Post-Lie Magnus Expansion and BCH-Recursion
We identify the Baker-Campbell-Hausdorff recursion driven by a weight λ=1 Rota-Baxter operator with the Magnus expansion relative to the post-Lie structure naturally associated with the corresponding Rota-Baxter algebra. Post-Lie Magnus expansion and BCH-recursion are reviewed before the proof of th...
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| description | We identify the Baker-Campbell-Hausdorff recursion driven by a weight λ=1 Rota-Baxter operator with the Magnus expansion relative to the post-Lie structure naturally associated with the corresponding Rota-Baxter algebra. Post-Lie Magnus expansion and BCH-recursion are reviewed before the proof of the main result.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 023, 16 pages
Post-Lie Magnus Expansion and BCH-Recursion
Mahdi J. Hasan AL-KAABI a, Kurusch EBRAHIMI-FARD b and Dominique MANCHON c
a) Mathematics Department, College of Science, Mustansiriyah University,
Palestine Street, P.O. Box 14022, Baghdad, Iraq
E-mail: mahdi.alkaabi@uomustansiriyah.edu.iq
b) Department of Mathematical Sciences, Norwegian University of Science and Technology,
Trondheim, Norway
E-mail: kurusch.ebrahimi-fard@ntnu.no
c) Laboratoire de Mathématiques Blaise Pascal, CNRS et Université Clermont-Auvergne
(UMR 6620), 3 place Vasarély, CS 60026, F63178 Aubière, France
E-mail: dominique.manchon@uca.fr
Received August 26, 2021, in final form March 10, 2022; Published online March 23, 2022
https://doi.org/10.3842/SIGMA.2022.023
Abstract. We identify the Baker–Campbell–Hausdorff recursion driven by a weight λ = 1
Rota–Baxter operator with the Magnus expansion relative to the post-Lie structure naturally
associated to the corresponding Rota–Baxter algebra. Post-Lie Magnus expansion and BCH-
recursion are reviewed before the proof of the main result.
Key words: post-Lie algebra; pre-Lie algebra; Rota–Baxter algebra; Magnus expansion;
BCH-formula; rooted trees
2020 Mathematics Subject Classification: 16T05; 16T10; 16T30; 17A30
1 Introduction
The present paper consists in a quick survey of post-Lie algebras, Baker–Campbell–Haudorff
recursion, Rota–Baxter algebras and post-Lie Magnus expansion (Sections 2, 3 and 4), followed
by a new result in Section 5, which establishes an equality between two seemingly different
formal series: the Baker–Campell–Hausdorff recursion in a weight-one Rota–Baxter algebra, and
the post-Lie Magnus expansion relative to the associated post-Lie algebra structure described
in [4]. Our motivation comes from a result obtained in 2006 by the second and the third author
together with Li Guo [15], identifying the BCH-recursion with the pre-Lie Magnus expansion in
the weight-zero case.
Interest in Magnus-type expansions results from their appearance in the context of numerical
integration methods for Lie group valued problems [23]. In [12], we studied the relation between
the classical and the post-Lie Magnus expansions by looking at a non-autonomous matrix-valued
initial value problem from the viewpoint of the theory of numerical Lie group integrators. Post-
Lie algebras and the post-Lie Magnus expansion play a central part in the latter. In this context
post-Lie algebras characterise the relation between two Lie algebras (one coming from the Jacobi
Lie bracket and the other from the torsion Lie bracket, in the context of a flat connection with
constant torsion). This relation can be lifted to the level of the (completed) enveloping algebra
(of the Lie algebra implied by the torsion Lie bracket). In a nutshell, the post-Lie Magnus
expansion naturally appears in the context of backward error analysis for the Lie–Euler method.
This is consistent with the fact that the pre-Lie Magnus expansion plays an analogous role with
respect to backward error analysis for the Euler method.
mailto:mahdi.alkaabi@uomustansiriyah.edu.iq
mailto:kurusch.ebrahimi-fard@ntnu.no
mailto:dominique.manchon@uca.fr
https://doi.org/10.3842/SIGMA.2022.023
2 M.J.H. Al-Kaabi, K. Ebrahimi-Fard and D. Manchon
The well-known Baker–Campbell–Hausdorff (BCH) formula BCH(x, y) is a formal power
series, which lives in the completion of the free Lie algebra L(x, y) generated (over a base
field K of characteristic zero) by the two non-commutating variables x and y. It is defined by
exp(x) exp(y) = exp
(
BCH(x, y)
)
= exp
(
x+ y + B̃CH(x, y)
)
or
BCH(x, y) = log
(
exp(x) exp(y)
)
= x+ y + B̃CH(x, y).
It plays a prominent role in modern mathematics [3, 7].1
A fruitful connection between the BCH-series and the notion of Rota–Baxter algebra has been
explored in [13, 14, 15]. The latter originated in the seminal 1960 article [5] by the American
mathematician G. Baxter, which in turn was motivated by F. Spitzer’s 1956 article [31]. Baxter’s
algebra was further developed foremost in the commutative realm in the 1960s and ’70s by
P. Cartier, G.-C. Rota and F.V. Atkinson, among others, from algebraic, combinatorial and
analytic viewpoints. We refer the reader to the review article [20] as well as the monograph [22]
for details.
A weight-λ Rota–Baxter operator on an associative K-algebra A is a K-linear map R:
A −→ A, satisfying the Rota–Baxter identity of weight λ ∈ K:
R(x)R(y) = R
(
R(x)y + xR(y) + λxy
)
, x, y ∈ A. (1.1)
The pair (A,R) is a weight λ Rota–Baxter algebra.2 For example, the indefinite Riemann
integral satisfies (1.1) when the weight λ = 0 (integration by parts). The linear map R̃ :=
−λidA −R is also Rota–Baxter of weight λ, and satisfies together with R the mixed identity
R(x)R̃(y) = R̃
(
R(x)y
)
+R
(
xR̃(y)
)
, x, y ∈ A.
Starting from a Rota–Baxter operator R of weight λ, the BCH-recursion [15] is defined by
χλ(a) := a+
1
λ
B̃CH
(
R
(
χλ(a)
)
, R̃
(
χλ(a)
))
, a ∈ A. (1.2)
It lies at the heart of the solution of an exponential factorisation problem [15] and thereby
permits the generalisation of a classical result for commutative Rota–Baxter algebras, known as
Spitzer’s identity [31], to non-commutative Rota–Baxter algebras. The resulting non-commuta-
tive Spitzer identity says that for a ∈ A the exponential
X := exp
(
R
(
χλ
(
log(1 + tλa)
λ
)))
solves the fixed point equation
X = 1 + tR(aX) (1.3)
in the algebraA[[t]] of formal series with coefficients inA. Here the formal parameter t commutes
with all elements in A. More precisely, iterating the fixed point equation (1.3) yields the rather
non-trivial equality
1 + tR(a) + t2R
(
aR(a)
)
+ t3R
(
aR
(
aR(a)
))
+ · · · = exp
(
R
(
χλ
(
log(1 + tλa)
λ
)))
.
1The remainder Baker–Campbell–Hausdorff series, B̃CH(x, y), is denoted by BCH(x, y) in [15]. We adopt here
a more conventional notation.
2The convention for the weight is with the opposite sign in [15].
Post-Lie Magnus Expansion and BCH-Recursion 3
Thanks to the commuting parameter t, the last equality can be seen as between formal power
series and therefore encompasses at each order a specific relation between coefficients. For
instance, at order two, that is, comparing the coefficients of t2, we have the identity
2R
(
aR(a)
)
= R(a)R(a)−R
(
[R(a), a] + λa2
)
,
which is easily verifiable in a Rota–Baxter algebra of weight λ by using the Rota–Baxter iden-
tity (1.1) on the right-hand side. We note that the fixed point equation (1.3) is reminiscent of the
integral fixed point equation naturally associated to a linear matrix-valued initial value problem;
the indefinite Riemann integral is a weight-zero Rota–Baxter map. Indeed, the series (1.2) turns
out to be closely related to a well-known Lie algebra expansion due to W. Magnus [24]. This
connection to the so-called Magnus expansion was studied in reference [15] in the case of the
weight being zero (λ = 0). The adequate algebraic setting is provided through the notion of pre-
Lie algebra, which is naturally defined on any non-commutative Rota–Baxter algebra. In [17] it
was shown that the pre-Lie Magnus expansion can be expressed in terms of the BCH-recursion
as follows
Ω′
�
(a) := a+
∑
n>0
Bn
n!
L
(n)
�
[
Ω′
�
(a)
]
(a) = χλ
(
log(1 + λa)
λ
)
. (1.4)
Here Bn is the n-th Bernoulli number and L�[x](y) = L
(1)
� [x](y) := x�y is the left-multiplication
operator defined in terms of the aforementioned (left) pre-Lie product, denoted �, on a non-
commutative Rota–Baxter algebra. Note that the weight λ is absorbed in the definition of the
pre-Lie product. In the weight-zero case, (1.4) boils down to
Ω′
�
(a) = χ0(a). (1.5)
In particular, for the indefinite Riemann integral, the pre-Lie product is defined for – matrix-
valued – functions A, B as (A�B)(t) :=
[ ∫ t
0 A(s)ds,B(t)
]
. When inserted in (1.4), one recovers
Magnus’ original expansion [24].
Recall that any Rota–Baxter algebra with nonzero weight gives rise to a post-Lie algebra
structure [4]. In this work, we describe a close relationship between the BCH-recursion (1.2) in
the nonzero weight case and the Magnus expansion in its post-Lie version [16, 18, 19, 26]. Our
main result (Theorem 5.3) shows that the post-Lie Magnus expansion and the BCH-recursion
in (1.2) coincide in the context of a Rota–Baxter algebra of weight 1 endowed with its naturally
associated post-Lie structure. This is an extension to nonzero weight of one of the main results
of [15], resumed by (1.5), identifying the weight zero BCH-recursion with the pre-Lie Magnus
expansion. The special role of weight one here simply comes from the definition of the post-Lie
structure (2.9), (2.10), and any Rota–Baxter algebra with nonzero weight can be set to weight
one by an appropriate rescaling of the Rota–Baxter operator.
We close this introduction by noting that the Magnus expansion, in its various forms (classical
[24, 27], pre-Lie [1, 10, 17] and post-Lie [16, 18, 19, 26]), has been studied in applied mathematics,
control theory, physics and chemistry. See reference [6] for details on the classical Magnus
expansion in applied mathematics. The reader can also find a brief summary in the recent
work [12].
This paper consists of four sections accompanied by two appendices. In Section 2, we review
some basic topics related to post-Lie algebras and their universal enveloping algebras. The
post-Lie structure defined on any Rota–Baxter algebra is recalled from [4]. Section 3 contains
the description of the Baker–Campbell–Hausdorff recursion and its inverse, as well as their
properties. Several important details on the post-Lie Magnus expansion and its inverse are
included in Section 4. Section 5 is the main part of this work, in which the identification of the
post-Lie Magnus expansion with the BCH-recursion is proven. Finally, the two Appendices A
and B contain low-order computations of the post-Lie Magnus expansion and its inverse.
4 M.J.H. Al-Kaabi, K. Ebrahimi-Fard and D. Manchon
2 Post-Lie algebras
A post-Lie algebra is a Lie algebra (L, [· , ·]) together with a bilinear mapping � : L × L −→ L,
which is compatible with the Lie bracket in the following sense
x� [y, z] = [x� y, z] + [y, x� z], (2.1)
[x, y]� z = a�(x, y, z)− a�(y, x, z), (2.2)
for any x, y, z ∈ L. Here, a�(x, y, z) is the associator defined by
a�(x, y, z) = x� (y � z)− (x� y)� z.
Any Lie algebra can be seen as a post-Lie algebra by setting the second product � to zero.
Another possibility is to take for the second product � the opposite of the Lie bracket.
A (left) pre-Lie algebra is an abelian post-Lie algebra, i.e., a post-Lie algebra with Lie bracket
set to zero. The defining relation is the left pre-Lie identity
0 = a�(x, y, z)− a�(y, x, z). (2.3)
We refer the reader to [25] for a short survey on pre-Lie algebras. The post-Lie operation �
permits to produce two other operations:
[[x, y]] := x� y − y � x+ [x, y],
x� y := x� y + [x, y],
for all x, y ∈ L. From (2.1) and (2.2), one can see that
(
L, [[· , ·]]
)
forms a Lie algebra, denoted L̃.
In the case of an abelian post-Lie algebra, this amounts to Lie admissibility of pre-Lie algebras.
The triple
(
L,−[· , ·],�
)
forms another post-Lie algebra [12, 28] sharing the same double Lie
bracket, i.e.,
[[x, y]] = x� y − y � x+ [x, y] = x� y − y � x− [x, y].
For more details on post-Lie algebras, we refer to [11, 16, 18, 28] and references therein.
2.1 The universal enveloping algebra of a post-Lie algebra
Inspired by the work of J.-M. Oudom and D. Guin in the pre-Lie context [30], the authors
in [16] consider the enveloping algebra
(
U(L), ·
)
of the Lie algebra
(
L, [· , ·]
)
underlying a post-
Lie algebra
(
L, [· , ·],�
)
. The post-Lie product � is then extended to L ⊗ U(L) → U(L) by
requiring x� 1 := 0 and
x� (x1 · · ·xn) :=
n∑
i=1
x1 · · ·xi−1(x� xi)xi+1 · · ·xn,
for all x, x1, . . . , xn ∈ L. Here, 1 denotes the unit in U(L). Recall that the enveloping alge-
bra U(L) together with the product · and the unshuffle coproduct has the structure of a non-
commutative, co-commutative Hopf algebra. The unshuffle coproduct ∆ is defined for all letters
x ∈ L ↪→ U(L), by ∆(x) := x⊗ 1+ 1⊗ x and extended multiplicatively. We employ Sweedler’s
notation, ∆(X) := X(1) ⊗ X(2), for the coproduct of any X ∈ U(L). The final definition of
the extended post-Lie product on U(L), together with its properties, is given by the next two
propositions.
Post-Lie Magnus Expansion and BCH-Recursion 5
Proposition 2.1 ([16, Proposition 3.1]). There is a unique extension of the post-Lie product �
from L to U(L) satisfying:
1�X = X,
xX � y = x� (X � y)− (x�X)� y,
X � Y Z =
(
X(1) � Y
)(
X(2) � Z
)
,
for all x, y ∈ L, and X,Y, Z ∈ U(L).
Proposition 2.2 ([16, Proposition 3.2]). The extended post-Lie product � on U(L) possesses
the following properties:
X � 1 = ϵ(X),
ϵ(X � Y ) = ϵ(X)ϵ(Y ),
∆(X � Y ) = (X(1) � Y(1))⊗ (X(2) � Y(2)),
xX � Y = x� (X � Y )− (x�X)� Y,
X � (Y � Z) = (X(1)
(
X(2) � Y
)
)� Z,
for all x ∈ L and X,Y, Z ∈ U(L), where ϵ : U(L) → K is the counit map.
From the last equality in Proposition 2.2, an associative product, known as Grossman–Larson
product, can be defined on U(L) as follows
X ∗ Y := X(1)
(
X(2) � Y
)
, (2.4)
for all X,Y ∈ U(L). As a main example, for any x ∈ L and Y ∈ U(L), we find
x ∗ Y = x� Y + xY, (2.5)
since any element of L is primitive. The Grossman–Larson product (2.4) defines together with
the coproduct ∆ another structure of Hopf algebra on U(L). The corresponding antipode will be
denoted by S∗. The Hopf algebras
(
U(L), ∗,∆
)
and
(
U(L̃), .,∆
)
are isomorphic [15, Section 3],
[30, Section 2].
Remark 2.3. Conversely, the product of the enveloping algebra can be expressed in terms of
the Grossman–Larson product and the unshuffle coproduct as follows
XY = X(1) ∗
(
S∗X(2) � Y
)
. (2.6)
This is seen by plugging (2.4) into the right-hand side of (2.6).
2.2 Free post-Lie algebras
F. Chapoton and M. Livernet presented in [9] the free pre-Lie algebra in terms of (non-planar)
decorated rooted trees. Similarly, H. Munthe-Kaas and A. Lundervold gave in [28] an explicit
description of the free post-Lie algebra in terms of formal Lie brackets of planar decorated
rooted trees. Let us briefly review this construction: a magma is a set M together with a binary
operation, without any further properties. For any (non-empty) set E, the set of all parenthesized
words on the alphabet E is the free magma over E, denoted M(E). A practical presentation of
it can be given in terms of planar rooted trees. Indeed, consider the set T pl
E of all planar rooted
trees with vertices decorated by E, and let ◦↘ denote the left Butcher product defined on T pl
E as
σ◦↘τ = Be
+(στ1τ2 · · · τk),
6 M.J.H. Al-Kaabi, K. Ebrahimi-Fard and D. Manchon
for σ, τ1, τ2, . . . , τk ∈ T pl
E and τ := Be
+(τ1τ2 · · · τk). Here, Be
+ is the operation defined by graft-
ing a monomial τ1τ2 · · · τk of E-decorated rooted trees on a common root decorated by some
element e in E, to obtain a new tree. For example (in the undecorated context)
6 Mahdi J. Hasan AL-KAABI, Kurusch EBRAHIMI-FARD and Dominique MANCHON
Remark 2.3. Conversely, the product of the enveloping algebra can be expressed in terms of the Grossman–
Larson product and the unshuffle coproduct as follows:
XY = X(1) ∗ (S ∗X(2) � Y). (2.8)
This is seen by plugging (2.6) into the right-hand side of (2.8).
2.2 Free post-Lie Algebras
F. Chapoton and M. Livernet presented in [9] the free pre-Lie algebra in terms of (non-planar) decorated
rooted trees. Similarly, H. Munthe-Kaas and A. Lundervold gave in [29] an explicit description of the free
post-Lie algebra in terms of formal Lie brackets of planar decorated rooted trees. Let us briefly review this
construction: a magma is a set M together with a binary operation, without any further properties. For any
(non-empty) set E, the set of all parenthesized words on the alphabet E is the free magma over E, denoted
M(E). A practical presentation of it can be given in terms of planar rooted trees. Indeed, consider the set T pl
E
of all planar rooted trees with vertices decorated by E, and let ◦ց denote the left Butcher product defined on
T pl
E as:
σ◦ցτ = Be
+(στ1τ2 · · · τk),
for σ, τ1, τ2, . . . , τk ∈ T pl
E and τ := Be
+(τ1τ2 · · · τk). Here, Be
+ is the operation defined by grafting a monomial
τ1τ2 · · · τk of E-decorated rooted trees on a common root decorated by some element e in E, to obtain a new
tree. For example (in the undecorated context):
◦ց = , ◦ց = , ◦ց = .
Denote by T pl
E the linear span of the set T pl
E . Besides the left Butcher product, ◦ց, this space has another
magmatic product defined through left grafting, denotedց and defined by
σց τ =
∑
v vertex of τ
σցv τ, (2.9)
where σ ցv τ is the tree obtained by grafting the root of the tree σ onto the vertex v of the tree τ, such that
σ becomes the leftmost branch starting from vertex v. See for example references [2, 8]. Computing some
examples (in the undecorated context) we find:
ց = + , ց = + + .
By freeness universal property, there is a unique morphism of magmatic algebras
Ψ : (T pl
E ,
◦ց) −→ (T pl
E ,ց)
τ 7−→ eτ := Ψ(τ)
such that Ψ(•a) = •a for any a ∈ E, which is a linear isomorphism. A detailed account of the map Ψ can be
found in [2].
LetL(T pl
E ) be the free Lie algebra generated by T pl
E . It can be endowed with a structure of post-Lie algebra
by extending the aforementioned left grafting,ց, as follows:
σց [ τ, τ′] = [σց τ, τ′] + [ τ, σց τ′] ,
Denote by T pl
E the linear span of the set T pl
E . Besides the left Butcher product, ◦↘, this space
has another magmatic product defined through left grafting, denoted ↘ and defined by
σ ↘ τ =
∑
v vertex of τ
σ ↘v τ, (2.7)
where σ ↘v τ is the tree obtained by grafting the root of the tree σ onto the vertex v of
the tree τ , such that σ becomes the leftmost branch starting from vertex v. See for example
references [2, 8]. Computing some examples (in the undecorated context) we find
6 Mahdi J. Hasan AL-KAABI, Kurusch EBRAHIMI-FARD and Dominique MANCHON
Remark 2.3. Conversely, the product of the enveloping algebra can be expressed in terms of the Grossman–
Larson product and the unshuffle coproduct as follows:
XY = X(1) ∗ (S ∗X(2) � Y). (2.8)
This is seen by plugging (2.6) into the right-hand side of (2.8).
2.2 Free post-Lie Algebras
F. Chapoton and M. Livernet presented in [9] the free pre-Lie algebra in terms of (non-planar) decorated
rooted trees. Similarly, H. Munthe-Kaas and A. Lundervold gave in [29] an explicit description of the free
post-Lie algebra in terms of formal Lie brackets of planar decorated rooted trees. Let us briefly review this
construction: a magma is a set M together with a binary operation, without any further properties. For any
(non-empty) set E, the set of all parenthesized words on the alphabet E is the free magma over E, denoted
M(E). A practical presentation of it can be given in terms of planar rooted trees. Indeed, consider the set T pl
E
of all planar rooted trees with vertices decorated by E, and let ◦ց denote the left Butcher product defined on
T pl
E as:
σ◦ցτ = Be
+(στ1τ2 · · · τk),
for σ, τ1, τ2, . . . , τk ∈ T pl
E and τ := Be
+(τ1τ2 · · · τk). Here, Be
+ is the operation defined by grafting a monomial
τ1τ2 · · · τk of E-decorated rooted trees on a common root decorated by some element e in E, to obtain a new
tree. For example (in the undecorated context):
◦ց = , ◦ց = , ◦ց = .
Denote by T pl
E the linear span of the set T pl
E . Besides the left Butcher product, ◦ց, this space has another
magmatic product defined through left grafting, denotedց and defined by
σց τ =
∑
v vertex of τ
σցv τ, (2.9)
where σ ցv τ is the tree obtained by grafting the root of the tree σ onto the vertex v of the tree τ, such that
σ becomes the leftmost branch starting from vertex v. See for example references [2, 8]. Computing some
examples (in the undecorated context) we find:
ց = + , ց = + + .
By freeness universal property, there is a unique morphism of magmatic algebras
Ψ : (T pl
E ,
◦ց) −→ (T pl
E ,ց)
τ 7−→ eτ := Ψ(τ)
such that Ψ(•a) = •a for any a ∈ E, which is a linear isomorphism. A detailed account of the map Ψ can be
found in [2].
LetL(T pl
E ) be the free Lie algebra generated by T pl
E . It can be endowed with a structure of post-Lie algebra
by extending the aforementioned left grafting,ց, as follows:
σց [ τ, τ′] = [σց τ, τ′] + [ τ, σց τ′] ,
By freeness universal property, there is a unique morphism of magmatic algebras
Ψ:
(
T pl
E , ◦↘
)
−→
(
T pl
E ,↘
)
,
τ 7−→ eτ := Ψ(τ),
such that Ψ(•a) = •a for any a ∈ E, which is a linear isomorphism. A detailed account of the
map Ψ can be found in [2].
Let L
(
T pl
E
)
be the free Lie algebra generated by T pl
E . It can be endowed with a structure of
post-Lie algebra by extending the aforementioned left grafting, ↘, as follows
σ ↘ [τ, τ ′] = [σ ↘ τ, τ ′] + [τ, σ ↘ τ ′],
[σ, τ ] ↘ τ ′ = a↘(σ, τ, τ ′)− a↘(τ, σ, τ ′),
for all σ, τ, τ ′ ∈ T pl
E . The triple
(
L
(
T pl
E
)
, [· , ·],↘
)
is the free post-Lie algebra generated by E [32]
(see also [28, 29]).
Recall that an E-decorated planar forest f = τ1 · · · τn is a (non-commutative) product of
E-decorated planar rooted trees τi ∈ T pl
E , i = 1, . . . , n. Denote by F pl
E the set of all E-decorated
planar forests, and by Fpl
E its linear span. The space Fpl
E forms together with the concatenation
product the free associative algebra generated by T pl
E . The left grafting, ↘, defined by (2.7)
on T pl
E can be generalized to a grafting of forests as follows:
� Left grafting a tree on a forest is also defined by (2.7). We thus have
σ ↘ ff ′ = (σ ↘ f)f ′ + f(σ ↘ f ′),
for any tree σ ∈ T pl
E and any two forests f, f ′ ∈ F pl
E .
� The left grafting of a forest f = τ1 · · · τk onto a forest f ′ is the sum of forests obtained by
summing over all ways of successively left grafting the trees τk, . . . , τ1 to any node of f ′.
The well-known (planar) Grossman–Larson product on Fpl
E is defined by [21]
f ⋆ f ′ := B−
(
f ↘ Be
+(f
′)
)
, (2.8)
Post-Lie Magnus Expansion and BCH-Recursion 7
where B− is the left inverse operation of Be
+, which removes the root of a tree and thus produces
a forest. This product endows the space Fpl
E with a structure of an non-commutative associative
unital algebra, called the Grossman–Larson algebra. This algebra acts naturally on T pl
E by
extended left grafting
(f ⋆ f ′) ↘ τ := f ↘ (f ′ ↘ τ),
for all f, f ′ ∈ Fpl
E and τ ∈ T pl
E . The universal enveloping algebra U
(
L
(
T pl
E
))
of the free post-Lie
algebra L
(
T pl
E
)
is the free associative algebra on T pl
E , and can therefore be identified with Fpl
E .
The terminology is justified by the following
Proposition 2.4 ([16, Proposition 3.5]). With the identification recalled above, the Grossman–
Larson product ∗ on U
(
L
(
T pl
E
))
is identical to the Grossman–Larson product ⋆ on
(
Fpl
E , ⋆
)
.
2.3 Post-Lie structure on a Rota–Baxter algebra
Recall that a unital algebra is said to be complete filtered if it is equipped with a separating
complete filtration
A = A0 ⊇ A1 ⊇ A2 ⊇ · · · ⊇ An ⊇ · · ·
by ideals [13]. Separation means that the intersection of the An’s is equal to {0}, and complete-
ness refers to the topology associated with the filtration, so that any series a =
∑
n≥1 an with
an ∈ An converges in A. The filtration is moreover supposed to be compatible with the product,
i.e., An1An2 ⊆ An1+n2 for any n1, n2 ≥ 0. A Rota–Baxter algebra is said to be complete filtered
if it is equipped with a separating complete filtration by Rota–Baxter ideals, i.e., by ideals An
stable by the Rota–Baxter operator.
The Rota–Baxter Algebra
(
A,R
)
of weight λ has a structure of a post-Lie algebra defined
by the following operations:
[x, y]λ := λ[x, y], (2.9)
x� y := [R(x), y], (2.10)
for all x, y ∈ A. We leave it to the reader to show that the operations in (2.9), (2.10) satisfy
the post-Lie identities (2.1) and (2.2) (see [4, Section 5.2]). As expected, the post-Lie algebra(
A, [· , ·]λ,�
)
reduces to a (left) pre-Lie algebra in the case of a weight zero Rota–Baxter algebra.
Indeed, if λ = 0, the product � defined by (2.10) verifies the left pre-Lie identity (2.3).
3 Baker–Campbell–Hausdorff (BCH)-recursion
We give here a brief account of the Baker–Campbell–Hausdorff recursion, which was defined and
explored in [13, 14, 15]. Let A = K ⟨⟨x, y⟩⟩ be the free complete associative K-algebra of formal
power series generated by non-commuting variables x and y. The Baker–Campbell–Hausdorff
expansion BCH(x, y) is the element in A satisfying the following equation
exp(x) exp(y) = exp
(
BCH(x, y)
)
.
The first terms are given by
BCH(x, y) = x+ y + B̃CH(x, y)
= x+ y +
1
2
[x, y] +
1
12
[x, [x, y]]− 1
12
[y, [x, y]]− 1
24
[x, [y, [x, y]]] + · · · ,
where [x, y] := xy − yx is the usual commutator of x and y in A. See, e.g., [7] for details.
8 M.J.H. Al-Kaabi, K. Ebrahimi-Fard and D. Manchon
Proposition 3.1 ([15, Proposition 1]). Let A be a complete filtered K-algebra, and let R be
a K-linear map preserving the filtration of A. There exists a unique (usually non-linear) map
χ : A1 −→ A1, such that (χ− idA)(An) ⊂ A2n, for all n ≥ 1, and
BCH
(
R(χ(x)), R̃(χ(x))
)
= x, (3.1)
for all x ∈ A1, where R̃ := idA −R. This map is bijective, and its inverse is
χ−1(x) = BCH
(
R(x), R̃(x)
)
= x+ B̃CH
(
R(x), R̃(x)
)
. (3.2)
As a consequence of (3.1), we have the exponential factorization
exp
(
R(χ(x))
)
exp
(
R̃(χ(x))
)
= exp(x), (3.3)
for any x ∈ A1. Note also that (3.2) yields the non-linear BCH-recursion
χ(x) := x− B̃CH
(
R(χ(x)), R̃(χ(x))
)
, (3.4)
for all x ∈ A1.
Lemma 3.2 ([15]). Let A be a complete filtered algebra, and let R : A −→ A be a linear map
preserving the filtration. The following holds:
1. The map χ given by (3.4), can be simplified:
χ(x) = x+ B̃CH
(
−R(χ(x)), x
)
∀x ∈ A1. (3.5)
2. If R is an idempotent algebra homomorphism, then the map χ in (3.5) is further simplified,
namely χ(x) = x+ B̃CH(−R(x), x).
Proof. See [15, Lemmas 6 and 7]. ■
The BCH-recursion in the Rota–Baxter algebra framework is given as follows in the case
where the weight λ is different from zero:
Proposition 3.3 ([15, Proposition 11]). Let
(
A,R
)
be a complete filtered Rota–Baxter algebra
of weight λ ̸= 0, and set R̃ := −λidA −R. The λ-weighted BCH-recursion is written
χλ(x) = x+
1
λ
B̃CH
(
R(χλ(x)), R̃(χλ(x))
)
, (3.6)
for all x ∈ A1. It can be simplified to
χλ(x) = x− 1
λ
B̃CH
(
−R(χλ(x)), λx
)
.
Its inverse is given by
χ−1
λ (x) = x− 1
λ
B̃CH
(
R(x), R̃(x)
)
.
Moreover, the factorization obtained in (3.3) becomes
exp
(
R(χλ(x))
)
exp
(
R̃(χλ(x))
)
= exp(−λx). (3.7)
Post-Lie Magnus Expansion and BCH-Recursion 9
The expansion χλ can be written as the infinite sum
χλ(x) =
∑
n≥1
χ
(n)
λ (x),
where χ
(n)
λ ∈ An is the n-th homogenous component of the BCH-recursion. Here, we write the
components χ
(n)
λ up to order n = 4 using the post-Lie algebra notation (2.9) and (2.10)
χ
(1)
λ (x) = x,
χ
(2)
λ (x) =
1
2λ
[
R
(
χ
(1)
λ (x)
)
, R̃
(
χ
(1)
λ (x)
)]
=
1
2λ
[
R
(
χ
(1)
λ (x)
)
,
(
−λidA −R
)(
χ
(1)
λ (x)
)]
= −1
2
[
R(x), x
]
= −1
2
x� x,
χ
(3)
λ (x) =
1
2λ
([
R
(
χ
(1)
λ (x)
)
, R̃
(
χ
(2)
λ (x)
)]
+
[
R
(
χ
(2)
λ (x)
)
, R̃
(
χ
(1)
λ (x)
)])
+
1
12λ
([
R
(
χ
(1)
λ (x)
)
,
[
R
(
χ
(1)
λ (x)
)
, R̃
(
χ
(1)
λ (x)
)]]
−
[
R̃
(
χ
(1)
λ (x)
)
,
[
R
(
χ
(1)
λ (x)
)
, R̃
(
χ
(1)
λ (x)
)]])
=
1
4
(x� x)� x+
1
12
x� (x� x) +
1
12
[x� x, x]λ,
χ
(4)
λ (x) =
λ− 1
24
x�
(
(x� x)� x
)
− λ+ 1
24
(x� x)� (x� x) +
λ− 3
24
(
(x� x)� x
)
� x
− λ+ 1
24
(
x� (x� x)
)
� x+
1
24
[
x, x� (x� x) + (x� x)� x
]
λ
.
These coefficients are recursively computed using (3.4).
4 Magnus expansion
W. Magnus [24] considered the problem of expressing the solution of the matrix-valued linear
initial value problem Ẏ (t) = M(t)Y (t), Y (0) = Y0 as an exponential [6, 27]
Y (t) = exp
(
Ω(M)(t)
)
Y0.
The Magnus expansion, Ω(M)(t) = log(Y (t)), is determined by the particular differential equa-
tion
Ω̇(M) := M +
∑
n>0
Bn
n!
ad
(n)
Ω(M)(M) (4.1)
= dexp−1
Ω(M)(M)
:=
adΩ(M)
eadΩ(M) − 1
(M), (4.2)
with Ω(M)(0) = 0. Here, Bn are the Bernoulli numbers and ad
(n)
M1
(M2) := ad
(n−1)
M1
([M1,M2]),
ad
(0)
M1
(M2) = M2. Defining the pre-Lie product, (M1 �M2)(t) :=
[ ∫ t
0 M1(s)ds,M2(t)
]
, we can
rewrite (4.2) using the left-multiplication operators L�(x) := x � − defined in terms of the
pre-Lie product:
Ω̇(M) =
L�[Ω̇(M)]
eL�[Ω̇(M)] − 1
(M).
10 M.J.H. Al-Kaabi, K. Ebrahimi-Fard and D. Manchon
4.1 Post-Lie Magnus expansion
We consider now the universal enveloping algebra Fpl
E := U
(
L
(
T pl
E
))
of the free post-Lie algebra(
L(T pl
E ), [· , ·],↘
)
, graded by the number of vertices of the forests. Denote by
̂U
(
L
(
T pl
E
))
its
completion with respect to the grading. Any element of the completion can be written as
a so-called Lie–Butcher series [16, 28, 29]
α =
∑
f∈Fpl
E
⟨α, f⟩ f,
where ⟨· , ·⟩ : ̂U
(
L
(
T pl
E
))
⊗ U
(
L
(
T pl
E
))
→ K is the natural pairing defined on any pair (f, f ′) of
forests by
⟨f, f ′⟩ =
{
0, f ̸= f ′,
1, f = f ′.
The unshuffle coproduct, ∆, is naturally extended to the completion. The set Prim
(
F pl
E
)
consists
in primitive elements (infinitesimal characters), whereas G
(
F pl
E
)
denotes the set of group-like
elements (characters)
Prim(F pl
E ) :=
{
α ∈ ̂U
(
L
(
T pl
E
))
| ∆(α) = 1⊗ α+ α⊗ 1
}
= L̂
(
T pl
E
)
,
G(F pl
E ) :=
{
α ∈ ̂U
(
L
(
T pl
E
))
| ∆(α) = α⊗ α
}
.
Both products on U
(
L
(
T pl
E
))
– the concatenation and the Grossman–Larson product (2.8) –
can also be extended to products on the completion
̂U
(
L
(
T pl
E
))
. As a result, two different
exponential functions can be defined on
̂U
(
L
(
T pl
E
))
, namely,
exp∗(f) =
∞∑
n=0
f∗n
n!
= 1+ f +
1
2
f ∗ f +
1
6
f ∗ f ∗ f + · · · ,
exp(f) =
∞∑
n=0
fn
n!
= 1+ f +
1
2
ff +
1
6
fff + · · · .
Both these exponential functions map Prim
(
F pl
E
)
bijectively onto G
(
F pl
E
)
. See [16] for details.
The post-Lie Magnus expansion χ is the bijective map from L̂
(
T pl
E
)
onto itself defined by
exp∗
(
χ(f)
)
= exp(f),
namely,
χ(f) = log∗
(
exp(f)
)
. (4.3)
Introducing a formal commuting indeterminate t, it can also be described as
χ(ft) =
∑
n≥1
χ(n)(f)tn,
where χ(n)(f) is the n-th order component of the post-Lie Magnus expansion χ. The latter is
defined recursively by χ(1)(f) = f , and [16, 18, 19]
χ(n)(f) :=
fn
n!
−
n∑
k=2
1
k!
∑
p1+···+pk=n
pi>0
χ(p1)(f) ∗ χ(p2)(f) ∗ · · · ∗ χ(pk)(f). (4.4)
Post-Lie Magnus Expansion and BCH-Recursion 11
The computation of the coefficients χ(n)(f) for the first five values of n is displayed in Appendix A
below. They have been obtained by hand by the recursive formula (4.4), using (2.5) repeatedly.
Comparing with the computations at the end of Section 3, one observes that, up to order n = 4,
the coefficient χ(n)(f) coincides with the coefficient χ
(n)
λ (f) of the BCH-recursion in the weight
λ = 1 case. We shall prove this fact at any order in Theorem 5.3 below.
Remark 4.1. Formula (4.3) defines the post-Lie Magnus expansion in any complete filtered
post-Lie algebra L. If the underlying Lie algebra is Abelian, then the post-Lie Magnus expansion
is reduced to the so-called pre-Lie Magnus expansion. The latter already appears in [1] and
encompasses classical Magnus expansion [24].
Remark 4.2. We may deduce a Magnus-type differential equation similar to (4.1) for the post-
Lie Magnus expansion (4.3), by differentiating exp∗
(
χ(ft)
)
= exp(ft) with respect to t. This
results in
χ̇(ft) = dexp∗−1
−χ(ft)
(
exp∗(−χ(ft))� f
)
, χ(0) = 0.
4.2 Inverse post-Lie Magnus expansion
The inverse post-Lie Magnus expansion θ is the bijective map from L̂
(
T pl
E
)
onto itself given by
the following formula
θ(f) = log(exp∗(f)) (4.5)
or
exp(θ(f)) = exp∗(f).
The homogeneous component θ(n) = θ(n)(f) of degree n of the expansion
θ(ft) =
∑
n≥1
θ(n)(f)tn
is given by θ(1)(f) = f and the following recursive formula [16]
θ(n)(f) =
1
n
(
n−1∑
j=1
1
j!
∑
k1+···+kj=n−1
ki>0
(
θ(k1)θ(k2) · · · θ(kj)
)
� f
+
n−1∑
j=1
Bj
j!
∑
k1+···+kj=n−1
ki>0
adθ(k1) · · · adθ(kj) f
+
n−1∑
j=2
((
j−1∑
q=1
Bq
q!
∑
k1+···+kq=j−1
ki>0
adθ(k1) · · · adθ(kq)
)
×
(
n−j∑
p=1
1
p!
∑
k1+···+kp=n−j
ki>0
(
θ(k1)θ(k2) · · · θ(kp)
)
� f
)))
, (4.6)
where adθ(i)(f) :=
[
θ(i), f
]
, and the Bi’s are the Bernoulli numbers. The computation of the
first θ(n)’s is given in Appendix B.
12 M.J.H. Al-Kaabi, K. Ebrahimi-Fard and D. Manchon
Remark 4.3. The same fact, described in Remark 4.1, will be repeated again in the case of the
inverse post-Lie Magnus expansion. In other words, the formula in (4.6) for the inverse post-
Lie Magnus expansion is reduced, in the case of commutative post-Lie algebras, to the inverse
pre-Lie Magnus expansion formula described below (see also [17, 25])
W (x) :=
eL�[x] − 1
L�[x]
(x) =
∞∑
n=0
1
(n+ 1)!
L
(n)
� [x](x).
Modulo removal of a fictitious unit, W is also known as the pre-Lie exponential [1].
Remark 4.4. Similar to Remark 4.2, we may deduce a Magnus-type differential equation similar
to (4.1) for the inverse post-Lie Magnus expansion [16, 19]
θ̇(ft) = dexp−1
−θ(ft)
(
exp(θ(ft))� f
)
, θ(0) = 0.
5 Post-Lie Magnus expansion and BCH-recursion
We now show that the Baker–Campbell–Hausdorff recursion driven by a weight λ = 1 Rota–
Baxter operator identifies with the Magnus expansion relative to the post-Lie structure naturally
associated to the corresponding Rota–Baxter algebra.
Theorem 5.1. Let
(
A,R
)
be a complete filtered Rota–Baxter algebra of weight λ = 1. We have
the following equality in Û(A) for any x ∈ A and t ∈ K:
exp∗(tx) = exp
(
−tR̃(x)
)
exp
(
−tR(x)
)
, (5.1)
where R̃ = −idA−R, and ∗ is the associative product defined in (2.4), using the post-Lie product
x� y = [R(x), y].
The proof of this theorem will rely on the following proposition:
Proposition 5.2. In any complete filtered Rota–Baxter algebra
(
A,R
)
of weight λ = 1, we have
the following identity in Û(A):
dn
dtn
exp
(
−tR̃(x)
)
exp
(
−tR(x)
)
= exp
(
−tR̃(x)
)
x∗n exp
(
−tR(x)
)
. (5.2)
Proof. The proof goes by induction. The base case k = 0 is trivial. For k = 1, we have in Û(A):
d
dt
exp
(
−tR̃(x)
)
exp
(
−tR(x)
)
= exp
(
t(idA +R)(x)
)
(idA +R)(x) exp
(
−tR(x)
)
− exp
(
t(idA +R)(x)
)
R(x) exp
(
−tR(x)
)
= exp
(
−tR̃(x)
)
x exp
(
−tR(x)
)
.
Now, suppose that the statement is true in the case k = n− 1, i.e.,
dn−1
dtn−1
exp
(
−tR̃(x)
)
exp
(
−tR(x)
)
= exp
(
−tR̃(x)
)
x∗n−1 exp
(
−tR(x)
)
.
Post-Lie Magnus Expansion and BCH-Recursion 13
We then get
dn
dtn
exp
(
−tR̃(x)
)
exp
(
−tR(x)
)
=
d
dt
dn−1
dtn−1
exp
(
−tR̃(x)
)
exp
(
−tR(x)
)
=
d
dt
exp
(
−tR̃(x)
)
x∗n−1 exp
(
−tR(x)
)
= exp
(
−tR̃(x)
)
(idA +R)(x)x∗n−1 exp
(
−tR(x)
)
− exp
(
−tR̃(x)
)
x∗n−1R(x) exp
(
−tR(x)
)
= exp
(
−tR̃(x)
)(
xx∗n−1 +R(x)x∗n−1 − x∗n−1R(x)
)
exp
(
−tR(x)
)
= exp
(
−tR̃(x)
)(
xx∗n−1 + x� x∗n−1
)
exp
(
−tR(x)
)
= exp
(
−tR̃(x)
)(
x ∗ x∗n−1
)
exp
(
−tR(x)
)
= exp
(
−tR̃(x)
)
x∗n exp
(
−tR(x)
)
,
which means that (5.2) is true for k = n, and it is true for all n ≥ 0. This ends the proof. ■
Proof of Theorem 5.1. We have that
dn
dtn
exp∗(tx) = x∗n exp∗(tx),
thus,
dn
dtn
∣∣∣∣
t=0
exp∗(tx) =
dn
dtn
∣∣∣∣
t=0
exp
(
−tR̃(x)
)
exp
(
−tR(x)
)
for all n ≥ 0.
One can therefore conclude that both members of (5.1) do coincide as infinite formal series. ■
Theorem 5.3. The post-Lie Magnus expansion χ, described in (4.3), coincides with the weighted
BCH-recursion χλ recursively given by (3.6), with weight λ = 1.
Proof. From equation (3.7), specialized to λ = 1, and by setting θBCH := χ−1
1 , we obtain that
exp
(
−θBCH(tx)
)
= exp
(
tR(x)
)
exp
(
tR̃(x)
)
in Û(A), which is equivalent to
exp
(
θBCH(tx)
)
= exp
(
−tR̃(x)
)
exp
(
−tR(x)
)
. (5.3)
From (4.5), (5.1) and (5.3) we have
exp
(
θBCH(tx)
)
= exp∗(tx) = exp
(
θ(tx)
)
. (5.4)
Then the two θ’s, namely the inverse BCH-recursion in (5.3) and the inverse post-Lie Magnus
expansion (5.4), do coincide. ■
A Calculations on post-Lie Magnus expansion
The first five elements of the post-Lie Magnus expansion are
χ(1)(f) = f,
χ(2)(f) = −1
2
f � f,
χ(3)(f) =
1
12
f � (f � f) +
1
4
(f � f)� f +
1
12
[f � f, f ],
14 M.J.H. Al-Kaabi, K. Ebrahimi-Fard and D. Manchon
χ(4)(f) = − 1
12
(
(f � f)� (f � f) + (f � (f � f))� f + ((f � f)� f)� f
)
+
1
24
(
[f, f � (f � f)] + [f, (f � f)� f ]
)
,
χ(5)(f) = − 1
720
f �
(
f �
(
f � (f � f)
))
+
1
144
(
(f � f)�
(
f � (f � f)
)
− f �
(
((f � f)� f)� f
)
−f �
(
(f � (f � f))� f
)
− f �
(
f � ((f � f)� f)
)
+ 5
(
f � (f � f)
)
� (f � f)
+ 5
(
(f � f)� f
)
� (f � f) + 6
(
(f � f)� (f � f)
)
� f
+ 3
(
(f � (f � f)
)
� f)� f + 3
(
f � (f � (f � f))
)
� f
+ 3
(
f � ((f � f)� f)
)
� f + 3(f � f)�
(
(f � f)� f
)
+ 3
(
((f � f)� f)� f
)
� f
)
+
1
180
[
f, [f, f � (f � f)]− f � (f � (f � f))
]
− 1
120
[f � f, f � (f � f)]− 1
36
[f, (f � f)� (f � f)]− 1
72
[
f, f �
(
(f � f)� f
)
+
(
f � (f � f)
)
� f +
(
(f � f)� f
)
� f
]
− 1
360
[
f � f, [f, f � f ]
]
+
1
720
[
f,
[
f, [f, f � f ]
]]
.
B Computations on the inverse post-Lie Magnus expansion
Here, we calculate the first five inverse post-Lie Magnus elements:
θ(1)(f) = f,
θ(2)(f) =
1
2
f � f,
θ(3)(f) =
1
6
f � (f � f) +
1
12
[f, f � f ],
θ(4)(f) =
1
24
(
f �
(
f � (f � f)
)
+ [f, f � (f � f)]
)
,
θ(5)(f) =
1
120
f �
(
f �
(
f � (f � f)
))
+
1
80
[
f, f �
(
f � (f � f)
)]
+
1
720
([
f, [f, f � (f � f)]
]
−
[
f,
[
f, [f, f � f ]
]])
+
1
120
[f � f, f � (f � f)]− 1
240
[
f � f, [f, f � f ]
]
.
Acknowledgments
The first author was funded by the Iraqi Ministry of Higher Education and Scientific Research.
He would like to thank the Department of Mathematics at the University of Bergen, Norway,
for warm hospitality during a visit in 2021, which was partially supported by the project Pure
Mathematics in Norway, funded by Trond Mohn Foundation and Tromsø Research Founda-
tion. He would also like to thank Mustansiriyah University, College of Science, Mathematics
Department for their support in carrying out this work. The second author is supported by the
Research Council of Norway through project 302831 “Computational Dynamics and Stochas-
tics on Manifolds” (CODYSMA). The third author is supported by Agence Nationale de la
Recherche, projet CARPLO ANR20-CE40-0007. We thank the anonymous referees for their
remarks and suggestions which greatly helped us to improve the presentation of the present
paper.
Post-Lie Magnus Expansion and BCH-Recursion 15
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1 Introduction
2 Post-Lie algebras
2.1 The universal enveloping algebra of a post-Lie algebra
2.2 Free post-Lie algebras
2.3 Post-Lie structure on a Rota–Baxter algebra
3 Baker–Campbell–Hausdorff (BCH)-recursion
4 Magnus expansion
4.1 Post-Lie Magnus expansion
4.2 Inverse post-Lie Magnus expansion
5 Post-Lie Magnus expansion and BCH-recursion
A Calculations on post-Lie Magnus expansion
B Computations on the inverse post-Lie Magnus expansion
References
|
| id | nasplib_isofts_kiev_ua-123456789-211522 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T08:19:59Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Al-Kaabi, Mahdi J. Hasan Ebrahimi-Fard, Kurusch Manchon, Dominique 2026-01-05T12:24:08Z 2022 Post-Lie Magnus Expansion and BCH-Recursion. Mahdi J. Hasan Al-Kaabi, Kurusch Ebrahimi-Fard and Dominique Manchon. SIGMA 18 (2022), 023, 16 pages 1815-0659 2020 Mathematics Subject Classification: 16T05; 16T10; 16T30; 17A30 arXiv:2108.11103 https://nasplib.isofts.kiev.ua/handle/123456789/211522 https://doi.org/10.3842/SIGMA.2022.023 We identify the Baker-Campbell-Hausdorff recursion driven by a weight λ=1 Rota-Baxter operator with the Magnus expansion relative to the post-Lie structure naturally associated with the corresponding Rota-Baxter algebra. Post-Lie Magnus expansion and BCH-recursion are reviewed before the proof of the main result. The first author was funded by the Iraqi Ministry of Higher Education and Scientific Research. He would like to thank the Department of Mathematics at the University of Bergen, Norway, for warm hospitality during a visit in 2021, which was partially supported by the project Pure Mathematics in Norway, funded by the Trond Mohn Foundation and Tromsø Research Foundation. He would also like to thank Mustansiriyah University, College of Science, Mathematics Department, for their support in carrying out this work. The second author is supported by the Research Council of Norway through the project 302831 “Computational Dynamics and Stochastics on Manifolds” (CODYSMA). The third author is supported by Agence Nationale de la Recherche, projet CARPLO ANR20-CE40-0007. We thank the anonymous referees for their remarks and suggestions, which greatly helped us to improve the presentation of the present paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Post-Lie Magnus Expansion and BCH-Recursion Article published earlier |
| spellingShingle | Post-Lie Magnus Expansion and BCH-Recursion Al-Kaabi, Mahdi J. Hasan Ebrahimi-Fard, Kurusch Manchon, Dominique |
| title | Post-Lie Magnus Expansion and BCH-Recursion |
| title_full | Post-Lie Magnus Expansion and BCH-Recursion |
| title_fullStr | Post-Lie Magnus Expansion and BCH-Recursion |
| title_full_unstemmed | Post-Lie Magnus Expansion and BCH-Recursion |
| title_short | Post-Lie Magnus Expansion and BCH-Recursion |
| title_sort | post-lie magnus expansion and bch-recursion |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211522 |
| work_keys_str_mv | AT alkaabimahdijhasan postliemagnusexpansionandbchrecursion AT ebrahimifardkurusch postliemagnusexpansionandbchrecursion AT manchondominique postliemagnusexpansionandbchrecursion |