Q}}^{(1)}\triangleright({\mathrm{Q}}^{(1)}\triangleright{\mathrm{Q}}^{(1)})\big{)}_{n}\Big{)}$
(4.21) $\displaystyle\qquad\qquad\qquad\quad-{1\over
2}\sum_{n=1}^{N}\Big{(}({\mathrm{Q}}^{(2)}\triangleright{\mathrm{Q}}^{(1)})_{n}+({\mathrm{Q}}^{(1)}\triangleright{\mathrm{Q}}^{(2)})_{n}\Big{)}$
(4.22) $\displaystyle\qquad\qquad\qquad\quad-{1\over
4}\sum_{n=1}^{N}\Big{(}(({\mathrm{Q}}^{(1)})^{2}\triangleright{\mathrm{Q}}^{(1)})_{n}+({\mathrm{Q}}^{(1)}\triangleright({\mathrm{Q}}^{(1)})^{2})_{n}\Big{)}$
(4.23)
$\displaystyle\qquad\qquad\qquad\quad+\sum_{n=1}^{N}\big{(}{\mathrm{Q}}_{n}^{(3)}+{1\over
2}({\mathrm{Q}}_{n}^{(2)}{\mathrm{Q}}_{n}^{(1)}+{\mathrm{Q}}_{n}^{(1)}{\mathrm{Q}}_{n}^{(2)})+{1\over
6}({\mathrm{Q}}^{(1)}_{n})^{3}\big{)},\ \ldots$ (4.24)

The $N$-coprodutcs of the alternative generators ${\mathrm{Q}}^{(m)}_{a,b}$ of
the Yangian are extracted from expressions (4.24), by recalling
$(\mbox{id}\otimes\Delta^{(N)}){\mathrm{Q}}^{(n)}=\sum_{a,b}e_{a,b}\otimes\Delta^{(N)}({\mathrm{Q}}_{a,b}^{(n)})$
and $\
{\mathbb{Q}}_{a,b}^{(p)}(N)=\Delta^{(N)}({\mathrm{Q}}_{a,b}^{(p)}),\leavevmode\nobreak\
a,b\in\\{1,\ldots,{\cal N}\\}.$ Specifically, for the first couple of terms,
$n=1,2$ we obtain from (4.24), Remark 4.6,

  1. (1)

$\Delta^{(N)}({\mathrm{Q}}^{(1)}_{a,b})=\sum_{n=1}^{N}({\mathrm{Q}}^{(1)}_{a,b})_{n},$

  2. (2)

$\Delta^{(N)}({\mathrm{Q}}^{(2)}_{a,b})=\sum_{n=1}^{N}\Big{(}({\mathrm{Q}}^{(2)}_{a,b})_{n}-{1\over
2}\sum_{c=1}^{\cal
N}\big{(}({\mathrm{Q}}^{(1)}_{a,c}\succ{\mathrm{Q}}^{(1)}_{c,b})_{n}-({\mathrm{Q}}^{(1)}_{a,c}\prec{\mathrm{Q}}^{(1)}_{c,b})_{n}\big{)}\Big{)}.$

The notation and the requirements introduced in Remark 4.6 apply in this case.
∎

######  Remark 4.8.

In the special case where
${\mathbb{L}}(\lambda)=1+{1\over\lambda}{\mathbb{P}},$ i.e.
$L^{(1)}={\mathbb{P}}$ and $L^{(m)}=0$ $\forall m>0,$ and given relation
(4.5), (4.6), we conclude that expressions (4.24) reduce to the much more
concise formulas (3.26), and Remark 3.6 on the construction of braces from
pre-Lie algebras holds.

We present below two distinct examples/applications associated to the
alternative discrete expansion described in Subsection 3.1. More details on
these two examples will be presented in future works as they are both of
particular interest.

######  Example 4.9.

Let $R:{\mathbb{C}}^{\cal N}\otimes{\mathbb{C}}^{\cal N}\to{\mathbb{C}}^{\cal
N}\otimes{\mathbb{C}}^{\cal N}$ be a solution of the YBE, of the form
$R(\lambda)=r+\lambda^{-1}{\cal P},$ where $r$ is also a solution of the YBE
and ${\cal P}$ is the permutation operator (see e.g.[13, 14, 15]). Such
solutions can be obtained for instance from involutive set-theoretic solutions
of the YBE [18, 13]. Recall now the expression of the monodromy matrix
$T(\lambda)={\mathbb{L}}_{N}(\lambda)\ldots{\mathbb{L}}_{1}(\lambda),$ where
in this example ${\mathbb{L}}(\lambda)\to R(\lambda).$ Given that $R$
satisfies the YBE one shows that the monodromy matrix naturally satisfies the
(4.2). In this case expressions (3.34) and (3.33), (1)-(3) in Subsection 3.1
hold: $M\to r,$ $L\to{\cal P}$ and $\alpha=\lambda^{-1}.$

######  Example 4.10.

Another typical example in this class is the algebra
${\mathfrak{U}}_{q}(\mathfrak{gl}_{\cal N})$ [27, 28]. In this case
${\mathbb{L}}(\lambda)=L^{+}-e^{-2\lambda}L^{-},$ where $L^{+}=\sum_{a\leq
b}e_{a,b}\otimes L^{+}_{a,b}$ and $L^{-}=\sum_{a\geq b}e_{a,b}\otimes
L^{-}_{a,b}.$The elements $L^{\pm}_{a,b}\in B^{\pm},$ where $B^{\pm}$ are
subalgebras of ${\mathfrak{U}}_{q}(\mathfrak{gl}_{\cal N}),$ (upper/lower
Borel decomposition; for more details the interested reader is referred to
[27, 28, 25]). The associated $R$-matrix is an ${\cal N}^{2}\times{\cal
N}^{2}$ matrix and is of the form $R(\lambda)=R^{+}-e^{-2\lambda}R^{-},$ where
$R^{+}=\sum_{a\leq b}e_{a,b}\otimes R^{+}_{a,b}$ and $R^{-}=\sum_{a\geq
b}e_{a,b}\otimes R^{-}_{a,b},$ (see detailed expressions in [27, 28]). In this
case too expressions (3.34) and (3.33), (1)-(3) in Subsection 3.1 hold: $M\to
L^{+},$ $L\to L^{-}$ and $\alpha=-e^{-2\lambda}.$

#### Classical Integrability

Connections between (tri)dendriform, pre-Lie algebras and classical integrable
systems and the associated classical deformed algebras are naturally
identified given the findings of the previous subsection. The key point in the
description and construction of classical integrable systems from the
Hamiltonian point of view is the existence of a classical matrix
$r\in\mbox{End}({\mathbb{C}}^{\cal N}\otimes{\mathbb{C}}^{\cal N})$ that
satisfies the classical YBE [24, 42],

$\big{[}r_{12}(\lambda_{1}-\lambda_{2}),r_{13}(\lambda_{1}-\lambda_{3})\big{]}+\big{[}r_{12}(\lambda_{1}-\lambda_{2})+r_{13}(\lambda_{1}-\lambda_{3}),r_{23}(\lambda_{2}-\lambda_{3})\big{]}=0.$
(4.25)

The classical YBE can be seen as a linearization of the quantum YBE, i.e. we
set $R=1+\delta r+{\cal O}(\delta^{2}).$ Also, the classical Lax operator
${\mathbb{L}}(\lambda)\in\mbox{End}({\mathbb{C}}^{\cal N})$ associated to a
discrete space intregrable system satisfies the so called Sklyanin’s bracket
[24, 43, 44] (i.e. the classical analogue of the fundamental relation (4.2)):

$\big{\\{}{\mathbb{L}}_{n}(\lambda_{1})\underset{,}{\otimes}{\mathbb{L}}_{m}(\lambda_{2})\big{\\}}=\big{[}r(\lambda_{1}-\lambda_{2}),\
{\mathbb{L}}_{n}(\lambda_{1})\otimes{\mathbb{L}}_{m}(\lambda_{2})\big{]}\delta_{n,m}.$
(4.26)

As in the quantum case algebraic quantities in involution exist, due the
existence of an $r$-matrix. Indeed, the monodromy matrix
$T(\lambda)={\mathbb{L}}_{N}(\lambda)\ldots{\mathbb{L}}_{1}(\lambda)\in\mbox{End}({\mathbb{C}}^{\cal
N})$ also satisfies Sklyanin’s bracket and hence it is shown that
${\mathfrak{t}}(\lambda)=trT(\lambda)$ satisfies
$\big{\\{}{\mathfrak{t}}(\lambda),{\mathfrak{t}}(\mu)\big{\\}}=0,$ which,
given that
${\mathfrak{t}}(\lambda)=\sum_{m=1}^{N}\lambda^{-m}{\mathfrak{t}}^{(m)},$
leads to a family of quantities in involution
$\big{\\{}{\mathfrak{t}}^{(m)},{\mathfrak{t}}^{(k)}\big{\\}}=0,$ i.e.
classical integrability à la Liouville is shown [24].

In the case of the classical $\mathfrak{gl}_{\cal N}$ Yangian, the $r$-matrix
is given as $r(\lambda)={{\cal P}\over\lambda},$ where we recall that ${\cal
P}$ is the ${\cal N}^{2}\times{\cal N}^{2}$ permutation operator. Also we
recall, as in the quantum case,
${\mathbb{L}}(\lambda)=1+\sum_{m>1}\lambda^{-m}L^{(m)}$ and
$L^{(m)}=\sum_{a,b=1}^{\cal N}L_{a,b}e_{a,b}.$ By substituting ${\mathbb{L}}$
in (4.26) the classical Yangian relations are recovered, i.e. the classical
analogues of (4.3). Everything holds as in the quantum case, but
$\big{[},\big{]}\mapsto-\big{\\{},\big{\\}},$ i.e. $L^{(m)}_{a,b}$ are
commutative objects, and are the generators of the classical Yangian.
$T_{a,b}^{(m)}$ (non-local charges) are the classical analogues of the
coproducts $\Delta^{(N)}(L_{a,b}^{(m)})$.

In the continuum case the Lax operator
${\mathbb{A}}(x,\lambda)\in\mbox{End}({\mathbb{C}}^{\cal N})$ satisfies a
linear Sklyaninn’s bracket (ultra-local algebra):

$\big{\\{}{\mathbb{A}}(x,\lambda_{1})\underset{,}{\otimes}{\mathbb{A}}(y,\lambda_{2})\big{\\}}=\big{[}r(\lambda_{1}-\lambda_{2}),\
{\mathbb{A}}(x,\lambda_{1})\otimes
1+1\otimes{\mathbb{A}}(y,\lambda_{2})\big{]}\delta(x-y).$ (4.27)

In the case of the Yangian specifically $r(\lambda)={{\cal P}\over\lambda}$
and ${\mathbb{A}}(x,\lambda)={1\over\lambda}A(x).$ The monodromy matrix in
this case is defined as
T$(x,\alpha)=\overset{\curvearrowleft}{\mathrm{P}}\text{exp}\big{(}\int_{0}^{x}{\mathbb{A}}(\xi,\alpha)d\xi\big{)}$
(see also (1.2) and Remark 3.6) and satisfies the quadratic relation (4.26),
[24, 33]. Also, the expressions for the continuous Magnus expansion (1.10),
(1.12) and (2.17) hold ($\alpha=\lambda^{-1}$).

With this we conclude our analysis on the links between quantum and classical
algebras, arising in the context of integrable systems, and pre-Lie and
(tri)dendriform algebras. Further study on the quantum algebras associated to
the Examples 4.9 and 4.10 will follow in future investigations. Example 4.9 in
particular is of special interest given recent findings on the
characterization of the quantum algebra associated to involutive set-theoretic
solutions of the YBE as quasi-bialgebras [15, 16]. These quasi-bialgebras
naturally emerge after suitably twisting the Yangian [15]. We note that set-
theoretic solutions do not have a classical analogue, a fact that makes
Example 4.9 even more intriguing. Moreover, it is known that all involutive
set-theoretic solution of the YBE come from braces [9, 40]. This together with
the fact that braces are obtained from pre-Lie algebras, while pre-lie
algebras in turn are naturally connected to quantum algebras via the FRT
construction, give us a strong motivation to further investigate these
algebraic structures at the level of solutions, but also at the level of the
emerging quantum algebras.

### Acknowledgments

Support from the EPSRC research grant EP/V008129/1 is acknowledged.

## References

  * [1] M.J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, Nonlinear-Evolution Equations of Physical Significance, Phys. Rev. Lett. 31 (1973) 125.
  * [2] A. Agrachev, R. Gamkrelidze, Chronological algebras and non-stationary vector fields, J. Sov. Math. 17 (1981) 1650–1675.
  * [3] M. Aguiar, Infinitesimal Hopf algebras, Contemp. Math 267 (2000) 1-30;
M. Aguiar, Infinitesimal bialgebras, pre-Lie and dendriform algebras, in “Hopf
Algebras”, Lecture Notes in Pure and Applied Mathematics, 237 (2004) 1-33.

  * [4] J. Avan, A. Doikou and K. Sfetsos, Systematic classical continuum limits of integrable spin chains and emerging novel dualities, Nucl. Phys. B840 (2010) 469-490.
  * [5] C. Bai, Introduction to pre-Lie algebras, https://b-ok.org/book/2665901/384dc8, (2016).
  * [6] G. Baxter, An analytic problem whose solution follows from a simple algebraic identity, Pacific J. Math. 10 (1960) 731-742.
  * [7] R.J. Baxter, Exactly solved models in statistical mechanics, Academic Press (1982).
  * [8] S. Blanes, F. Casas, J.A. Oteo and J. Ros, The Magnus expansion and some of its applications, Phys. Reports 470 (2009) 151-238.
  * [9] F. Cedó, E. Jespers, J. Oknínski, Braces and the Yang-Baxter equation, Comm. Math. Phys. 327, (2014), 101–116.
  * [10] I.V. Cherednik, Factorizing particles on a lalf lines and root systems, Theor. Math. Phys. 61 (1984) 977-983.
  * [11] A. Doikou and I. Findlay, The quantum auxiliary linear problem $\&$ Darboux-Bäcklund transformations, Proc. of Science 376 (2020) 210.
  * [12] A. Doikou and I. Findlay, An algebraic approach to discrete time integrability, J. Phys. A54 (2021) 045202.
  * [13] A. Doikou and A. Smoktunowicz, From Braces to Hecke algebras $\&$ Quantum Groups, J. of Alg. and its Appl. (2022) 2350179.
  * [14] A. Doikou and A. Smoktunowicz, Set-theoretical Yang-Baxter and reflection equations $\&$ quantum group symmetries, Lett. Math. Phys. 111 (2021) 105.
  * [15] A. Doikou, Set-theoretic Yang–Baxter equation, braces and Drinfeld twists, J. Phys A54 (2021) 415201.
  * [16] A. Doikou, A. Ghionis and B. Vlaar, Quasi-bialgebras from set-theoretic type solutions of the Yang-Baxter equation, Lett. Math. Phys. 112, 78 (2022).
  * [17] V.G. Drinfeld and V.V. Sokolov, Equations of Korteweg-de Vries type and simple Lie algebras, Dokl. Akad. Nauk SSSR,258, No. 1, 11–16 (1981).
  * [18] V.G. Drinfeld, On some unsolved problems in quantum group theory, Lecture Notes in Math., vol. 1510, Springer-Verlag, Berlin (1992) 1-8.
  * [19] V.G. Drinfeld, Hopf algebras and the quantum Yang–Baxter equation, Sov. Math. Dokl. 32 (1985) 254.
V.G. Drinfeld, Quantum Groups, in Proc. of the I. C. M. Berkeley (1986).
V.G. Drinfeld, A new realization of Yangians $\&$ quantized affine algebras,
Sov. Math. Dokl. 36 (1988) 212.

  * [20] F.J. Dyson, The radiation theorem of Tomonaga, Swinger and Feynman, Phys. Rev. 75 (1949) 486–502.
  * [21] K. Ebrahimi-Fard, Loday-type algebras and the Rota–Baxter relation, Lett. Math. Phys. 61 (2002) 139–147.
  * [22] K. Ebrahimi-Fard and D. Manchon, The tridendriform structure of a Magnus expansion, Discr. and Cont. Dyn. Systems - Series A (DCDS-A) 34, 3 (2014) 1021-1040.
  * [23] K. Ebrahimi-Fard and L. Guo, Rota-Baxter algebras and dendriform algebras, J. Pure Appl. Alg. 212, 2 (2008) 320-339.
  * [24] L.D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, (1987) Springer-Verlag.
  * [25] L.D. Faddeev, N.Yu. Reshetikhin and L.A. Takhtajan, Quantization of Lie groups and Lie algebras, Leningrad Math. J. 1 (1990) 193.
  * [26] M. Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math. 78 (1963) 267–288.
  * [27] M. Jimbo, A q-difference analogue of U(g) and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985) 63.
  * [28] M. Jimbo, Quantum R-matrix for the generalized Toda system, Comm. Math. Phys.102 (1986) 537–547.
  * [29] P. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. on Pure and Appl. Math. 21 (5) (1968) 467–490.
  * [30] J.-L. Loday, Dialgebras, in Dialgebras and related operads, Lecture Notes in Math. 1763 (2002) 7-66.
  * [31] J.-L. Loday and M. Ronco, Trialgebras and families of polytopes, in “Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic K-theory”, Comtep. Math. 346 (2004) 369-398.
  * [32] J.-L. Loday, On the algebra of quasi-shuffles, Manusc. Mathematica 123 (2007) 1, 79-93.
  * [33] N.J. MacKay, Lattice quantization of Yangian charges, Phys.Lett. B349 (1995) 94-98.
  * [34] W. Magnus, On the exponential solution of differential equations for a linear operator, Commun. Pure Appl. Math. 7 (1954) 649–673.
  * [35] D. Manchon, A short survey on pre-Lie algebras, ESI Lectures in Mathematics and Physics, Noncommutative Geometry and Physics: Renormalisation, Motives, Index Theory, (2009) pp 89-102.
  * [36] V.B. Matveev and M.A. Salle, Darboux transformations and solitons, (1991) SpringerVerlag.
  * [37] A. Molev, M. Nazarov and G. Olshanski, Yangians and classical Lie algebras, Russ. Math. Surveys 51 (1996) 205.
  * [38] J. Pei, Ch. Bai and L. Guo, Rota-Baxter operators on sl(2,C) and solutions of the classical Yang-Baxter equation, J. Math. Phys. 55 (2014) 021701.
  * [39] G.-C. Rota, Baxter algebras and combinatorial identities I, Bull. Amer. Math. Soc. 75 (1969) 325-329.
G.-C. Rota, Baxter algebras and combinatorial identities II, Bull. Amer. Math.
Soc. 75 (1969) 330-334.

  * [40] W. Rump, Braces, radical rings, and the quantum Yang-Baxter equation, J. of Algebra 307, 1 (2007) 153–170.
  * [41] W. Rump, The brace of a classical group, Note Mat. 34 (2014) no. 1, 115–144.
  * [42] M.A. Semenov-Tyan-Shanskii, What is a classical r-matrix?, Funct. Anal. Appl. 17 (1983) 259-272.
  * [43] E.K. Sklyanin, Method of the inverse scattering problem and quantum nonlinear Schrödinger equation, Dokl. Acad. Nauk SSSR 244, No. 6 (1978) 1337.
  * [44] E.K. Sklyanin, On complete integrability of the Landau-Lifshitz equation, preprint Leningrad Branch of the Mathematics Institute, E-3-79 (1979).
  * [45] E.K. Sklyanin, Boundary conditions in integrable equations, Funct. Analysis and Appl. 21 (1987) 164-166;
E.K. Sklyanin, Boundary conditions for integrable quantum systems, J. Phys.
A21 (1988) 2375.

  * [46] A. Smoktunowicz, A new formula for Lazard’s correspondence for finite braces and pre-Lie algebras, J. of Algebra, 594, 15 (2022) 202-229;
A. Smoktunowicz, Algebraic approach to Rump’s results on relations between
braces and pre-Lie algebras, J. of algebra and its applications 21, 3 (2020)
12 p., 2250054.

  * [47] E.B. Vinberg, The theory of homogeneous convex cones, Transl. Moscow Math. Soc. 12 (1963), 340–403.
  * [48] C.N. Yang, Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Rev. Lett. 19 (1967) 1312.
  * [49] V.E. Zakharov and A.B. Shabat, Exact Theory of Two-dimensional Self-focusing and One-dimensional Self-modulation of Waves in Nonlinear Media  Sov. Phys. - JETP 34 (1972) 62-69.