1 Introduction
The first purpose of this paper is to study the universal enveloping of a post-Lie algebroid by introducing the notion of post-Hopf algebroids. The second purpose is to give applications of post-Hopf algebroids in braiding operators on Hopf algebroids and numerical analysis on manifolds.
1.1 Post-Lie algebras, post-Hopf algebras and post-Lie algebroids
The notion of a post-Lie algebra was introduced by Vallette in his study of Koszul duality of operads in [Reference Vallette49], and can be applied to the study of generalized Lax pairs [Reference Bai, Guo and Ni3] and regularity structures [Reference Bruned and Katsetsiadis10]. Munthe-Kaas and Lundervold found that post-Lie algebras also naturally appear in differential geometry and play important roles in the study of numerical integration on manifolds [Reference Munthe-Kaas and Lundervold41]. As the integration of post-Lie algebras, the notion of post-Lie groups was introduced in [Reference Bai, Guo, Sheng and Tang4], which can be used to construct set-theoretical solutions of the Yang-Baxter equation. See [Reference Al-Kaabi, Ebrahimi-Fard and Manchon1] for good examples of post-groups.
In [Reference Li, Sheng and Tang35], the notion of post-Hopf algebras was introduced as an abstraction from two basic examples, namely the group algebra of a post-group and the universal enveloping algebra of a post-Lie algebra. A novel property is that any cocommutative post-Hopf algebra gives rise to a subadjacent Hopf algebra with a generalized Grossman-Larson product. There is naturally a post-Lie algebra structure on the space of primitive elements, and a post-group structure on the set of group-like elements in an arbitrary post-Hopf algebra. Also, cocommutative post-Hopf algebras are essentially equivalent to cocommutative Hopf braces, and thus naturally provide Yang-Baxter operators [Reference Li, Sheng and Tang35, Reference Zheng, Zhang, Ma and Guo50]. See [Reference Fernández Vilaboa, González Rodríguez and Ramos Pérez17, Reference Sciandra48] for two different kinds of generalizations of post-Hopf algebras along this connection.
The notion of Lie algebroids was introduced by Pradines in 1967, which are generalizations of Lie algebras and tangent bundles. Just as Lie algebras are the infinitesimal objects of Lie groups, Lie algebroids are the infinitesimal objects of Lie groupoids. Lie algebroids and Lie groupoids play important roles in differential geometry, foliation theory, Poisson geometry and mathematical physics. See [Reference Mackenzie38] for general theories about Lie algebroids and Lie groupoids. As the geometric generalization of a post-Lie algebra, the notion of a post-Lie algebroid was also introduced by Munthe-Kaas and Lundervold in their study of numerical integration [Reference Munthe-Kaas and Lundervold41]. See also [Reference Munthe-Kaas, Stern and Verdier42] for further applications.
1.2 Hopf algebroids and pre-Hopf algebroids
Roughly speaking, bialgebroids, resp. Hopf algebroids, are the appropriate generalization of bialgebras, resp. Hopf algebras over a commutative ring (often a field) to algebraic structures over a noncommutative base algebra. Bialgebroids appeared independently at different times and places, in algebra, algebraic topology and Poisson geometry, in various versions of generality. The notion of a bialgebroid in its full generality was given by Takeuchi in the 1970s, under the name
$\times _R$
-bialgebras. Later his ideas were rediscovered independently by J.-H. Lu [Reference Lu36]. Similarly to the relation between Hopf algebras and groups, Hopf algebroids can be viewed as a quantization of groupoids. The term Hopf algebroids was first introduced by Haynes R. Miller in his 1975 doctoral dissertation for the purpose of describing the
$E_2$
-term of the Adams-Novikov spectral sequence, and also with other applications in algebraic topology. In [Reference Maltsiniotis39], Maltsiniotis gave a definition of Hopf algebroids with commutative base algebras and their images under the source and target maps lie in the centers of total algebras. In [Reference Lu36], Lu generalized the notion of Hopf algebroids, in which neither the total algebras nor the base algebras are required to be commutative. After that there appear several other different definitions of Hopf algebroids for various purposes; see [Reference Böhm and Szlachányi7, Reference Schauenburg47, Reference Han and Majid24].
In this paper, we adopt Lu’s definition of Hopf algebroids in [Reference Lu36], fit for our fundamental example: the universal enveloping algebra of a Lie algebroid (Lie-Rinehart algebra). So far the bialgebroid structure on the universal enveloping algebra of a Lie algebroid has been extensively studied with fruitful results; see, for example, [Reference Moerdijk and Mrčun40, Reference Bekaert, Kowalzig and Saracco5]. On the other hand, a natural construction of cocommutative Hopf algebroids from a finite groupoid was given in [Reference Ghobadi20], generalizing the group algebra of a group.
Recently, Bronasco and the first author uncovered in [Reference Bronasco and Laurent8] that the pre-Hopf algebroid and the Grossman-Larson Hopf algebroid are the underlying algebraic structures associated to the laws of composition and substitution of exotic aromatic S-series, and used them to provide the algebraic foundations of stochastic numerical analysis (see also [Reference Laurent and Vilmart34]). They later extended these structures with planar rooted trees and naturally obtained post-Hopf algebras in [Reference Bronasco, Laurent and Huguet9]. This allowed for the creation of high-order intrinsic numerical methods for stochastic differential equations on manifolds.
1.3 Main results and outline of the paper
The purpose of this paper is twofold. On the one hand, we study the universal enveloping algebra of a post-Lie-Rinehart algebra (post-Lie algebroid). On the other hand, we generalize the pre-Hopf algebroid introduced in [Reference Bronasco and Laurent8] to the post-algebraic structure context, and give applications in geometric numerical integration.
For these purposes, first we introduce the notion of post-Hopf algebroids, and show that a post-Hopf algebroid gives rise to a Hopf algebroid, which is called the Grossman-Larson Hopf algebroid. Through actions of post-Hopf algebras, we construct a class of examples, called action post-Hopf algebroids. We show that post-Hopf algebroids give rise to braiding operators on the associated Grossman-Larson Hopf algebroid. Then we introduce the notion of post-Lie-Rinehart algebras as the algebraic analog of post-Lie algebroids, and construct the free object using a magma algebra with a linear map to
$\text {Der}(R)$
, where R is a commutative associative algebra. The universal enveloping algebra of a post-Lie-Rinehart algebra is naturally a post-Hopf algebroid. Applications of this universal enveloping post-Hopf algebroid and the action post-Hopf algebroid in numerical analysis are provided for the study of volume-preserving Lie-group methods.
The article is organized as follows. In Section 2, we introduce the notion of (weak) post-Hopf algebroids and show the existence of a subadjacent Hopf algebroid associated with a post-Hopf algebroid, the so-called Grossman-Larson Hopf algebroid. A useful construction called the action post-Hopf algebroid is provided (Theorem 2.20). Also, a post-Hopf algebroid naturally induces a braiding operator on the Grossman-Larson Hopf algebroid (Theorem 2.24). In Section 3, first we introduce the notion of post-Lie-Rinehart algebras and give the construction of their free objects (Theorem 3.18). Then we show that the universal enveloping algebra of a post-Lie-Rinehart algebra is a (weak) post-Hopf algebroid (Theorem 3.20). In Section 4, we outline applications of post-Lie-Rinehart algebras and post-Hopf algebroids in numerical analysis.
Convention. In this paper, we fix an algebraically closed ground field
${\mathbf {k}}$
of characteristic 0 and a commutative associative unital algebra R. All the objects under discussion, including vector spaces, linear maps, algebras and tensor products, are taken over
${\mathbf {k}}$
by default if without emphasis.
2 Post-Hopf algebroids
In this section, we introduce the notion of post-Hopf algebroids, which are generalizations of post-Hopf algebras. In particular, we give the action post-Hopf algebroids using actions of post-Hopf algebras. We introduce the notion of braiding operators on Hopf algebroids, and show that post-Hopf algebroids give rise to braiding operators on the Grossman-Larson Hopf algebroids.
2.1 Bialgebroids and Hopf algebroids
First we adopt the following definition of Hopf algebroids with commutative base algebras as a special situation in [Reference Lu36]; see also [Reference Kaminker and Tang30, § 2.1].
Definition 2.1. Take the commutative unital algebra R as the base algebra, and H another unital algebra as the total algebra. A bialgebroid structure on H over R consists of the following data:
-
(i) An algebra homomorphism $\iota :R\to H$
as the source and also target map, giving the following canonical left R-module structure on H, $$ \begin{align*} &\lambda:R\otimes H\to H,\quad f\otimes x\mapsto \iota(f)x, \end{align*} $$
and we write $\iota (f)x$
as
$fx$
for short. -
(ii) An R-module map $\Delta :H\to H\otimes _R H,\ x \mapsto x_1\otimes_R x_2$
as the coproduct, satisfying-
(a) $(\Delta \otimes _R\mathrm {id}_H)\Delta = (\mathrm {id}_H\otimes _R\Delta )\Delta $
. -
(b) $\Delta (x)(\iota (f)\otimes 1_H-1_H\otimes \iota (f))=0$
for any
$f\in R$
and
$x\in H$
, so inducing an algebra structure on the image
$\Delta (H)$
. -
(c) $\Delta $
is an algebra homomorphism from H to
$\Delta (H)$
. In particular,
$\Delta (1_H)=1_H \otimes _R 1_H$
.
-
-
(iii) An R-module map $\varepsilon :H\to R$
as the counit, satisfying-
(a) $\varepsilon (1_H)=1_R$
. -
(b) $\ker \varepsilon $
is a left ideal of H, namely
$\varepsilon (x(y-\iota \varepsilon (y)))=0$
for any
$x,y\in H$
. -
(c) $\lambda (\varepsilon \otimes \mathrm {id}_H)\Delta =\mathrm {id}_H$
and
$\lambda (\varepsilon \otimes \mathrm {id}_H)\Delta ^{\mathrm {op}}=\mathrm {id}_H$
, where
$\Delta ^{\mathrm {op}}$
is the opposite of
$\Delta $
.
-
We say that H is cocommutative if
$\Delta =\Delta ^{\mathrm {op}}$
.
A Hopf algebroid is a bialgebroid H over R which admits an algebra antiisomorphism
$S:H\to H$
as the antipode map, satisfying
-
(i) $S\iota =\iota $
, -
(ii) $m_H(S\otimes \mathrm {id}_H)\Delta = \iota \varepsilon S$
, -
(iii) $m_H(\mathrm {id}_H\otimes S)\Delta = \iota \varepsilon $
.
Remark 2.2. It is straightforward to see that if
$\iota (R)$
lies in the center
$Z(H)$
of H, then
$H\otimes _R H$
is an R-algebra, and both the coproduct
$\Delta $
and the counit
$\varepsilon $
are R-algebra homomorphisms. In this situation, we call H an R-bialgebra.
Definition 2.3. A Hopf algebroid H over R is called a Hopf R-algebra, if
$\iota (R)$
lies in the center
$Z(H)$
of H, the antipode map S of H is R-linear and
$\varepsilon S=\varepsilon $
.
Note that all structural maps of a Hopf R-algebra are R-linear.
Example 2.4. Let
$(A,\Delta _A,\varepsilon _A,S_A)$
be a Hopf algebra. The tensor algebra
$H=R\otimes A$
has a natural Hopf R-algebra structure
$(H,\Delta _H,\varepsilon _H,S_H,\iota )$
, where
for any
$fa\in R\otimes A$
with
$f\in R$
,
$a\in A$
, and where we use the Sweedler notation. We call
$H=R\otimes A$
the tensor product Hopf R-algebra.
Lemma 2.5. If H is a cocommutative Hopf algebroid over R such that
$\iota (R)\subseteq Z(H)$
, then H is a Hopf R-algebra.
Proof. We only need to show that S is R-linear and
$\varepsilon S=\varepsilon $
. First for any
$f\in R$
,
$x\in H$
, we have
which implies that S is R-linear. Also, since the counit
$\varepsilon $
becomes an algebra homomorphism, we have
which implies that
$\varepsilon S=\varepsilon $
. Hence, H is a Hopf R-algebra by definition.
Definition 2.6. Given a bialgebroid H over R, an R-algebra A is called a module algebra over H via an action
$\rightharpoonup :H\otimes A\to A$
, if
for any
$f\in R$
,
$x\in H$
and
$a,b\in A$
. When
$R={\mathbf {k}}$
, it recovers the notion of a module algebra over a bialgebra.
For later use, we need the following construction of Hopf algebroids from smash product algebras. Let
$(A,\Delta _A,\varepsilon _A,S_A)$
be a cocommutative Hopf algebra such that R is an A-module algebra via an action
$\rightharpoonup $
. Then there is an algebra structure on
$R\otimes A$
given by
for all
$fa,gb\in R\otimes A$
. This algebra is called the smash product algebra and denoted by
$R\# A$
.
Theorem 2.7. With the above notations, the smash product algebra
$H=R\# A$
has a natural Hopf algebroid structure
$(H,\Delta _H,\varepsilon _H,S_H,\iota )$
over R, where the comultiplication
$\Delta _H$
, the counit
$\varepsilon _H$
and the antipode
$S_H$
are defined as follows:
for any
$f,g\in R$
and
$a,b\in A$
, and
$\iota :R\to H$
is the canonical inclusion of algebras. We call H an action Hopf algebroid over R.
Proof. According to [Reference Brzeziński and Militaru11, Theorem 4.1], given a braided commutative algebra A in the Yetter-Drinfeld module category
${_H}\mathscr Y\!\mathscr D^H$
over a Hopf algebra H, the smash product algebra
$A\# H$
naturally possesses a Hopf algebroid structure over A in the sense of Lu. Such a result is a generalization of [Reference Lu36, Theorem 5.1].
Now under our notations, we take the commutative algebra R as a module algebra over a given cocommutative Hopf algebra A and choose the trivial coaction of A on R, then R is clearly a braided commutative algebra in
${_A}\mathscr Y\!\mathscr D^A$
, and so [Reference Brzeziński and Militaru11, Theorem 4.1] can be applied to give the desired statement. Note that in this special situation, both the source map and the target map are the canonical inclusion, and we just denote it by
$\iota $
as previously.
2.2 Post-Hopf algebroids
In order to introduce the key notion of post-Hopf algebroids, we first recall post-Hopf algebras introduced in [Reference Li, Sheng and Tang35]. The group algebra of a post-group and the universal enveloping algebra of a post-Lie algebra are two fundamental examples.
Definition 2.8 [Reference Li, Sheng and Tang35, Definition 2.1]
A post-Hopf algebra is a pair
$(H,\rhd )$
, where H is a Hopf algebra and
$\rhd :H\otimes H\to H$
is a coalgebra homomorphism the following equalities:
for any
$x,y,z\in H$
, and the left multiplication
$\alpha _\rhd :H\to \mathrm {End}(H)$
defined by
$\alpha _{\rhd , x} y= x\rhd y$
for all
$x,y\in H,$
is convolution invertible in
$\mathrm {Hom}(H,\mathrm {End}(H))$
. Namely, there exists a unique
$\beta _\rhd :H\to \mathrm {End}(H)$
such that
Moreover, we have the following properties.
Lemma 2.9 [Reference Li, Sheng and Tang35, Lemma 2.4]
Let
$(H,\rhd )$
be a post-Hopf algebra. For any
$x,y\in H$
, we have
Theorem 2.10 [Reference Li, Sheng and Tang35, Theorem 2.5]
Let
$(H,\rhd )$
be a cocommutative post-Hopf algebra. Then
is a Hopf algebra, which is called the subadjacent Hopf algebra, where for all
$x,y\in H$
,
and
$*_\rhd $
is called the Grossman-Larson product.
Furthermore, H is a left
$H_\rhd $
-module bialgebra via the action
$\rhd $
, so
$\beta _{\rhd ,x}=\alpha _{\rhd ,S_\rhd (x)}$
.
Now we are in a position to define post-Hopf algebroids generalizing cocommutative post-Hopf algebras. First we introduce a weakened version, whose counterpart in the original Hopf algebra case was given by getting rid of the restrictive condition that the left multiplication
$\alpha _\rhd $
is convolution invertible.
Definition 2.11. A weak post-Hopf algebroid over R consists of a cocommutative Hopf R-algebra
$(H,\iota )$
, and a
${\mathbf {k}}$
-linear map
$\rhd :H\otimes H\to H$
such that
for any
$f\in R$
and
$x,y,z\in H$
.
Lemma 2.12. Let
$(H,\iota ,\rhd )$
be a weak post-Hopf algebroid over R. For any
$f\in R$
and
$x\in H$
, we have
Proof. First we see that
Namely, the image of
$\iota $
is stable under the left multiplication of
$\rhd $
. Also, we have
which finishes the proof.
Theorem 2.13. Given a weak post-Hopf algebroid
$(H,\iota ,\rhd )$
over R, there exists an associative product
$*_\rhd $
, called the Grossman-Larson product, on H defined by
such that the tuple
$H_\rhd =(H,*_\rhd ,\Delta ,\varepsilon ,\iota )$
forms a bialgebroid over R, which is called the Grossman-Larson bialgebroid over R.
Moreover, H is a module algebra over
$H_\rhd $
with respect to the product
$\rhd $
in the sense of Definition 2.6.
Proof. For any
$f\in R$
and
$x,y\in H$
, since
$\Delta $
is R-linear and
$\iota (R)\subseteq Z(H)$
, we have
which implies that the product
$*_\rhd $
is well-defined.
By Eqs. (15) and (16), it is clear that
$1_H*_\rhd x =x*_\rhd 1_H=x$
. Also, Eq. (14) and the cocommutativity of H imply that
Hence, we have
Thus,
$(H, *_\rhd )$
is an associative algebra also with the unit
$1_H$
.
By Eq. (16),
$\iota $
is an algebra homomorphism from R to the associative algebra
$(H,*_\rhd )$
:
Thus
$H_\rhd $
has the same R-module structure as H:
Also, we have
Combined with (24), we see that (ii) in Definition 2.1 holds for the tuple
$(H,*_\rhd ,\Delta ,\iota )$
.
Finally, we have
$\iota (\varepsilon (x_1))*_\rhd x_2 \stackrel{({23})}{=} \iota (\varepsilon (x_1))(1\rhd x_2) \stackrel {{({16})}}{=} \iota (\varepsilon (x_1))x_2=x$
, and
which imply that (iii) in Definition 2.1 holds. Hence, we have confirmed the desired bialgebroid structure
$H_\rhd $
over R.
It is straightforward to see that H is clearly a module algebra over
$H_\rhd $
with respect to the product
$\rhd $
by Definition 2.6.
Corollary 2.14. For a weak post-Hopf algebroid
$(H,\iota ,\rhd )$
over R, there is a left
$H_\rhd $
-module algebra action on R given by:
Proof. By Theorem 2.13 and Eq. (21),
$\iota (R)$
is a module subalgebra of H over
$H_\rhd $
. Since
$\varepsilon \iota =\mathrm {id}_R$
, there is an induced module algebra structure on R given as desired.
Remark 2.15. The
$H_\rhd $
-module algebra action on R given in Corollary 2.14 is consistent with the one given by Lu in [Reference Lu36, Definition 2.3] for any bialgebroid over R. Thus, for the Grossman-Larson bialgebroid
$H_\rhd $
, the
$H_\rhd $
-module algebra action on H given in Theorem 2.13 can be viewed as an expansion of Lu’s result.
Definition 2.16. A weak post-Hopf algebroid
$(H,\iota ,\rhd )$
over R is called a post-Hopf algebroid, if there exists an algebra antiautomorphism
$\theta $
of the Grossman-Larson bialgebroid
$H_\rhd $
such that
for any
$f\in R$
and
$x\in H$
.
Remark 2.17. The uniqueness of such a linear map
$\theta $
is an interesting problem. In particular, when
$R={\mathbf {k}}$
, a post-Hopf algebroid over R reduces to a cocommutative post-Hopf algebra, and in this case
$\theta =S_\rhd $
is unique, where
$S_\rhd $
is given by (13).
Lemma 2.18. Let
$(H,\iota ,\rhd ,\theta )$
be a post-Hopf algebroid over R. For any
$f\in R$
and
$x, y\in H$
, we have
Proof. First according to Eq. (27), we have
as
$\Delta (\iota (f))=\iota (f)\otimes _R 1_H=1_H\otimes _R\iota (f)$
. Then
Next since H is a Hopf R-algebra, we know that the antipode S is R-linear and
$\varepsilon S=\varepsilon $
, so
and then
Also, we have
Then taking
$y=1_H$
in Eq. (33) and applying Eq. (15), we see that
$x_1*_\rhd \iota (\varepsilon (\theta (x_2)))=x$
. Hence, we have shown all the desired equalities.
Corollary 2.19. Given a post-Hopf algebroid
$(H,\iota ,\rhd ,\theta )$
over R, the Grossman-Larson bialgebroid
$H_\rhd =(H,*_\rhd ,\Delta ,\varepsilon ,\iota )$
equipped with
$\theta $
forms a Hopf algebroid over R, which we call the Grossman-Larson Hopf algebroid over R.
Proof. Now it is clear that
$\theta \iota =\iota $
. So
$H_\rhd $
is a Hopf algebroid over R by Eqs. (27) and (28).
Theorem 2.20. Let
$(A,\rhd )$
be a cocommutative post-Hopf algebra and R be an
$A_\rhd $
-module algebra via an action
$\rightharpoonup $
, where
$A_\rhd =(A,*_\rhd ,1_A,\Delta ,\varepsilon ,S_\rhd )$
is the subadjacent Hopf algebra of
$(A,\rhd )$
. Then the tensor product Hopf R-algebra
$H=R\otimes A$
given in Example 2.4 has a natural post-Hopf algebroid structure
$(H,\Delta _H,\varepsilon _H,S_H,\iota ,\overline {\rhd },\theta )$
over R, where
for any
$f,g\in R$
and
$a,b\in A$
. We call
$(H,\iota ,\overline {\rhd },\theta )$
an action post-Hopf algebroid over R. In particular, the corresponding Grossman-Larson Hopf algebroid
$H_{\overline {\rhd }}=(H,*_{\overline {\rhd }},\Delta _H,\varepsilon _H,\theta ,\iota )$
is the action Hopf algebroid
$R\# A_\rhd $
defined as in Theorem 2.7.
Proof. First Eqs. (14)–(16) and (18) are easy to see. For any
$f,g,h\in R$
and
$a,b,c\in A$
, we have
which implies that Eq. (17) holds.
Eq. (19) follows from
Eq. (20) follows from
Hence, the tuple
$(H,\iota ,\overline {\rhd })$
is a weak post-Hopf algebroid over R.
Now we compute the Grossman-Larson product
$*_{\overline {\rhd }}$
on H as follows.
Next we check that the linear operator
$\theta $
defined by (36) is an algebra antiinvolution of
$H_{\overline {\rhd }}$
.
Meanwhile, we have
Therefore,
$\theta $
also satisfies all conditions (26)–(28). The tensor product Hopf R-algebra
$H=R\otimes A$
has the desired post-Hopf algebroid structure over R.
2.3 Braiding operators on Hopf algebroids
Guccione, Guccione and Vendramin introduced the notion of a braiding operator on a cocommutative Hopf algebra in [Reference Guccione, Guccione and Vendramin21, Definition 5.9]. Here we further generalize this notion to Hopf algebroids. We interpret any Hopf algebroid H over R as an R-bimodule via the source map
$\iota :R\to H$
, and we use the notation
$\otimes _R^b$
to denote the tensor product of R-bimodules, different from the former tensor product
$\otimes _R$
of left R-modules.
Definition 2.21. Given a Hopf algebroid H over R, an R-bimodule homomorphism
$r:H\otimes _R^b H\to H\otimes _R^b H$
is called a braiding operator on H if it satisfies
-
(i) $(\mathrm {id}_H\otimes \tau \otimes \mathrm {id}_H)(\Delta \otimes \Delta )r = (r\otimes r)(\mathrm {id}_H\otimes \tau \otimes \mathrm {id}_H)(\Delta \otimes \Delta )$
as R-bimodule homomorphisms from
$H\otimes _R^b H$
to
$(H\otimes _R^b H)\otimes _R (H\otimes _R^b H)$
, -
(ii) $mr=m$
, -
(iii) $r(m\otimes _R^b\mathrm {id}_H)=(\mathrm {id}_H\otimes _R^b m)(r\otimes _R^b \mathrm {id}_H)(\mathrm {id}_H\otimes _R^b r)$
, -
(iv) $r(\mathrm {id}_H\otimes _R^b m)=(m\otimes _R^b\mathrm {id}_H)(\mathrm {id}_H\otimes _R^b r)(r\otimes _R^b \mathrm {id}_H)$
, -
(v) $r(1_H\otimes _R^b x)=x\otimes _R^b 1_H,\ \forall x\in H$
, -
(vi) $r(x\otimes _R^b 1_H)=1_H\otimes _R^b x,\ \forall x\in H$
,
where
$\tau $
is the flip map and m is the multiplication of H, and we can take the tensor product
$\otimes _R$
since the R-bimodule
$H\otimes _R^b H$
is a left R-module.
Lemma 2.22. Given a braiding operator r on a Hopf algebroid H over R, we have
Proof. Indeed, we have
for any
$x,y\in H$
.
Remark 2.23. When
$R={\mathbf {k}}$
, the counit
$\varepsilon :H\to {\mathbf {k}}$
becomes an algebra map, and condition (i) and Eq. (37) imply that r is a coalgebra homomorphism, and thus a braiding operator on a Hopf algebra in the sense of [Reference Guccione, Guccione and Vendramin21].
Theorem 2.24. Let
$(H,\iota ,\rhd ,\theta )$
be a post-Hopf algebroid over R. There exists an R-bimodule homomorphism
$r=r_{H_\rhd }:H_\rhd \otimes _R^b H_\rhd \to H_\rhd \otimes _R^b H_\rhd $
defined by
which is a braiding operator on the Grossman-Larson Hopf algebroid
$H_\rhd $
over R.
Proof. First, note that the map r is well-defined, since the RHS of Eq. (38) is independent of any expression of the coproduct
$\Delta $
according to Eqs. (29) and (33) when x or y is of the form
$\iota (f)*_\rhd z$
or
$z*_\rhd \iota (f)$
. Moreover, r is an R-bimodule homomorphism. That is,
We then check that r is a braiding operator on
$H_\rhd $
. First note that both sides of the equality in condition (i) are well-defined R-bimodule homomorphisms, and then such an equality is due to Eqs. (14) and (26). Condition (ii) is clear by Eqs. (38), (14) and (27). Conditions (iii) and (iv) follow from
Conditions (v) and (vi) follow from
All the braiding conditions have been checked. The proof is finished.
3 The universal enveloping algebras of post-Lie-Rinehart algebras
In this section, first we recall the universal enveloping algebra of a Lie-Rinehart algebra. Then we introduce the notion of post-Lie-Rinehart algebras as the algebraic analog of post-Lie algebroids. We construct the free post-Lie-Rinehart algebra generated by a magma algebra with a fixed linear map to
$\mathrm {Der}(R)$
, which turns out to be an action post-Lie-Rinehart algebra. Finally we show that the universal enveloping algebra of a post-Lie-Rinehart algebra is a weak post-Hopf algebroid. In the particular case of action post-Lie-Rinehart algebras, the universal enveloping algebras are post-Hopf algebroids.
3.1 Lie-Rinehart algebras and their universal enveloping algebras
The structure of a Lie-Rinehart algebra appeared in Rinehart’s work [Reference Rinehart46] under the terminology of
$(R,A)$
-Lie algebra. Then, Huebschmann referred to such a structure as a Lie-Rinehart algebra [Reference Huebschmann25, Reference Huebschmann26]. Lie-Rinehart algebras are the algebraic counterpart of Lie algebroids, and have various important applications in Poisson geometry, symplectic geometry and quantization.
Definition 3.1. [Reference Rinehart46] Let L be an R-module. A tuple
$\mathscr {R}=(R,L,[-,-]_L,\rho )$
is called a Lie-Rinehart algebra, if
-
(i) L is a Lie algebra with its bracket $[-,-]_L$
. -
(ii) $\rho :L\to \mathrm {Der}(R)$
is an R-module homomorphism and a Lie algebra homomorphism, called the anchor. -
(iii) the following compatibility condition holds:
(39) $$ \begin{align} [x,fy]_L &= f[x,y]_L+\rho(x)(f)y,\quad\forall f\in R,\,x,y\in L. \end{align} $$
For convenience, we will write
$\rho (x)(f)$
as
$x(f)$
for short.
Example 3.2. Recall that a Lie algebroid structure on a vector bundle
$A\longrightarrow {\mathcal {M}} $
is a pair consisting of a Lie algebra structure
$[-,-]_A$
on the section space
$\Gamma (A)$
and a vector bundle morphism
$a_A:A\longrightarrow T{\mathcal {M}} $
from A to the tangent bundle
$T{\mathcal {M}} $
, called the anchor, satisfying
where
$\mathcal {C}^\infty ({\mathcal {M}} )$
is the commutative associative algebra of smooth functions on the manifold
${\mathcal {M}} $
. Let
$(A\longrightarrow {\mathcal {M}} ,[-,-]_A,a_A)$
be a Lie algebroid. Then the quadruple
$(\mathcal {C}^\infty ({\mathcal {M}} ),\Gamma (A),[-,-]_A,a_A)$
is a Lie-Rinehart algebra.
Example 3.3. For the Lie algebra
$\mathrm {Der}(R)$
of derivations on R, the triple
$(R,\mathrm {Der}(R),\mathrm {id})$
is a Lie-Rinehart algebra.
Example 3.4. Let
$\mathfrak {g}$
be a Lie algebra with a Lie algebra homomorphism
$\rho :\mathfrak {g}\to \mathrm {Der}(R)$
. Let
$L=R\otimes \mathfrak {g}$
as a free R-module. Define
and extend
$\rho $
to be an R-module homomorphism
$\overline {\rho }:L\to \mathrm {Der}(R)$
. Then the quadruple
$(R,L,[-,-]_L,\overline {\rho })$
is a Lie-Rinehart algebra, and we call it an action Lie-Rinehart algebra.
Definition 3.5. A Lie algebra L is called a Lie R-algebra if L is an R-module and its Lie bracket
$[-,-]_L$
is R-linear. In other words, a Lie R-algebra is a Lie-Rinehart algebra with trivial anchor.
Note that the authors in [Reference Moerdijk and Mrčun40] defined the universal enveloping algebra of a Lie algebroid
$(A\longrightarrow M,[-,-]_A,a_A)$
as the universal enveloping algebra of its underlying Lie-Rinehart algebra
$(\mathcal {C}^\infty ({\mathcal {M}} ),\Gamma (A),[-,-]_A,a_A)$
. Now we recall the definition of the universal enveloping algebra of a Lie-Rinehart algebra.
Definition 3.6 [Reference Moerdijk and Mrčun40]
For a Lie-Rinehart algebra
$\mathscr {R}=(R,L,[-,-]_L,\rho )$
, the universal enveloping algebra of
$\mathscr {R}$
is a triple
$(\mathscr U_R(L),\iota _R,\iota _L)$
, where
$\mathscr U_R(L)$
is a unital algebra,
$\iota _R:R\to \mathscr U_R(L)$
is a homomorphism of unital algebras and
$\iota _L:L\to \mathscr U_R(L)$
is a homomorphism of Lie algebras such that
and satisfying the following universal property: if A is a unital algebra,
$\kappa _R:R\to A$
is a homomorphism of unital algebras and
$\kappa _L:L\to A$
is a homomorphism of Lie algebras such that
then there exists a unique homomorphism of unital algebras
$\bar \kappa :\mathscr U_R(L)\to A$
such that
$\bar \kappa \iota _R=\kappa _R$
and
$\bar \kappa \iota _L=\kappa _L$
.
We have the following construction of the universal enveloping algebra of a Lie-Rinehart algebra.
Proposition 3.7 [Reference El Kaoutit and Saracco16, § 3.2]
For a Lie-Rinehart algebra
$\mathscr {R}=(R,L,[-,-]_L,\rho )$
, let
$T_R(L\otimes R)=R \oplus \bigoplus _{n\geq 1}(L\otimes R)^{\otimes _R^b n}$
be the tensor R-algebra over the R-bimodule
$L\otimes R$
, and let
$j_R:R\to T_R(L\otimes R)$
and
$j_L:L\to T_R(L\otimes R)$
be the canonical embeddings, that is,
Define
and
$j_R$
(resp.
$j_L$
) induces the map
$\iota _R:R\to \mathscr U_R(L)$
(resp.
$\iota _L:L\to \mathscr U_R(L)$
). Then
$(\mathscr U_R(L),\iota _R,\iota _L)$
is the universal enveloping algebra of
$\mathscr {R}$
.
Remark 3.8. The structural map
$\iota _R:R\to \mathscr U_R(L)$
is injective, while
$\iota _L:L\to \mathscr U_R(L)$
is also injective if L is R-projective, by the PBW theorem proven in [Reference Rinehart46, Theorem 3.1].
For the universal enveloping algebra
$\mathscr U_R(L)$
of a Lie-Rinehart algebra
$\mathscr {R}=(R,L,[-,-]_L,\rho )$
, using its left R-module structure to define the tensor product
$\otimes _R$
, then we have the comultiplication as a left R-module map and a
${\mathbf {k}}$
-algebra homomorphism
where
$\mathscr U_R(L)\,\bar {\otimes }_R\, \mathscr U_R(L) $
is the kernel of the map
The counit
$\varepsilon : \mathscr U_R(L) \to R$
is the left R-module map such that
Then
$(\mathscr U_R(L),\cdot ,\Delta ,\varepsilon ,\iota _R)$
is a bialgebroid over R, also called an
$R/{\mathbf {k}}$
-bialgebra in [Reference Moerdijk and Mrčun40] or a left bialgebroid over R in [Reference Bekaert, Kowalzig and Saracco5].
Proposition 3.9. Let L be a Lie R-algebra with Lie bracket
$[-,-]_L$
. Then the universal enveloping algebra
$\mathscr U_R(L)$
of L is a Hopf R-algebra.
Proof. Note that a Lie R-algebra is a Lie-Rinehart algebra with trivial anchor. So
$\iota _R(R)$
lies in the center of
$\mathscr U_R(L)$
according to Eq. (41). Let
$T_R(L)=R \oplus \bigoplus _{n\geq 1}L^{\otimes _R n}$
be the tensor R-algebra over the R-module L and let
$j_L:L\to T_R(L)$
be the canonical embedding. We have the following R-algebra isomorphism
Hence, we can define the antipode map
as an R-algebra antiisomorphism, and
$\mathscr U_R(L)$
becomes a Hopf R-algebra.
As a special example of action Hopf algebroids defined in Theorem 2.7, we have
Example 3.10. Let
$\mathscr {R}$
be the action Lie-Rinehart algebra
$(R,L=R\otimes \mathfrak {g},\rho )$
given in Example 3.4. Since
$\rho $
naturally defines a module algebra action
$\rightharpoonup $
of
$\mathscr U(\mathfrak {g})$
over R, the universal enveloping algebra
$(\mathscr U_R(L),\iota _R,\iota _L)$
of
$\mathscr {R}$
is a cocommutative Hopf algebroid over R isomorphic to the action Hopf algebroid
$R\# \mathscr U(\mathfrak {g})$
over R.
3.2 Post-Lie-Rinehart algebras and free objects
In this subsection, we introduce the notion of a post-Lie-Rinehart algebra as the algebraic counterpart of a post-Lie algebroid introduced in [Reference Munthe-Kaas and Lundervold41], and study its free object.
The notion of a post-Lie algebra was introduced by Vallette in his study of Koszul duality of operads in [Reference Vallette49].
Definition 3.11. [Reference Vallette49] A post-Lie algebra
$(L,[-,-]_L,\rhd )$
consists of a Lie algebra
$(L,[-,-]_L)$
and a binary product
$\rhd :L\otimes L\to L$
such that
A post-Lie algebra homomorphism between two post-Lie algebras is a Lie algebra homomorphism compatible with post-Lie products.
Any post-Lie algebra
$(L,[-,-]_L,\rhd )$
has a subadjacent Lie algebra
$L_\rhd :=(L,[-,-]_{ \rhd })$
defined by
The notion of a post-Lie algebroid was introduced by Munthe-Kaas and Lundervold in their study of numerical integration [Reference Munthe-Kaas and Lundervold41]. Now we introduce the notion of a post-Lie-Rinehart algebra. The relationship between post-Lie-Rinehart algebras and post-Lie algebroids is similar to the relationship between Lie-Rinehart algebras and Lie algebroids.
Definition 3.12. Let L be a Lie R-algebra with Lie bracket
$[-,-]_L$
and
$\rhd :L\otimes L\to L$
be a bilinear map. A tuple
$\mathscr {R}=(R,L,[-,-]_L,\rho ,\rhd )$
is called a post-Lie-Rinehart algebra, if
-
(i) $(L,[-,-]_L,\rhd )$
is a post-Lie algebra, -
(ii) $\rho :L_\rhd \to \mathrm {Der}(R)$
is an R-module homomorphism and a Lie algebra homomorphism for the subadjacent Lie algebra
$L_\rhd $
of
$(L,[-,-]_L,\rhd )$
, -
(iii) the following compatibility conditions hold:
(44) $$ \begin{align} fx\rhd y &= f(x\rhd y), \end{align} $$
(45) $$ \begin{align}x\rhd fy &= f(x\rhd y)+x(f)y \end{align} $$
for any $f\in R,\,x,y\in L$
, where
$x(f)=\rho (x)(f)$
as before.
A post-Lie-Rinehart algebra homomorphism between two post-Lie-Rinehart algebras is an R-module homomorphism and a post-Lie algebra homomorphism compatible with anchors.
Proposition 3.13. Given a post-Lie-Rinehart algebra
$\mathscr {R}=(R,L,[-,-]_L,\rho ,\rhd )$
, the tuple
$\mathscr {R}_\rhd =(R,L,[-,-]_{\rhd },\rho )$
is a Lie-Rinehart algebra, which is called the subadjacent Lie-Rinehart algebra of
$\mathscr {R}$
.
Proof. By definition, we only need to check the compatibility condition (39) as follows:
for any
$f\in R$
and
$x,y\in L$
.
Example 3.14. Recall from [Reference Munthe-Kaas and Lundervold41] that a post-Lie algebroid structure on a vector bundle
$A\longrightarrow {\mathcal {M}} $
is a triple that consists of a
$\mathcal {C}^\infty ({\mathcal {M}} )$
-linear Lie algebra structure
$[-,-]_A$
on
$\Gamma (A)$
, a bilinear operation
$\triangleright _A:\Gamma (A)\otimes \Gamma (A)\longrightarrow \Gamma (A)$
and a vector bundle morphism
$a_A:A\longrightarrow T{\mathcal {M}} $
, called the anchor, such that
$(\Gamma (A),[-,-]_A,\triangleright _A)$
is a post-Lie algebra, and for all
$f\in \mathcal {C}^\infty ({\mathcal {M}} )$
and
$u,v\in \Gamma (A)$
, the following relations are satisfied:
-
(i) $u\triangleright _A(fv)=f(u\triangleright _A v)+a_A(u)(f)v,$
-
(ii) $(fu)\triangleright _A v=f(u\triangleright _A v).$
It is straightforward to check that
$(\mathcal {C}^\infty ({\mathcal {M}} ),\Gamma (A),[-,-]_A,a_A,\triangleright _A)$
is a post-Lie-Rinehart algebra.
As a generalization of [Reference Li, Sheng and Tang35, Theorem 2.7], there is a post-Lie-Rinehart algebra structure on the space of primitive elements of a weak post-Hopf algebroid.
Given a weak post-Hopf algebroid
$(H,\iota ,\rhd )$
over R, let
Since H is a Hopf R-algebra, it is straightforward to see that
$(P(H),[-,-])$
is a Lie R-algebra with the commutator as its Lie bracket. By the same argument as [Reference Li, Sheng and Tang35, Theorem 2.7], the product
$\rhd $
can be restricted to
$P(H)$
such that
$(P(H),[-,-],\rhd )$
is a post-Lie algebra. Moreover, we have the following result.
Proposition 3.15. Let
$(H,\iota ,\rhd )$
be a weak post-Hopf algebroid over R. Then
$(R,P(H),[-,-],\rho ,\rhd )$
is a post-Lie-Rinehart algebra, where the anchor
$\rho $
is given by Eq. (25), that is,
Proof. First note
$\rho (x)\in \mathrm {Der}(R)$
for any
$x\in P(H)$
by conditions (16) and (19). It remains to show that
$\rho $
is an R-module homomorphism and a Lie algebra homomorphism, and Eqs. (44) and (45) hold. The fact that
$\rho $
is an R-module homomorphism follows from
For all
$f\in R$
and
$x,y\in P(H)$
, we have
which implies that
$\rho $
is a Lie algebra homomorphism.
On the other hand, Eq. (18) clearly implies Eq. (44). Also, we have
Namely, Eq. (45) holds. Hence,
$(R,P(H),[-,-],\rho ,\rhd )$
is a post-Lie-Rinehart algebra.
Next we introduce a useful construction of post-Lie-Rinehart algebras, originally given in [Reference Guo, Lang and Sheng22, Prop. 4.23] via a geometric terminology, namely action post-Lie algebroids.
Proposition 3.16. Let
$(\mathfrak {g},[-,-]_{\mathfrak {g}},\rhd )$
be a post-Lie algebra with its subadjacent Lie algebra
$\mathfrak {g}_\rhd $
. Let
$\rho :\mathfrak {g}_\rhd \to \mathrm { Der}(R)$
be a Lie algebra homomorphism. Then we have an extended post-Lie algebra structure on
$L=R\otimes \mathfrak {g}$
defined as follows,
for any
$f_1,f_2\in R$
and
$x_1,x_2\in \mathfrak {g}$
, where we write
$\rho (x)(f)$
simply as
$x(f)$
.
Extend
$\rho :\mathfrak {g}\to \mathrm {Der}(R)$
to be a linear map
$\overline {\rho }:L\to \mathrm {Der}(R)$
defined by
Then, the tuple
$(R,L,[-,-]_L,\overline {\rho },\overline {\rhd })$
is a post-Lie-Rinehart algebra, and we call it an action post-Lie-Rinehart algebra.
The corresponding subadjacent Lie-Rinehart algebra
$L_{\overline {\rhd }}:= (R,L,[-,-]_{\overline {\rhd }},\overline {\rho })$
is given by
Remark 3.17. The special case of Proposition 3.16 for a Lie algebra
$\mathfrak {g}$
with zero post-Lie product generalizes the post-Lie algebras of derivations in [Reference Jacques and Zambotti29, Theorem 3.1] and the corresponding post-Lie-Rinehart algebras in [Reference Jacques and Zambotti29, Remark 3.10].
At the end of this subsection, we construct the free post-Lie-Rinehart algebra generated by a magma algebra with a fixed linear map to
$\mathrm {Der}(R)$
.
Let
$(V,\curvearrowright )$
be a magma algebra. It is well-known that the free post-Lie algebra
$\mathrm {PostLie}(V)$
generated by
$(V,\curvearrowright )$
is the free Lie algebra
$(\mathrm {Lie}(V),[-,-]_{\mathrm {Lie}(V)})$
equipped with the extended post-Lie product
$\rhd _V$
from
$\curvearrowright $
[Reference Foissy19]. Namely, for any post-Lie algebra
$(\mathfrak {g},[-,-]_{\mathfrak {g}},\rhd )$
with a magma algebra homomorphism
$\phi :(V,\curvearrowright )\to (\mathfrak {g},\rhd )$
, there exists a unique post-Lie algebra homomorphism
$ \tilde {\phi }:(\mathrm {Lie}(V),[-,-]_{\mathrm { Lie}(V)},\rhd _V)\to (\mathfrak {g},[-,-]_{\mathfrak {g}},\rhd )$
such that
$\phi =\tilde \phi i_V$
, where
$i_V:V\to \mathrm {Lie}(V)$
is the natural inclusion.
In order to define a free post-Lie-Rinehart algebra generated by
$(V,\curvearrowright )$
, a linear map
$f_V:V\to \mathrm {Der}(R)$
needs to be given first. Now we extend it to be a Lie algebra homomorphism
$\rho _V:\mathrm {PostLie}(V)_{\rhd _V}\to \mathrm {Der}(R)$
by the following recursion,
for any
$x,y\in V$
and
$X\in \mathrm {Lie}(V)$
.
By Proposition 3.16, we have the action post-Lie-Rinehart algebra
Theorem 3.18. The action post-Lie-Rinehart algebra
$\mathrm {PostLR}(V)$
has the following universal property:
For any post-Lie-Rinehart algebra
$\mathscr {R}=(R,L,[-,-]_L,\rho ,\rhd )$
, and a magma algebra homomorphism
$\varphi :(V,\curvearrowright )\to (L,\rhd )$
such that
$\rho \varphi =f_V$
, there exists a unique post-Lie-Rinehart algebra homomorphism
$\tilde \varphi :\mathrm {PostLR}(V)\to L$
such that
$\varphi =\tilde \varphi (1_R\otimes i_V)$
, where
$i_V:V \to \mathrm {Lie}(V)$
is the natural embedding. That is, we have the following commutative diagram:
Thus, the post-Lie-Rinehart algebra
$\mathrm {PostLR}(V)$
is the free post-Lie-Rinehart algebra generated by
$(V,\curvearrowright ,f_V)$
.
Proof. First by the universal property of
$\mathrm {PostLie}(V)$
, there exists a unique post-Lie algebra homomorphism
$\psi :\mathrm {PostLie}(V)\to (L,[-,-]_L,\rhd )$
such that
$\varphi =\psi i_V$
. It induces a Lie algebra homomorphism
$\rho \psi :\mathrm {PostLie}(V)_{\rhd _V}\to \mathrm {Der}(R)$
satisfying
$\rho \psi i_V=\rho \varphi =f_V,$
so we have
$\rho \psi =\rho _V$
, since
$\mathrm {Lie}(V)$
is generated by V.
Now define a linear map
$\tilde \varphi :R\otimes \mathrm {Lie}(V)\to L$
by
It is clear that
$\tilde \varphi $
is R-linear and
$\tilde \varphi (1_R\otimes i_V)=\psi i_V=\varphi $
. Also, we have
which implies that
$\tilde \varphi $
is a post-Lie algebra homomorphism. Moreover, we have
which implies that
$\tilde \varphi :\mathrm {PostLR}(V)\to L$
is a post-Lie-Rinehart algebra homomorphism. Meanwhile, a post-Lie-Rinehart algebra homomorphism from
$\mathrm {PostLR}(V)$
to L is determined by its restriction on V, so the desired homomorphism
$\tilde \varphi $
is unique.
Example 3.19. The free post-Lie algebra on one generator is realized as
$\mathrm {PostLie}({\mathbf {k}}{\mathcal {OT}})$
, where
${\mathbf {k}}{\mathcal {OT}}$
is the magma algebra of planar rooted trees with the left grafting operation
$\curvearrowright $
[Reference Munthe-Kaas and Lundervold41].
According to Theorem 3.18, given any linear map
$f_{{\mathbf {k}}{\mathcal {OT}}}:{\mathbf {k}}{\mathcal {OT}}\to \mathrm {Der}(R)$
, the action post-Lie-Rinehart algebra
$\mathrm {PostLR}({\mathbf {k}}{\mathcal {OT}})=(R,R\otimes \mathrm {Lie}({\mathbf {k}}{\mathcal {OT}}),[-,-]_{R\otimes \mathrm { Lie}({\mathbf {k}}{\mathcal {OT}})},\overline {\rho }_{{\mathbf {k}}{\mathcal {OT}}},\overline {\rhd }_{{\mathbf {k}}{\mathcal {OT}}})$
is the free post-Lie-Rinehart algebra generated by
$({\mathbf {k}}{\mathcal {OT}},\curvearrowright ,f_{{\mathbf {k}}{\mathcal {OT}}})$
.
We mention that the free object in the category of pre-Lie-Rinehart algebras with trace has been constructed via aromatic nonplanar trees in [Reference Fløystad, Manchon and Munthe-Kaas18], and such algebraic structures can also be characterized by universal geometric properties [Reference Munthe-Kaas and Verdier43, Reference Laurent and Munthe-Kaas33].
3.3 The universal enveloping algebra of a post-Lie-Rinehart algebra
Given a post-Lie-Rinehart algebra
$\mathscr {R}=(R,L,[-,-]_L,\rho ,\rhd )$
, since L is a Lie R-algebra, we have a Hopf R-algebra
$(\mathscr U_R(L),\Delta ,\varepsilon ,S,\iota _R)$
by Proposition 3.9. Now we provide our main theorem in this section, which generalizes [Reference Li, Sheng and Tang35, Theorem 2.8].
Theorem 3.20. For a post-Lie-Rinehart algebra
$\mathscr {R}=(R,L,[-,-]_L,\rho ,\rhd )$
, the Hopf R-algebra
$(\mathscr U_R(L),\Delta ,\varepsilon ,S,\iota _R)$
has a
${\mathbf {k}}$
-linear product
$\rhd $
recursively defined by
for any
$f,g\in R$
,
$x,y\in L$
and
$X,Y\in \mathscr U_R(L)$
. Then the tuple
$(\mathscr U_R(L),\Delta ,\varepsilon ,S,\iota _R,\rhd )$
is a weak post-Hopf algebroid over R, which is called the universal enveloping algebra of
$\mathscr {R}$
.
Proof. According to the Oudom-Guin construction in [Reference Ebrahimi-Fard, Lundervold and Munthe-Kaas14], the post-Lie product
$\rhd $
can be extended to the universal enveloping algebra
$\mathscr U(L)$
of the Lie algebra L via Eqs. (48)–(49) with
$f=1_R$
, and also (50) and (51) with
$X,Y\in \mathscr U(L)$
. As the first part, we show that such a construction can be generalized for the post-Lie-Rinehart algebra
$\mathscr {R}$
, namely the post-Lie product
$\rhd $
of L can be extended further to the R-algebra
$\mathscr U_R(L)$
via formulas (48)–(51).
Let
$\mathscr U(L)=T(L)/(xy-xy-[x,y]_L\,|\,x,y\in L)={\mathbf {k}}\oplus \bar {\mathscr U}(L)$
, where
$\bar {\mathscr U}(L)=L\mathscr U(L)$
. Correspondingly, we have a direct sum of
${\mathbf {k}}$
-linear subspaces
$\mathscr U_R(L)=R \oplus \bar {\mathscr U}(L)/\mathscr J$
, where
$\mathscr J$
is the ideal of
$\mathscr U(L)$
given by
First note that the action
$\rhd $
of R on
$\mathscr U_R(L)$
via Eq. (48) is well-defined, since
$\mathscr U_R(L)$
is an R-module by the left multiplication of R. So we still need to verify that
$\mathscr U_R(L)$
acts on R properly by Eqs. (49) and (51), and then the linear map
is also well-defined by Eqs. (50) and (51). In order to do so, we just check that
and then
$\bar {\mathscr U}(L) \rhd \mathscr J\subseteq \mathscr J$
,
$\mathscr J\rhd R = \{0\}$
and
$\mathscr J \rhd \bar {\mathscr U}(L)\subseteq \mathscr J$
successively.
For the proof of Eq. (52), we apply Eqs. (42), (50) and (51) to see that
for any
$f\in R$
,
$w,x,y\in L$
and
$X,Y\in T(L)$
. So by induction on the degree of polynomials it reduces to checking that
By Eqs. (49)–(51), we find that
which vanishes by Eqs. (43), (50) and the anchor
$\rho :L_\rhd \to \mathrm {Der}(R)$
as a Lie homomorphism. Hence, Eq. (52) holds.
Now it remains to check that
$\bar {\mathscr U}(L) \rhd \mathscr J\subseteq \mathscr J$
, and then
$\mathscr J\rhd R = \{0\}$
and
$\mathscr J \rhd \bar {\mathscr U}(L)\subseteq \mathscr J$
. First one can use Eqs. (50) and (51) repeatedly to see that
$\bar {\mathscr U}(L)\rhd \mathscr J\subseteq \mathscr J$
, since the following initial step holds:
for any
$f\in R$
and
$w,x,y\in L$
.
Next we check that
$\mathscr J\rhd R = \{0\}$
and
$\mathscr J \rhd \bar {\mathscr U}(L)\subseteq \mathscr J$
simultaneously by induction on the degree of polynomials in
$\mathscr J$
. First fix
$J\in \mathscr J$
and assume that
$J'\rhd R=\{0\}$
(resp.
$J'\rhd \bar {\mathscr U}(L)\subseteq \mathscr J$
) for any polynomial
$J'\in \mathscr J$
of degree not greater than J. Then for any
$x\in L$
and
$X\in R$
(resp.
$X\in \bar {\mathscr U}(L)$
),
As shown previously
$\bar {\mathscr U}(L)\rhd \mathscr J\subseteq \mathscr J$
, we have
$x\rhd (J\rhd X)$
,
$(x\rhd J)\rhd X=0$
(resp.
$x\rhd (J\rhd X), (x\rhd J)\rhd X\in \mathscr J$
) by the assumption, so
$xJ\rhd X=0$
(resp.
$xJ\rhd X\in \mathscr J$
). As a result, it is enough to show that
$((fx)y-x(fy))\mathscr U(L)\rhd R =\{0\}$
and
$((fx)y-x(fy))\mathscr U(L)\rhd \bar {\mathscr U}(L)\subseteq \mathscr J$
.
For any
$f,g\in R$
,
$x,y\in L$
, we have
so
$((fx)y-x(fy))\rhd R=\{0\}$
. On the other hand, if
$w\in L$
, then
Then assuming that
$((fx)y-x(fy))\rhd X\in \mathscr J$
for
$X\in \bar {\mathscr U}(L)$
, we see that
as
$(fx)Xy-xX(fy)\in \mathscr J$
for any
$x,y\in L$
and
$X\in \bar {\mathscr U}(L)$
by the definition of
$\mathscr J$
. So we also have
$((fx)y-x(fy))\rhd \bar {\mathscr U}(L)\subseteq \mathscr J$
.
Then for any
$Y\in R$
(resp.
$Y\in \bar {\mathscr U}(L)$
), we obtain that
by induction on the degree of polynomials in
$\mathscr J$
, since the above elements underlined all belong to
$\mathscr J$
. Hence, the proof that
$\mathscr J\rhd R = \{0\}$
and
$\mathscr J \rhd \bar {\mathscr U}(L)\subseteq \mathscr J$
is complete.
For the second part, we show that the Hopf R-algebra
$(\mathscr U_R(L),\Delta ,\varepsilon ,S,\iota _R)$
with the
${\mathbf {k}}$
-linear product
$\rhd $
satisfies conditions (14)–(20), so
$(\mathscr U_R(L),\iota _R,\rhd )$
is a weak post-Hopf algebroid.
Recall that
$\mathscr U_R(L)=R \oplus \bar {\mathscr U}(L)/\mathscr J$
, where
$\mathscr U(L)={\mathbf {k}}\oplus \bar {\mathscr U}(L)$
is the universal enveloping algebra of the post-Lie algebra
$(L,\rhd )$
. First we see that conditions (15)–(17) clearly hold for
$\mathscr U_R(L)$
by definitions of the product
$\rhd $
and the counit
$\varepsilon $
of
$\mathscr U_R(L)$
. For instance, Eqs. (48), (49) and (51) tell us that
so Eq. (15) holds. By the recursion formulas (50) and (51), the compatibility condition (44) for the post-Lie-Rinehart algebra
$\mathscr {R}$
is extended to condition (18) for
$\mathscr U_R(L)$
.
In order to verify conditions (14), (19) and (20) for
$\mathscr U_R(L)$
, we only need to check the situations involving elements in the base algebra R, since they already hold for the original post-Hopf algebra
$\mathscr U(L)$
. For any
$f,g\in R$
,
$X,Y\in \mathscr U_R(L)$
,
so Eq. (14) holds. By induction on the degree of polynomials in
$\mathscr U_R(L)$
, we check that
so Eq. (19) holds. Also, we have
Hence, Eq. (20) also holds.
Remark 3.21. When
$R={\mathbf {k}}$
, this result recovers [Reference Li, Sheng and Tang35, Theorem 2.8], which shows that for a post-Lie algebra
$(\mathfrak {g},[-,-]_{\mathfrak {g}},\rhd )$
, its universal enveloping algebra
$(\mathscr U(\mathfrak g),\rhd )$
is a post-Hopf algebra.
Now we consider an action post-Lie-Rinehart algebra
$(R,L=R\otimes \mathfrak {g},[-,-]_L,\overline {\rho },\overline {\rhd })$
given in Proposition 3.16 from a post-Lie algebra
$(\mathfrak {g},[-,-]_{\mathfrak {g}},\rhd )$
with an action
$\rho :\mathfrak {g}_\rhd \to \mathrm { Der}(R)$
. The action
$\rho $
can be lifted to a module algebra action of
$\mathscr U(\mathfrak {g}_\rhd )$
on R by the universal property of
$\mathscr U(\mathfrak {g}_\rhd )$
. On the other hand, for the post-Hopf algebra
$(\mathscr U(\mathfrak {g}),\rhd )$
, there is a natural Hopf algebra isomorphism
$\Phi :\mathscr U(\mathfrak {g}_\rhd )\to \mathscr U(\mathfrak {g})_\rhd $
, called the Oudom-Guin isomorphism [Reference Ebrahimi-Fard, Lundervold and Munthe-Kaas14]. Pulling back by
$\Phi ^{-1}$
, we obtain a module algebra action
$\rightharpoonup $
of
$\mathscr U(\mathfrak {g})_\rhd $
on R. Then by Theorem 2.20, we have an action post-Hopf algebroid
$(R\otimes \mathscr U(\mathfrak {g}),\Delta ,\varepsilon ,S,\iota ,\overline {\rhd },\theta )$
over R.
Corollary 3.22. With the above notations, the universal enveloping algebra
$\mathscr U_R(L)$
of the action post-Lie-Rinehart algebra
$(R,L=R\otimes \mathfrak {g},[-,-]_L,\overline {\rho },\overline {\rhd })$
is a post-Hopf algebroid, which is isomorphic to the action post-Hopf algebroid
$(R\otimes \mathscr U(\mathfrak {g}),\Delta ,\varepsilon ,S,\iota ,\overline {\rhd },\theta )$
over R.
Proof. First by Proposition 3.9,
$(\mathscr U_R(L),\Delta ,\varepsilon ,S)$
is a Hopf R-algebra, since
$L=R\otimes \mathfrak {g}$
is a Lie R-algebra. Also by Theorem 3.20, we know that
$(\mathscr U_R(L),\Delta ,\varepsilon ,S,\iota ,\overline {\rhd })$
is a weak post-Hopf algebroid.
Moreover, we have a natural identification between
$\mathscr U_R(L)$
and the tensor product Hopf R-algebra
$R\otimes \mathscr U(\mathfrak {g})$
. Elements in
$\mathscr U_R(L)$
have the form
$\sum _i f_ix_i$
with
$f_i\in R$
and
$x_i\in \mathscr U(\mathfrak {g})$
, and the post-Hopf product
$\overline {\rhd }$
of
$\mathscr U_R(L)$
extended from the post-Lie product
$\overline {\rhd }$
of L is just given by Eqs. (35). On the other hand, an algebra antiautomorphism
$\theta $
of
$\mathscr U_R(L)$
can be given by (36), so that
$\mathscr U_R(L)$
is the action post-Hopf algebroid
$(R\otimes \mathscr U(\mathfrak {g}),\Delta ,\varepsilon ,S,\iota ,\overline {\rhd },\theta )$
over R as in Theorem 2.20.
Given a magma algebra
$(V,\curvearrowright )$
, there exists a unique extended product
$\rhd $
on the tensor algebra
$T(V)$
such that the coshuffle Hopf algebra
$(T(V),\cdot ,\Delta ^{\mathrm {cosh}},S,\rhd _V)$
is a post-Hopf algebra [Reference Li, Sheng and Tang35, Theorem 2.9]. Now for any linear map
$f_V:V\to \mathrm {Der}(R)$
, it can be extended to a Lie algebra homomorphism from
$\mathrm {Lie}(V)$
to
$\mathrm {Der}(R)$
. Since
$T(V)$
is the universal enveloping algebra of the Lie algebra
$\mathrm {Lie}(V)$
, it follows that R is a
$T(V)$
-module algebra. Then using the Hopf algebra isomorphism
$K:T(V)_{\rhd _V}\to T(V)$
, which is called Gavrilov’s K-map ([Reference Al-Kaabi, Ebrahimi-Fard, Manchon and Munthe-Kaas2, Theorem 3]), we obtain the following
$T(V)_{\rhd _V}$
-module algebra action
$\rightharpoonup $
on R recursively given by:
for any
$f\in R$
and
$v,v_1,\dots ,v_n\in V$
. Then by Theorem 2.20, we obtain the action post-Hopf algebroid over R,
By Corollary 3.22, we obtain the following characterization of the universal enveloping algebra of the free post-Lie-Rinehart algebra.
Corollary 3.23. The universal enveloping algebra of the free post-Lie-Rinehart algebra generated by a magma algebra
$(V,\curvearrowright )$
together with a linear map
$f_V:V\to \mathrm {Der}(R)$
given in Theorem 3.18 is the action post-Hopf algebroid
$T_R(V)=(R \otimes T(V),\iota ,\overline {\rhd }_V,\theta )$
over R.
Example 3.24. As shown in Example 3.19, the free post-Lie-Rinehart algebra generated by the triple
$({\mathbf {k}}{\mathcal {OT}},\curvearrowright ,f_{{\mathbf {k}}{\mathcal {OT}}})$
is the action post-Lie-Rinehart algebra
According to Corollary 3.23, the universal enveloping algebra of
$\mathrm {PostLR}({\mathbf {k}}{\mathcal {OT}})$
is the action post-Hopf algebroid
$T_R({\mathbf {k}}{\mathcal {OT}})=(R \otimes T({\mathbf {k}}{\mathcal {OT}}),\iota ,\overline {\rhd }_{{\mathbf {k}}{\mathcal {OT}}},\theta )$
over R, where the post-Hopf algebra
$(T({\mathbf {k}}{\mathcal {OT}}),\rhd _{{\mathbf {k}}{\mathcal {OT}}})$
is the universal enveloping algebra of the free post-Lie algebra
$\mathrm { PostLie}({\mathbf {k}}{\mathcal {OT}})$
, and its subadjacent Hopf algebra
$T({\mathbf {k}}{\mathcal {OT}})_{\rhd _{{\mathbf {k}}{\mathcal {OT}}}}$
is the Grossman-Larson Hopf algebra of ordered planar forests.
4 Post-Lie-Rinehart algebras and post-Hopf algebroids for numerical analysis
In this section, we detail the natural post-Lie-Rinehart algebras and post-Hopf algebroid structures appearing in numerical analysis.
Let
${\mathcal {M}} $
be a smooth manifold equipped with a frame basis
$(E_d)$
, defined globally for clarity. The smooth vector fields
$\mathfrak {X}({\mathcal {M}} )$
are naturally equipped with the Jacobi bracket
$[\![ -,-]\!]_J$
and the (curvature-free) Weitzenböck connection,
where we use the Einstein summation convention. Let us further assume that the connection
$\rhd $
has constant torsion
$\nabla T=0$
, or equivalently that
${\mathcal {M}} $
is locally a Lie group. In this case, the torsion defines the bracket:
The two brackets are linked by the identity
Then, the connection algebra
$(\mathfrak {X}({\mathcal {M}} ),[-,-],\rhd )$
is a post-Lie algebra [Reference Munthe-Kaas and Lundervold41].
As a specific application of Theorem 3.20 and Theorem 2.13, we have the following result.
Proposition 4.1. The tuple
$(\mathcal {C}^\infty ({\mathcal {M}} ),\mathfrak {X}({\mathcal {M}} ),[-,-],\rho ,\rhd )$
is a post-Lie-Rinehart algebra and its universal enveloping algebra
$(\mathscr U_{\mathcal {C}^\infty ({\mathcal {M}} )}(\mathfrak {X}({\mathcal {M}} )),\cdot ,\Delta ^{\mathrm {cosh}},\rhd )$
is a weak post-Hopf algebroid over
$\mathcal {C}^\infty ({\mathcal {M}} )$
, where the anchor map
$\rho $
is given by
Moreover,
$(\mathscr U_{\mathcal {C}^\infty ({\mathcal {M}} )}(\mathfrak {X}({\mathcal {M}} )),*_\rhd ,\Delta ^{\mathrm {cosh}})$
is the associated Grossman-Larson bialgebroid.
We rely on specific substructures of the universal enveloping algebra
$\mathscr U_{\mathcal {C}^\infty ({\mathcal {M}} )}(\mathfrak {X}({\mathcal {M}} ))$
in numerical analysis. Let
$F\in \mathfrak {X}({\mathcal {M}} )$
be a given smooth Lipschitz vector field. Then, F generates a post-Lie subalgebra
$(\mathfrak {X}_F({\mathcal {M}} ),[-,-],\rhd )$
:
This yields the Hopf subalgebras
$(\mathscr U(\mathfrak {X}_F({\mathcal {M}} )),\cdot ,\Delta ^{\mathrm {cosh}})$
and
$(\mathscr U(\mathfrak {X}_F({\mathcal {M}} )),*_\rhd ,\Delta ^{\mathrm {cosh}})$
, commonly represented with ordered planar forests [Reference Lundervold and Munthe-Kaas37]. Analogously, let
$\mathrm {End}_F(\mathfrak {X}({\mathcal {M}} ))$
be the space of endomorphisms of
$\mathfrak {X}({\mathcal {M}} )$
generated from F,
$\rhd $
and
$[-,-]$
:
Let
$\mathcal {C}^\infty _F({\mathcal {M}} )$
be the subalgebra of
$\mathcal {C}^\infty ({\mathcal {M}} )$
generated by the following functions, with elements called aromas in numerical analysis [Reference Chartier and Murua12, Reference Iserles, Quispel and Tse28, Reference Bogfjellmo6]:
Following Corollaries 3.22 and 2.19, we obtain the following action Hopf algebroid structures using the anchor map
$\rho $
.
Theorem 4.2. The tuple
$(\mathcal {C}^\infty _F({\mathcal {M}} ),\mathcal {C}^\infty _F({\mathcal {M}} )\otimes \mathfrak {X}_F({\mathcal {M}} ),[-,-],\rho ,\rhd )$
is an action post-Lie-Rinehart algebra, with elements called aromatic vector fields. Its universal enveloping algebra is the action post-Hopf algebroid
$(\mathcal {C}^\infty _F({\mathcal {M}} )\otimes \mathscr U(\mathfrak {X}_F({\mathcal {M}} )),\cdot ,\Delta ^{\mathrm {cosh}},\rhd ,\theta )$
over
$\mathcal {C}^\infty _F({\mathcal {M}} )$
, with the associated Grossman-Larson Hopf algebroid
$(\mathcal {C}^\infty _F({\mathcal {M}} )\otimes \mathscr U(\mathfrak {X}_F({\mathcal {M}} )),*_\rhd ,\Delta ^{\mathrm {cosh}},\theta )$
over
$\mathcal {C}^\infty _F({\mathcal {M}} )$
.
Remark 4.3. Let
$\mathrm {Div}\colon \mathfrak {X}({\mathcal {M}} )\rightarrow \mathcal {C}^\infty ({\mathcal {M}} )$
be the divergence operator associated to the connection
$\rhd $
:
The divergence restricts to aromatic vector fields
$\mathrm {Div}\colon \mathfrak {X}_F({\mathcal {M}} )\rightarrow \mathcal {C}^\infty _F({\mathcal {M}} )$
, but aromas are not in general all generated by the divergence:
$\mathrm {Div}(\mathfrak {X}_F({\mathcal {M}} ))\subsetneq \mathcal {C}^\infty _F({\mathcal {M}} )$
, as observed in [Reference Fløystad, Manchon and Munthe-Kaas18].
Let the flow
$\varphi _t(y_0)$
,
$t\in \mathbb R$
be the solution of the initial value problem on
${\mathcal {M}} $
:
The Taylor expansion of the pullback by the exact flow of (53) is described by the Grossman-Larson exponential:
A numerical flow for solving (53) is a smooth map
$\psi _t\colon {\mathcal {M}} \rightarrow {\mathcal {M}} $
with
$\psi _0=\mathrm {id}$
defined for t small enough. Most standard numerical flows
$\psi _t$
have Taylor expansions that rewrite with formal series in t over the Hopf algebra
$\mathscr U(\mathfrak {X}_{tF}({\mathcal {M}} ))$
, which we denote by
$\overline {\mathscr U(\mathfrak {X}_{tF}({\mathcal {M}} ))}$
. This is the case for Lie-group methods [Reference Iserles, Munthe-Kaas, Nørsett and Zanna27], and, in particular, for the Lie-Euler method, which follows the geodesics for
$\rhd $
:
The Taylor expansion of the pullback by Lie-group methods is typically expressed using the concatenation product. For instance, the Taylor expansion of the Lie-Euler flow (54) is described by the concatenation exponential
where the concatenation product acts on functions as the following differential operator
A method
$\psi _t$
is of order p if the Taylor expansions of
$\varphi _t^*\phi $
and
$\psi _t^*\phi $
match for all terms of order at most p. The Lie-Euler method (54) is only of order one, for instance.
Remark 4.4. The Grossman-Larson product
$*_\rhd $
corresponds to the composition of flows, in the sense that
The description of the composition of flows in simpler cases has been extensively studied in the numerical literature using tree structures and Butcher series for the description of the Grossman-Larson Hopf algebra. In particular, the dual of the Grossman-Larson Hopf algebra
$T({\mathbf {k}}{\mathcal {OT}})_{\rhd _{{\mathbf {k}}{\mathcal {OT}}}}$
is isomorphic to the Munthe-Kaas-Wright Hopf algebra [Reference Munthe-Kaas and Wright44, Reference Ebrahimi-Fard and Rahm15]. In the pre-Lie context, the action pre-Hopf algebroid of aromatic forests [Reference Chartier and Murua12, Reference Iserles, Quispel and Tse28] is closely related to the Butcher-Connes-Kreimer Hopf algebra over aromatic forests [Reference Bogfjellmo6, Reference Bronasco and Laurent8].
Let us now define aromatic numerical flows by relying on the post-Hopf algebroid structure of
$\mathcal {C}^\infty _F({\mathcal {M}} )\otimes \mathscr U(\mathfrak {X}_F({\mathcal {M}} ))$
(see also [Reference Munthe-Kaas and Verdier43, Reference Bogfjellmo6]). Note that the differential operators defined using functions in
$\mathcal {C}^\infty _F({\mathcal {M}} )$
do still act on all smooth functions in
$\mathcal {C}^\infty ({\mathcal {M}} )$
.
Definition 4.5. A numerical flow
$\psi _t$
is aromatic if the Taylor expansion of the pullback
$\psi _t^*\phi $
is a formal series in
$\overline {\mathcal {C}^\infty _{tF}({\mathcal {M}} )\otimes \mathscr U(\mathfrak {X}_{tF}({\mathcal {M}} ))}$
:
Example 4.6. Aromatic methods are straightforwardly obtained from Lie-group methods by substituting
$F=f^i E_i$
with an aromatic vector field
$t\hat {F}_t\in \mathcal {C}^\infty _{tF}({\mathcal {M}} )\otimes \mathfrak {X}_{tF}({\mathcal {M}} )$
. For instance, apply the Lie-Euler method (54) with the preprocessed aromatic vector field
Then, the resulting flow
$\psi _t$
is aromatic.
The application of aromatic flows in geometric numerical integration is better understood through backward error analysis [Reference Hairer, Lubich and Wanner23, Reference Lundervold and Munthe-Kaas37, Reference Rahm45]. Under technical assumptions satisfied by large classes of numerical methods, a numerical flow may be understood as the exact flow of a formal modified differential equation
$y'=\tilde {F}_t(y)$
driven by a modified vector field
$\tilde {F}_t$
, that is,
The geometric properties of the numerical method
$\psi _t$
are read directly from
$\tilde {F}_t$
. A case of importance in the literature lies in divergence-free features
$\mathrm {Div}(F)=\mathrm {Div}(\tilde {F}_t)=0$
and is linked with volume-preserving flows. In this context, one is interested in the creation of consistent divergence-free numerical methods. The design of such methods is an open problem of numerical analysis, even in the pre-Lie context. It is shown in [Reference Iserles, Quispel and Tse28, Reference Dotsenko and Laubie13] that nonexact numerical flows described by
$\overline {\mathscr U(\mathfrak {X}_{tF}({\mathcal {M}} ))}$
cannot satisfy
$\mathrm {Div}(\tilde {F}_t)=0$
, so that the use of aromatic numerical flows is crucial to the design of divergence-free methods.
Application 4.7. While the creation of exact divergence-free methods is an open problem, one can create pseudo-divergence-free methods by using the post-Hopf algebroid
$\mathcal {C}^\infty _F({\mathcal {M}} )\otimes \mathscr U(\mathfrak {X}_F({\mathcal {M}} ))$
. In particular, a calculation yields that the aromatic method from Example 4.6 satisfies (at least formally) third order of preservation:
$\mathrm {Div}(\tilde {F}_t)=O(t^3)$
. In contrast, the original Lie-Euler method (54) satisfies only the first order estimate
$\mathrm {Div}(\tilde {F}_t)=O(t)$
.
Given an aromatic flow, the computation of the associated modified vector field
$\tilde {F}_t$
is not direct and relies on the so-called substitution law. In the pre-Lie case, it is described with nonplanar aromatic trees in [Reference Bogfjellmo6, Reference Bronasco and Laurent8] and the divergence-free modified aromatic vector fields
$\tilde {F}_t$
are fully characterized in [Reference Laurent, McLachlan, Munthe-Kaas and Verdier32, Reference Laurent31, Reference Dotsenko and Laubie13]. The extension of these results to the post-Lie context and the design and study of convenient algebraic structures for representing aromatic vector fields and aromas are key to the creation of divergence-free aromatic numerical methods and are matters for future work.
Acknowledgments
We give warmest thanks to Xiao Han for helpful comments.
Competing interests
The authors have no competing interests to declare.
Funding statement
This research is supported by NSF of Jilin Province (20260101013JJ), BABRF of Guangdong Province (2026A1515012750), NSFC (12471060, W2412041) and ANR-25-CE40-2862-01 (MaStoC - Manifolds and Stochastic Computations).
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