The free tracial post-Lie-Rinehart algebra of planar aromatic trees for the design of divergence-free Lie-group methods

The free tracial p ost-Lie-Rinehart algebra of planar aromatic trees for the design of div ergence-free Lie-group metho ds A drien Busnot Lauren t 1 , Hans Mun the-Kaas 2 and V enkatesh G. S. 2 Marc h 31, 2026 Abstract Aromatic Butc her series were successfully in tro duced for the study and design of n umeri- cal integrators that preserve v olume while solving differen tial equations in Euclidean spaces. They are naturally asso ciated to pre-Lie-Rinehart algebras and pre-Hopf algebroids struc- tures, and aromatic trees were sho wn to form the free tracial pre-Lie-Rinehart algebra. In this pap er, we presen t the generalisation of aromatic trees for the study of div ergence-free in tegrators on manifolds. W e introduce planar aromatic trees, sho w that they span the free tracial p ost-Lie-Rinehart algebra, and apply them for deriving new Lie-group metho ds that preserv e geometric divergence-free features up to a high order of accuracy . K eywor ds: geometric numerical in tegration, Lie-Butcher series, aromatic trees, divergence- free, volume-preserv ation. AMS subje ct classific ation (2020): 41A58, 65L06, 37M15, 05C05, 16T05. 1 In tro duction Butc her trees and series were in tro duced in the 60’s for the creation of high-order integration metho ds [8, 9, 10, 23]. Sev eral extensions of the initial formalism were then in tro duced for the dev elopment of geometric numerical integration. In particular, planar trees and Lie-Butcher series w ere used for the design of Lie-group metho ds [24] and aromatic trees and aromatic B-series sho wed to b e crucial to ols for the study of volume-preserving integrators [15, 25]. The creation of n umerical metho ds on manifolds that preserve geometric inv arian ts is an activ e field of research. In particular, there are, to the b est of our knowledge, no existing work discussing the preserv ation of volume for intrinsic metho ds on manifolds. In general, the Lie-group metho ds (Lie-R unge- Kutta, Crouc h-Grossman or R unge-Kutta-Munthe-Kaas metho ds) and more generally LB-series metho ds, do not preserve v olume, ev en in the Euclidean case [15, 25]. The characterisation of Euclidean numerical volume-preserv ation is done through the use of bac kw ard error analysis [13, 23, 14, 11] and rewrites as the design of methods whose mo dified v ector field satisfies div p ˜ f q “ 0 . It is natural to consider aromatic mo difications of LB-series metho ds to obtain div ergence-free features for the numerical metho d at least up to a high order. A simple wa y to ac hieve high order of volume-preserv ation is obtained b y considering high order-metho ds, but this approach is costly and sub-optimal. Approaches for Euclidean pseudo-volume-preserv ation are first discussed in [3, 42]. There are other Euclidean techniques in the literature for building 1 Univ Rennes, INRIA (Researc h team MINGuS), IRMAR (CNRS UMR 6625) and ENS Rennes, F rance. A drien.Busnot-Laurent@inria.fr. 2 Departmen t of Mathematics and Statistics, UiT – The Arctic Univ ersity of Norwa y , T romsø, Norw ay . Hans.Mun the-Kaas@uib.no, Subbarao.V.Guggilam@uit.no. 1 v olume-preserving metho ds [54, 51], but their complexity blo ws up with the dimension of the problem, opp osite to the aromatic B-series approac h. The extension of the Euclidean results to in trinsic metho ds on manifolds ha ve not b een considered to the b est of our kno wledge. In this pap er, we in tro duce planar aromatic trees, study their algebraic structure, and apply them for the study and design of volume-preserving Lie-group metho ds. The analysis is successfully applied for the design of aromatic Lie-group metho ds with low con vergence order and high order of volume preserv ation. T o motiv ate the algebraic formalism below, and its relation to geometry and n umerical in tegration, we recall briefly the basic setup of Lie group integration and related numerical metho ds. Let E Ñ M b e a vector bundle ov er a manifold and R “ C 8 p M , R q and L “ Γ p E q the sections of the bundle (e.g. vector fields, tensor fields). If L is also equipp ed with a Lie brac ket r´ , ´s and a Lie algebra homomorphism ρ : L Ñ X p M q to the vector fields with the Jacobi brac ket, w e call L a Lie algebr oid . An imp ortan t example is the action algebroid obtained from the action of a Lie group on a manifold, which is the generic setup of numerical Lie group in tegration [40]. In numerical algorithms we define basic flo ws on M whic h can b e computed exactly , from which w e dev elop more adv anced in tegration algorithms. Such basic flows can often b e described as geo desics of a connection. F or the action algebroid, the canonical connection on L is inv ariant with zero curv ature and parallel torsion. This is called a p ost-Lie algebr oid [44]. In [45] an intimate relationship b et ween p ost-Lie algebroids and action algebroids is developed, and [1] explains why p ost-Lie algebroids are also fundamental in the understanding of geo desic flo ws of general non-in v arian t connections. There is a dual picture on the algebraic side, where the manifold M is replaced b y the comm utative ring of scalar functions R “ C 8 p M , R q , and vector bundles are studied through their space of sections L “ Γ p E q , whic h is algebraically describ ed as an R -mo dule, with the pro duct of a scalar function with a section defined p oin t wise. The Serre-Sw an theorem states that the category of finitely generated pro jectiv e mo dules o ver R is a faithful represen tation of the category of vector bundles E Ñ M . Informally: ‘pr oje ctive mo dules ar e like ve ctor bund les’ . The geometric concept of Lie algebroids are describ ed on the algebraic side as Lie-Rinehart algebras, and p ost-Lie algebroids as tr acial p ost-Lie-Rinehart algebr as , whic h is the main algebraic ob ject of study in this pap er. The trace condition, defined b elow, is fulfilled for an y finitely generated pro jectiv e mo dule, as those arising in the Serre-Sw an theorem. F rom the algebraic p oin t of view, planar trees generate the free p ost-Lie algebra [44], with n umerous applications in v arious fields (see, for instance, [57, 2, 26]). On the other hand, aromatic trees yield the free tracial pre-Lie-Rinehart algebra [20], and the use of aromas w as recen tly extended to a v ariety of contexts [31, 58]. The natural generalisation of these structures is p ost-Lie-Rinehart algebras, and ha ve b een recently mentioned in [22, 26, 7, 53]. In n umerical analysis, such structures are naturally used with their univ ersal en veloping algebras. P ost-Lie and pre-Lie-Rinehart algebras respectively give rise to p ost-Hopf algebras [46] and pre-Hopf algebroids [3, 5]. F or p ost-Lie-Rinehart algebras, the universal en veloping algebra yields a p ost- Hopf algebroid [7]. W e show in the present pap er that the newly defined planar aromatic trees generate the free tracial p ost-Lie-Rinehart algebra. The article is organized as follows. In Section 2, w e recall the definition and properties of Lie-Rinehart algebra. W e sp ecify suc h structures given an affine connection to define tracial Lie- Rinehart algebras and the divergence in S ection 3. In Section 4, w e define the main algebraic structures of this article, that are, p ost-Lie-Rinehart algebras. Section 5 introduces the planar aromatic trees, their algebraic prop erties, and shows that they are the free tracial post-Lie- Rinehart algebra. The approach is applied successfully in Section 6 for the design of Lie-group 2 metho ds that preserve div ergence-free features up to a high order. F uture w orks are discussed in Section 7. 2 Lie-Rinehart Algebras and their cohomology 2.1 Lie-Rinehart Algebras Let k b e a field of characteristic 0 . Definition 2.1. A k - Lie-Rinehart is a tuple p R , L, ρ q wher e R is a c ommutative k -algebr a and p L, r¨ , ¨sq is a k -Lie algebr a such that: (i) L is a R -mo dule, (ii) R is a L -mo dule wher e the action of L on R is given by the anchor map ρ viz, a Lie morphism, ρ : L Ý Ñ D er p R, R q , wher e D er p R , R q stands for Lie algebr a of derivations fr om R to R , (iii) (L eibniz R ule) : F or al l f P R and X , Y P L , r X , f Y s “ p X .f q Y ` f r X, Y s . Remark 2.2. F or the sake of br evity, the action of L on R via the anchor map is denote d as the fol lowing: X .f “ ρ p X qp f q , X P L, f P R . 2.2 Chev alley-Eilen b erg Cohomology Giv en a Lie-Rinehart algebra p L, R q , let the Chev alley-Eilen b erg co c hain complex denoted by CE p L, End R p L q , ω q “ p Ź ‚ L ˚ , End R p L qq where (i) End R p g q is the k -Lie algebra of R -linear endomorphisms of the R -mo dule g . The Lie brac ket of End R p g q is denoted by J ¨ , ¨ K , (ii) ω P g ˚ b End R p g q and is called a End R p g q -Lie algebra v alued c onne ction 1-form satisfying p ω p r .g 1 q ´ r .ω p g 1 qqp g 2 q “ p ρ p g 2 qp r qq g 1 , (2.1) (iii) g ˚ is the dual of the R -mo dule g , (iv) Ź ‚ g ˚ is the differen tial graded exterior algebra of the R -mo dule g ˚ . Giv en ω , then End R p g q is a g -mo dule with g -action: g .f “ J ω p¨qp g q , f p¨q K viz., p g .f qp g 1 q “ ω p f p g 1 qqp g q ´ f pp ω p g 1 qqqp g q “ J ω p¨qp g q , f p¨q K p g 1 q , for g , g 1 P g and f P End R p g q . It is straightforw ard to chec k via Jacobi identit y that g . p h.f q ´ h. p g .f q “ J g , h K .f and generalizes the adjoint action twiste d by connection form. 3 The Chev alley-Eilen b erg co c hain complex is End R p g q d Ý Ñ g ˚ b End R p g q d Ý Ñ 2 ľ g ˚ b End R p g q d Ý Ñ ¨ ¨ ¨ where the differen tial d is defined as: for all γ P Ź n g ˚ b End R p g q and x 1 , x 2 , . . . , x n P g : dγ p x 1 , x 2 , . . . , x n ` 1 q “ ÿ σ P Sh p 1 ,n q sgn p σ q x σ p 1 q ¨ γ p x σ p 2 q , x σ p 3 q , . . . , x σ p n ` 1 q q ´ ÿ σ P Sh p 2 ,n ´ 1 q sgn p σ q γ pr x σ p 1 q , x σ p 2 q s , x σ p 3 q , . . . , x σ p n ` 1 q q , where Sh p p, q q denote p p, q q -shuffles in the p erm utation group of p ` q letters. The curvatur e is a 2 -form R P Ź 2 g ˚ b End R p g q given by R “ dω ` 1 2 r ω ^ ω s . 2.3 Reductiv e Splitting and T orsion Sa y End R p g q has a reductive decomp osition viz., End R p g q “ h ‘ m as R -mo dules such that (i) h is a Lie subalgebra of g , (ii) m is a h -mo dule under adjoin t action of h i.e, J h , m K Ď m . The 1 -form whence splits as ω “ ω h ` ω m where ω h P g ˚ b h and ω m P g ˚ b m resp ectiv ely . The ω h is called princip al c onne ction form and ω m is called soldering form or vielb ein . Consequently , the Lie algebra v alued 2 -form curv ature R splits as R “ dω ` 1 2 r ω ^ ω s “ d p ω h ` ω m q ` 1 2 rp ω h ` ω m q ^ p ω h ` ω m qs “ dω h ` dω m ` 1 2 tr ω h ^ ω h s ` 2 r ω h ^ ω m s ` r ω m ^ ω m s u . Since m is not necessarily a Lie subalgebra, the term r ω m , ω m s can ha ve v alues in b oth h and m . W e obtain r ω m ^ ω m s “ r ω m ^ ω m s h ` r ω m ^ ω m s m . The curv ature form R thus splits into terms in h and m as R “ dω h ` 1 2 r ω h ^ ω h s ` r ω m ^ ω m s h l jh n terms in h ` dω m ` r ω h ^ ω m s ` 1 2 r ω m ^ ω m s m l jh n terms in m “ R ` T . The terms R is the intrinsic curvatur e 2 -f orm and T is the torsion . In the case where the Lie ideal m is an Ab elian Lie algebra, then the expressions for in trinsic curv ature and torsion reduce resp ectiv ely to R “ dω h ` 1 2 r ω h ^ ω h s , T “ dω m ` r ω h ^ ω m s . 4 3 Affine and Isometric Connections The connection form ω P g ˚ b End R p g q is an affine connection form when End R p g q – aff p n q where aff p n q is the Lie algebra of affine group. Note that, aff p n q has a reductive decomp osition: aff p n q “ gl p n q ‘ R n . 3.1 Levi-Civita and W eitzenbö ck Connection The connection form ω P g ˚ b End R p g q with End R p g q – iso p n q , the Lie algebra of isometries of the affine space R n . Recall that iso p n q ã Ñ aff p n q and has reductiv e decomp osition as iso p n q “ so p n q ‘ R n , where so p n q is the Lie subalgebra and R n is so p n q ideal. The b oth Levi-Civita and W eitzenbö ck connections are isometric connections which are explained shortly . The 2-form curv ature in isometric connection splits into intrinsic curv ature and torsion as R “ R ljhn v alued in so p n q ` T ljhn v alued in R n . The L evi-Civita c onne ction is the isometric connection where T “ 0 (torsion v anishes) and W eitzenb ö ck c onne ction is the isometric connection where R “ 0 (in trinsic curv ature v anishes). The Bianchi identit y for the Levi-Civita connection whence reduces to d R “ d R “ 0 , while for the W eitzenbö c k connection the Bianchi iden tity is d R “ d T “ 0 . An affine/linear connection on the Lie-Rinehart algebra can also be defined or rather de- scrib ed in a mor e elementary manner follo wing [20] which the current pap er also adheres to. Definition 3.1. A line ar c onne ction on a R -mo dule N with r esp e ct to Lie-R inehart algebr a p R, L q is a R -line ar map ∇ : L Ý Ñ End k p N q (3.1) X ÞÝ Ñ ∇ X (3.2) such that for al l f P R and X , Y P L : ∇ X p f Y q “ p X .f q Y ` f ∇ X Y . (3.3) Observ e that the asso ciativ e algebra End p N q can b e endo wed a Lie algebra structure via the comm utator J ¨ , ¨ K . Definition 3.2. L et N b e a R -mo dule e quipp e d with a c onne ction ∇ with r esp e ct to Lie-R inehart algebr a p R, L q . A n m r ank k -multiline ar map W : N b m Ý Ñ M is c al le d p ar al lel if for al l X P L and Y 1 , Y 2 , . . . , Y m ∇ X p W p Y 1 , Y 2 , . . . , Y m qq “ W p ∇ X Y 1 , Y 2 , . . . , Y m q ` W p Y 1 , ∇ X Y 2 , . . . , Y m q ` ¨ ¨ ¨ (3.4) ¨ ¨ ¨ ` W p Y 1 , Y 2 , . . . , ∇ X Y m q . 5 Definition 3.3. The curvatur e of the c onne ction ∇ on the R -mo dule N is the alternating biline ar map define d as R : L ^ L Ý Ñ E nd p N q p X , Y q ÞÝ Ñ R p X , Y q , wher e R p X , Y q “ J ∇ X , ∇ Y K ´ ∇ r X,Y s . Remark 3.4. The curvatur e R “ 0 if and only if N is a mo dule over the Lie algebr a L and the action is given by the c onne ction ∇ . Sinc e N is acte d by b oth R and L of the Lie-Rinehart p air p R, L q in this c ase, N is terme d as a mo dule over the Lie-Rinehart algebr a p R, L q . Definition 3.5 (Lie-Rinehart mo dule) . A R -mo dule N is a mo dule over the Lie-R inehart p air p R, L q or simply c al le d an p R , L q -mo dule), with a c onne ction ∇ if • N is a mo dule over the Lie algebr a L wher e the action is given by the c onne ction ∇ . ∇ r X 1 ,X 2 s Y “ p ∇ X 1 ˝ ∇ X 2 ´ ∇ X 2 ˝ ∇ X 1 q Y for al l X 1 , X 2 P L and Y P N . • The action of the c ommutaive algebr a R on N and the action on Lie algebr a L on N satisfy the Lie-Rinehart p air c omp atibility. F or al l X 1 , X 2 P L , f P R and Y P N : ∇ r X 1 ,f X 2 s Y “ p X 1 .f q ∇ X 2 Y ` f ∇ r X 1 ,X 2 s Y Example 3.6. F or a Lie-R inehart p air p R, L q , a primary example of Lie-Rinehart mo dule over p R, L q is the mo dule R . L o oking at R as R -mo dule, define the c onne ction ∇ : L Ý Ñ End k p R q X ÞÝ Ñ ∇ X , wher e ∇ X p f q “ X.f for al l X P L and f P R . The flatness of the c onne ction ∇ fol lows fr om the definition of p R, L q b eing a Lie-R inehart p air. Henc e, R is a Lie-Rinehart mo dule over p R, L q . Definition 3.7 (Morphism of Lie-Rinehart mo dules) . L et M , N b e p R, L q -mo dules with c onne c- tion ∇ , ∇ 1 r esp e ctively. A R -mo dule moprhism α : M Ý Ñ N is a morphism of p R, L q -mo dules if α is a morphism of L -mo dules viz. for al l X P L and Y P M α p ∇ X Y q “ ∇ 1 X p α p Y qq . F or a connection ∇ defined on the R -mo dule L with resp ect to Lie-Rinehart pair p R, L q , the torsion T of the connection ∇ is defined as T : L ^ L Ý Ñ L p X , Y q ÞÝ Ñ T p X , Y q , where T p X, Y q “ ∇ X Y ´ ∇ Y X ´ r X , Y s . Let N b e a R -mo dule endow ed with a connection ∇ with resp ect to the Lie-Rinehart pair p R, L q . The R -mo dule End R p N q is canonically equipp ed with the connection ˆ ∇ (with resp ect to p R, L q ), defined as p ˆ ∇ X ν qp Y q “ ∇ X p ν p Y qq ´ ν p ∇ X Y q , (3.5) 6 for all X P L, Y P N and ν P End R p N q . F rom p 3 . 5 q , it is straightforw ard to verify that for all X P L the connection ˆ ∇ X acts a deriv ation on the algebra End R p N q viz., for all ν, µ P End R p N q : ˆ ∇ X p ν ˝ µ q “ ˆ ∇ X p ν q ˝ µ ` ν ˝ ˆ ∇ X p µ q . (3.6) Prop osition 3.8. [20] If N is a Lie-Rinehart mo dule with r esp e ct to the Lie-Rinehart p air p R, L q , then End R p N q is also a Lie-R inehart mo dule over p R , L q . 3.2 T racial Lie-Rinehart Algebras Definition 3.9. L et L b e the R -mo dule endowe d with c onne ction ∇ over the Lie-Rinehart algebr a p R, L q and then define k -line ar maps d : L Ý Ñ End R p L q X ÞÝ Ñ dX, wher e dX p Z q “ ∇ Z X for al l Z, X P L and δ : L Ý Ñ End R p L q X ÞÝ Ñ δ X, wher e δ X p Y q “ T p Y , X q for al l X , Y P L . The k -linear map δ satisfies the follo wing prop erties 1. F or all X , Y P L : δ X p Y q ` δ Y p X q “ 0 , 2. If T is parallel, then for all X , Y 1 , Y 2 d p δ Y 1 p Y 2 qqp X q “ δ p d Y 1 p X qqp Y 2 q ´ δ p d Y 2 p X qqp Y 1 q . Define El R p L q as the R -algebra of elementary R -mo dule endomorphisms generated by the set of R -linear maps ␣ ˆ ∇ X 1 ˆ ∇ X 2 ¨ ¨ ¨ ˆ ∇ X k d Y , ˆ ∇ X 1 ˆ ∇ X 2 ¨ ¨ ¨ ˆ ∇ X k δ Y : k ě 0 , X 1 , X 2 , . . . , X k , Y P L ( . By definition, El R p L q is a subalgebra of the algebra of End R p L q . If L is a Lie-Rinehart mo dule, El R p L q is a Lie-Rinehart submo dule of the Lie-Rinehart mo dule End R p L q . Definition 3.10. A Lie-Rinehart algebr a p R, L q is tracial if L is a Lie-R inehart mo dule over p R, L q with c onne ction ∇ and a R -line ar map τ : El R p L q Ý Ñ R such that (i) τ is a homomorphism of p R , L q -mo dules fr om p El R , ˆ ∇ q to p R, ∇ q , that is, for nu P El R and X P L : τ p ˆ ∇ X ν q “ ∇ X p τ p ν qq “ X . p τ p ν qq (ii) for al l ν, µ P El R p L q τ p ν ˝ µ q “ τ p µ ˝ ν q , viz., τ is invariant under the action of cyclic p ermutations. Definition 3.11. The divergence map on p L, ∇ q in a tr acial Lie-Rinehart p air p R, L, τ q is div : L Ý Ñ R X ÞÝ Ñ div p X q “ p τ ˝ d qp X q . (3.7) Ther efor e, we have div “ τ ˝ d . 7 4 P ost-Lie-Rinehart algebras Let V b e a k -v ector space and let Ź : V b 2 Ý Ñ V b e a magmatic pro duct defined on V . Then, the A sso ciator map is defined as Ass Ź : V b 3 Ý Ñ V p a, b, c q ÞÝ Ñ a Ź p b Ź c q ´ p a Ź b q Ź c. P ost-Lie-Rinehart algebras w ere in tro duced recently in [22, 26]. They were further studied in [7], where their link with p ost-Hopf algebroids is discussed. Definition 4.1. A p ost-Lie-R inehart algebr a is a Lie-R inehart algebr a p R , L q endowe d with flat c onne ction ∇ : L Ý Ñ End k p L q with p ar al lel torsion or e quivalently ∇ : L b L Ý Ñ L p X , Y q ÞÝ Ñ X Ź Y : “ ∇ X Y . Hence for all X , Y , Z P L : (i) Ass Ź p X , Y , Z q ´ Ass Ź p Y , X , Z q “ T p Y , X q Ź Z ( R “ 0 ô L is a Lie-Rinehart mo dule), (ii) X Ź T p Y , Z q “ T pp X Ź Y q , Z q ` T p Y , p X Ź Z qq (torsion T is parallel). Example 4.2 (The co v arian t deriv ation algebra) . L et p M , ∇ q b e a manifold with a gener al TM - c onne ction ∇ (not ne c essarily flat nor c onstant torsion). A sso ciate d with this ge ometric sp ac e is a natur al p ost-Lie-R inehart algebr a which is governing the algebr a of flows of ve ctor fields and ge o desics on M . W e give a brief intr o duction and r efer to [1] for details. Define the tensor algebr a of ve ctor fields T p X p M qq : “ R ‘ X p M q ‘ p X p M q d X p M qq ‘ p X p M q d X p M q d X p M qq ‘ ¨ ¨ ¨ “ 8 à r “ 0 T r 0 , wher e X p M q d X p M q : “ X p M q b R X p M q . W e fol low [48, 18] and extend the c onne ction to an R -line ar pr o duct Ź on T p X p M qq , as the Guin-Odoum c onstruction: for φ P R , X , Y P X p M q and W 1 , W 2 P T p X p M qq let X Ź Y “ ∇ X Y , φ Ź W 1 “ φW 1 , X Ź p W 1 d W 2 q “ p X Ź W 1 q d W 2 ` W 1 d p X Ź W 2 q , p X d W 1 q Ź W 2 “ X Ź p W 1 Ź W 2 q ´ p X Ź W 1 q Ź W 2 . The pr o duct Ź is R multi-line ar in left ar gument and R line ar in the right. L et ∆ : T p X p M qq Ñ T p X p M qq b T p X p M qq denote the de-shuffle c o-pr o duct and define the asso- ciative Gr ossman—L arson pr o duct ˚ : T p X p M qq b T p X p M qq Ñ T p X p M qq as A ˚ B “ ÿ ∆ p A q A p 1 q d ` A p 2 q Ź B ˘ . Then W 1 Ź p W 2 Ź W 3 q “ p W 1 ˚ W 2 q Ź W 3 . 8 W e c al l D p M q : “ p T p X p M qq , d , Ź , ∆ , ˚q the cov arian t deriv ation algebra on p M , ∇ q . It acts natur al ly as higher or der c ovariant derivations on the tensor bund les T r s over M . The primitive elements g “ t W P D p M q : ∆ p W q “ W b I ` I b W u acts as first or der c ovariant derivations, in p articular g acts on R “ T 0 0 as derivations, which defines an anchor map ρ : g Ñ Der p R, R q “ X p M q . The lifte d c onne ction Ź on g is flat with p ar al lel torsion, henc e p g , r´ , ´s , Źq is a p ost-Lie algebr a with r A, B s “ A d B ´ B d A , and p g , J ´ , ´ K q is a Lie algebr a with J A, B K “ A ˚ B ´ B ˚ A “ r A, B s ` A Ź B ´ B Ź A. The tuple p R, L “ g , ρ, Źq is a tr acial p ost-Lie-R inehart algebr a with enveloping algebr a D p M q . The exp onential with r esp e ct to d yields pul lb ack series along ge o desics, while the exp onential with r esp e ct to ˚ yields pul lb ack series along the exact flow of a ve ctor field. It might b e a surprise that a gener al TM -c onne ction ∇ with torsion T ∇ and curvatur e R ∇ lifts to a flat p ost-Lie c onne ction Ź on g . The curvatur e R ∇ and torsion T ∇ app e ar thr ough a split exact se quenc e of Lie algebr as. L et r´ , ´s J b e the Jac obi br acket on ve ctor fields X p M q and for X , Y P X p M q define the curvatur e form s p X , Y q : “ J X , Y K ´ r X , Y s J , thus s is a tensor of mixe d typ e s P T 2 2 ‘ T 1 2 . Then R ∇ p X , Y q Z “ s p X, Y q Ź Z . L et I “ x s p X, Y q : X , Y P X p M qy b e the two-side d ide al of the Gr ossman—L arson Lie algebr a p g , J ´ , ´ K q gener ate d by the curvatur e forms. Then ther e is a split short exact se quenc e 0 I g X p M q 0 . ρ i Thus the G-L Lie algebr a p g , J ´ , ´ K q de c omp oses in a semi dir e ct pr o duct g – I ¸ X p M q , wher e the left p art c ontains the curvatur e forms and the right p art the torsion forms of ∇ . A s an example, we have for X , Y P X p M q that ρ pr X , Y sq “ ´ T ∇ p X , Y q and p 1 ´ ρ qpr X , Y sq “ s p X, Y q . W e r efer to [1] and forthc oming p ap ers for mor e details. Example 4.3. L et M n b e an n -dimensional r e al manifold and U b e an op en subset of M n . L et R “ C 8 p U q and L “ Γ p T M | U q b e the sp ac e of lo c al se ctions of the tangent bund le r estricte d to U with nonholonomic b asis p e 1 , e 2 , . . . , e n q e quipp e d with a flat c onne ction with p ar al lel torsion ∇ : L b 2 Ý Ñ L p X , Y q ÞÝ Ñ X Ź Y , such that for al l i, j, k “ 1 , 2 , . . . , n , r e i , e j s “ n ÿ k “ 1 c k ij e k , 9 wher e c k ij ar e structure constants and e i Ź e j “ ∇ e i p e j q “ n ÿ k “ 1 Γ k ij e k , wher e Γ k ij ar e connection co efficien ts . The torsion tensor is henc e c ompute d as T p e i , e j q “ n ÿ k “ 1 p Γ k ij ´ Γ k j i ´ c k ij q e k . Thus p C 8 p U q , Γ p T U q , ∇ q is a p ost-Lie-Rinehart algebr a and for al l X , Y P L with X “ ř n i “ 1 X i e i and Y “ ř n j “ 1 Y j e j wher e X i , Y j P R : X Ź Y “ n ÿ j “ 1 ˜ ÿ i “ 1 X i p e i .Y j q ¸ e j ` n ÿ k “ 1 ˜ n ÿ i “ 1 n ÿ j “ 1 X i Y j Γ k ij ¸ e k . Thus for a ve ctor field Y “ ř n j “ 1 Y j e j , the c orr esp onding endomorphism d Y maps e i ÞÝ Ñ n ÿ j “ 1 ` e i .Y j ˘ e j ` n ÿ k “ 1 ˜ n ÿ j “ 1 Y j Γ k ij ¸ e k . Ther efor e the diver genc e of Y is given by div p Y q “ τ p d Y q “ n ÿ i “ 1 ` e i .Y i ˘ ` n ÿ i “ 1 ˜ n ÿ j “ 1 Γ i ij Y j ¸ . Prop osition 4.4. In a p ost-Lie-Rinehart algebr a p R, L q , the algebr a of elementary R -mo dule endomorphisms, El R p L q is gener ate d by t dX , δ X : X P L u . Pr o of. The pro of is done by induction on the length n of arbitrary generators of El R p L q of the form ˆ ∇ Y n ˆ ∇ Y n ´ 1 ¨ ¨ ¨ ˆ ∇ Y 1 dX and ˆ ∇ Y n ˆ ∇ Y n ´ 1 ¨ ¨ ¨ ˆ ∇ Y 1 δ X for all X , Y 1 , Y 2 , . . . , Y n P L . (Base Case) n “ 1 : F or X , Y , Z P L (i) p ˆ ∇ Y dX qp Z q “ ∇ Y p dX p Z qq ´ dX p ∇ Y Z q “ Y Ź p Z Ź X q ´ p Y Ź Z q Ź X “ Z Ź p Y Ź X q ´ p Z Ź Y q Ź X ` T p Z , Y q Ź X “ ! d p Y Ź X q ´ dX ˝ d Y ` dX ˝ δ Y ) p Z q “ ! d p Y Ź X q ´ dX ˝ p d Y ´ δ Y q ) p Z q . (ii) p ˆ ∇ Y δ X qp Z q “ ∇ Y p δ X p Z qq ´ δ X p ∇ Y p Z qq “ ∇ Y p T p Z , X qq ´ T p ∇ Y Z, X q “ Y Ź T p Z , X q ´ T p Y Ź Z, X q “ T p Z , Y Ź X q “ δ p dX p Y qqp Z q . 10 T o chec k for n “ 2 , ˆ ∇ Y 2 ˆ ∇ Y 1 dX “ ˆ ∇ Y 2 p d p Y Ź X q ´ dX ˝ p d Y ´ δ Y q p from Case n “ 1 q p 3 . 6 q “ ˆ ∇ Y 2 d p Y Ź X q ´ ˆ ∇ Y 2 dX ˝ p d Y ´ δ Y q , where all the terms are generated by t dX , δ X : X P L u in view of the case n “ 1 . Successively , the cases for n “ 2 , 3 , . . . can b e pro ven in the same wa y . F or, ˆ ∇ Y 2 ˆ ∇ Y 1 δ X “ ˆ ∇ Y 2 p δ p dX p Y 1 qq , and is reduced to Case n “ 1 . Lik ewise, the cases for n “ 2 , 3 , . . . can iteratively b e reduced to the Case n “ 1 . 5 Planar aromatic trees and the free tracial p ost-Lie-Rinehart algebra This section gives a brief review of planar ro oted trees and introduces planar aromatic trees, whic h build up the explicit description of the free tracial p ost-Lie-Rinehart algebra. 5.1 Planar ro oted trees and free p ost-Lie-Rinehart algebra The planar ro oted trees are a conv enien t graphical representation of the free magma [44]. Definition 5.1. A planar tr e e, gather e d in the set T , is define d by induction: P T , p t 1 ¨ ¨ ¨ t p q ñ P T , t i P T , wher e p t 1 ¨ ¨ ¨ t p q ñ is an abstr act notation for the or der e d list of sub-tr e es t 1 , . . . , t p . The notation stands for a vertex of the tr e e, that ar e arbitr arily lab ele d (with differ ent lab els). The set of vertic es V p t q of a tr e e t “ p t 1 ¨ ¨ ¨ t p q ñ c ontains the r o ot and the vertic es of its subtr e es t k . The numb er of vertic es of a tr e e, denote d | t | is c al le d the or der of the tr e e. L et C b e a finite set of de c or ations, also c al le d c olours in numerics. A de c or ate d planar tr e e is a tuple p t, ϕ q with a tr e e t and a map ϕ : V Ý Ñ C . The de c or ate d tr e es ar e gather e d in the set T C and the sp ac e T C “ Span p T C q . Similarly, the subset/subsp ac e of de c or ate d planar tr e es of or der N ar e denote d T N C and T N C . The index C is omitted for examples and numerical applications, where one is mainly inter- ested in one colour C “ t u . Planar trees are drawn with the ro ot at the b ottom. The planar trees of order up to five in T are the following: ; ; , ; , , , , ; , , , , , , , , , , , , , . W e emphasize that the following trees are differen t as they are defined by induction as “ p ¨ p ñ qq ñ , “ pp ñ q ¨ q ñ . The left grafting product ñ : T C ˆ T C Ñ T C is defined on trees and then extended by bilinearit y as t 2 ñ t 1 “ ÿ v P V p t 1 q t 2 ñ v t 1 , 11 where t 2 ñ v t 1 attac hes the ro ot of t 2 to the no de v of t 2 on the left. F or instance, we find ñ “ ` ` ` . Let us consider the free Lie algebra spanned by decorated planar trees p Lie p T C q , r´ , ´sq . The grafting pro duct is extended on Lie p T C q by t 3 ñ r t 2 , t 1 s “ r t 3 ñ t 2 , t 1 s ` r t 2 , t 3 ñ t 1 s , r t 3 , t 2 s ñ t 1 “ Ass ñ p t 3 , t 2 , t 1 q ´ Ass ñ p t 2 , t 3 , t 1 q . Prop osition 5.2 ([44]) . The sp ac e p Lie p T C q , r´ , ´s , ñ q is the fr e e p ost-Lie algebr a over the set C , that is, for every set map F : C Ý Ñ L wher e p L, r´ , ´s , Źq is a p ost-Lie algebr a, ther e exists a unique p ost-Lie (epi-)morphism F : Lie p T C q Ý Ñ L extending the map F . C p L, r´ , ´s , Źq p Lie p T C q , r´ , ´s , ñ q F ι F Remark 5.3. L et us now c onsider the universal enveloping algebr a U p Lie p T C qq , e quipp e d with its pr o duct. It natur al ly is a c o c ommutative Hopf algebr a with the unshuffle c opr o duct ∆  and is isomorphic to the Hopf algebr a p T p T C q , ¨ , ∆  q by the Cartier-Quil len-Milnor-Mo or e the or em [37]. The gr afting pr o duct is extende d to T p T C q using the (p ost-Lie) Guin-Oudum c onstruction [48, 18] and p T p T C q , ⊛ , ∆  q , wher e ⊛ is the Gr ossman-L arson pr o duct. The enveloping algebr a of a p ost-Lie algebr a endows a c ointer acting structur e. The tuple p T p T C q , ¨ , ∆  q is a Hopf algebr a in the c ate gory of p T p T C q , ⊛ , ∆  q -c omo dules. This c ointer acting structur e is terme d as a p ost- Hopf algebr a [33]. Note that this c onstruction is c omp atible with the notation for the inductive definition of planar tr e es in Definition 5.1. The Hopf algebr as asso ciate d to planar tr e es have b e en extensively studie d in the numeric al liter atur e [24, 46, 34, 19], wher e they ar e use d for r epr esenting the T aylor exp ansion of Lie-gr oup metho ds. F or the Lie algebra Lie p T C q , the left adjoint Lie morphism for t P Lie p T C q is giv en b y δ t : Lie p T C q Ý Ñ End p Lie p T C qq η ÞÝ Ñ r t, η s Let T 0 C b e the set of decorated planar trees with one free edge. W e denote ˆ T 0 C the subset of decorated planar trees with the free edge attached to the ro ot. F or instance, we find P T 0 t , u , P ˆ T 0 t , u Ă T 0 t , u . The trees with a free edge can b e seen as endomorphisms in End p Lie p T C qq that act on trees by grafting on the free edge, : ÞÑ The (endomorphism) algebra generated by the Lie p T C q (via the left adjoint δ ¨ ) and T 0 C is defined b y ˝ “ , ˝ δ “ ´ . 12 This gives us in particular : r , s ÞÑ ´ . This yields that T 0 C “ Span p T 0 C q equipped with the comp osition ˝ is a subalgebra of p End p T C q , ˝q . Moreo ver, the subalgebra p T 0 C , ˝q is generated b y p ˆ T 0 C , ˝q (with ˆ T 0 C “ Span p ˆ T 0 C q ) as there is a unique wa y to decomp ose an element of T 0 C in to a comp osition of elemen ts of ˆ T 0 C : “ ˝ ˝ . Let the map d : T C Ñ T 0 C that sums ov er all p ossible w ays to graft from the left a free edge to the input: d “ ` ` ` . (5.1) Lemma 5.4. The algebr a of elementary mo dule morphisms El k p Lie p T C qq for the p ost-Lie- R inehart algebr a p k , Lie p T C q , r´ , ´s , ñ q is the algebr a gener ate d by ˆ T 0 C , the sub algebr a of tr e es with a fr e e e dge at the r o ot and the Lie sub algebr a δ p Lie p T C qq . Pr o of. This follows from (mimicking) Prop osition 4.4. The grafting op eration ñ on Lie p T C q is extended as the action of the Lie algebra on the trees with a free edge ñ : Lie p T C q ˆ T 0 C Ý Ñ T 0 C as for all t 1 , t 2 P Lie p T C q and t P T 0 C as t 1 ñ a “ ÿ v P V p a q t 1 ñ v t, r t 1 , t 2 s ñ t “ Ass ñ p t 1 , t 2 , t q ´ Ass ñ p t 2 , t 1 , t q . (5.2) The grafting of a planar tree t 1 on a planar tree with a free edge t is the sum of all p ossible grafting t 1 on all the v ertices of t . The important remark is that the fr e e e dge of a is not disturb e d at al l. F or example, one finds ñ “ ` ` . 5.2 Planar aromatic trees and free tracial p ost-Lie-Rinehart algebras Definition 5.5. A planar ar oma is a list p t 1 , . . . , t n q ö with t i P ˆ T 0 C that has cyclic invarianc e: p t 1 , . . . , t n q ö “ p t 2 , . . . , t n , t 1 q ö “ ¨ ¨ ¨ “ p t n , t 1 , . . . , t n ´ 1 q ö . W e imp ose the c onvention that the fr e e e dge in e ach t p is indic ate d with a cr oss ˆ . L et A C b e the symmetric algebr a gener ate d by planar ar omas, whose elements ar e c al le d multi-ar omas. The A C -mo dule of ar omatic tr e es is AT C “ A C b T C and the A C -Lie algebr a is Lie A C p AT C q “ A C b Lie p T C q e quipp e d with the A C -biline ar br acket r´ , ´s . These ve ctor sp ac es ar e gr ade d by the numb er of vertic es. 13 An example of planar aroma is the following: p ˆ , ˆ , ˆ q ö “ p ˆ , ˆ , ˆ q ö “ p ˆ , ˆ , ˆ q ö . In the Euclidean setting where there are no planar features, the lo cation of the cross would not matter. W e added the notation ˆ for the sake of clarity as we shall no w extend the map d to aromas, which shall yield additional free edges. The planar aromatic trees are the main ob ject of in terest for our numerical interests. F or a P A C , t P Lie p T C q , the asso ciated aromatic tree a b t is written as at for simplicity . The grafting pro duct is extended by the Leibniz rule: ñ : Lie A C p AT C q ˆ Lie A C p AT C q Ñ Lie A C p AT C q p a 1 t 1 q ñ p a 2 t 2 q “ a 1 a 2 p t 1 ñ t 2 q ` a 1 p t 1 ñ a 2 q t 2 , where the grafting of t P Lie p T C q on a multiaroma satisfies t ñ 1 “ 0 , t ñ p t 1 , . . . , t p q ö “ p t ñ t 1 , . . . , t p q ö ` ¨ ¨ ¨ ` p t 1 , . . . , t ñ t p q ö , r t 1 , t 2 s ñ a “ Ass ñ p t 1 , t 2 , a q ´ Ass ñ p t 2 , t 1 , a q , t ñ p a 1 a 2 q “ p t ñ a 1 q a 2 ` a 1 p t ñ a 2 q . The asso ciated anchor map is ρ : Lie A C p AT C q Ñ Der p A C q , ρ p x qp y q “ x ñ y . The following prop osition is an embo diment of the discussion so far in this subsection Prop osition 5.6. The sp ac e of planar ar omatic for ests p A C , Lie A C p AT C q , r´ , ´s , ρ, ñ q is a p ost-Lie-R inehart algebr a. Sa y the tree t 1 is with a free edge (other than the crossed free edge defining the aroma), then w e ha ve an aroma with a free edge. p t 1 , . . . , t p q ö : t P Lie p T C q Ñ p t 1 p t q , . . . , t p q ö . Here t 1 p t q denotes the grafting of t on the non-crossed free edge of t 1 . F or instance, we find p ˆ , ˆ , ˆ q ö p q “ p ˆ , ˆ , ˆ q ö , where we recall that the free edges with a cross are the one defining the aroma and cannot b e grafted up on. Thus hereafter unless otherwise mentioned, free edges for aromatic planar trees should imply that edges are non-crossed. Let Lie 0 A C p AT C q b e the Lie algebra of aromatic trees where exactly one aroma or one tree has a free edge. The algebra p Lie 0 A C p AT C q , ˝q forms a subalgebra of p End A C p Lie A C p AT C qq , ˝q b y imp osing A C -linearit y . The map d extends on aromatic trees by the Leibniz rule: d : Lie A C p AT C q Ñ Lie 0 A C p AT C q Ă End A C p Lie A C p AT C qq , d p at q “ d p a q t ` ad p t q , where we extend the map d defined in (5.1) on multiaromas and commutators b y d p 1 q “ 0 , 14 d p a 1 a 2 q “ d p a 1 q a 2 ` a 1 d p a 2 q , d pp t 1 , . . . , t n q ö q “ p d p t 1 q , . . . , t n q ö ` ¨ ¨ ¨ ` p t 1 , . . . , d p t n qq ö , d pr t 1 , t 2 sq “ r d p t 1 q , t 2 s ` r t 1 , d p t 2 qs . Here d p t 1 q is the sum of all p ossible graftings of a free edge from the left on to all the vertices of the planar tree t 1 and the the cr osse d fr e e e dge is not disturb e d at al l. F or example, d p ˆ q “ ˆ Lemma 5.7. Extend δ t o Lie A C p AT C q by A C -line arity. Then, the elementary endomorphisms of the p ost-Lie-R inehart algebr a of planar ar omatic for ests ar e given by El A C p Lie A C p AT C qq “ Lie 0 A C p AT C q ‘ δ p Lie A C p AT C qq . Using the unique decomp osition of elemen ts of T 0 C in to comp ositions of elements of ˆ T 0 C , we define the trace and the divergence maps. Examples of computations are presented in T able 1 and App endix A. Definition 5.8. The tr ac e is the A C -line ar map τ : El A C p Lie A C p AT C qq Ñ A C that vanishes on vanishes on δ p Lie A C p AT C qq and is given on Lie 0 A C p AT C q by the fol lowing. If the fr e e e dge is on a tr e e t : τ p t q “ p t 1 , . . . , t n q ö , t “ t 1 ˝ ¨ ¨ ¨ ˝ t n , t 1 , . . . , t n P ˆ T 0 C . If the fr e e e dge is on an ar oma: τ pp t 1 , . . . , t n q ö t q “ p t 1 , . . . , t n q ö p t q . If the fr e e e dge is in a c ommutator: τ pr t 1 , t 2 sq “ ˆ τ pr t 1 , t 2 sqp id q , wher e the auxiliary map ˆ τ satisfies ˆ τ p t 1 ˝ ¨ ¨ ¨ ˝ t n qp δ x q “ p t 1 , . . . , t n ˝ δ x q ö t, t 1 , . . . , t n P ˆ T 0 C , x P Lie p T C q , ˆ τ pr t 1 , t 2 sqp δ x q “ ˆ τ p t 1 qp δ x ˝ δ t 2 q , t 1 P Lie 0 p T C q , t 2 P Lie p T C q . The diver genc e map is then define d by div : Lie A C p AT C q Ñ A C , div “ τ ˝ d. Planar aromatic trees yield an imp ortan t example of tracial p ost-Lie-Rinehart algebra. Prop osition 5.9. The tuple p A C , Lie A C p AT C q , r´ , ´s , ρ, ñ , τ q is a k -tr acial p ost-Lie-R inehart algebr a. The main result of this section is that the Lie algebra of planar aromatic trees is a free ob ject, whic h generalises [20] to the p ost-Lie con text. Theorem 5.10. The fr e e k -tr acial p ost-Lie-Rinehart algebr a over the set C is the sp ac e of planar ar omatic tr e es p A C , Lie A C p AT C q , r´ , ´s , ρ, ñ , τ q , that is, for every set map F : C Ý Ñ L 15 t P Lie A p AT q d p t q div p t q p ˆ q ö ` p ˆ , ˆ q ö ` p ˆ q ö p ˆ q ö p ˆ q ö ` p ˆ q ö p ˆ q ö p ˆ q ö ` p ˆ q ö ` ` p ˆ , ˆ , ˆ q ö ` p ˆ , ˆ q ö ` p ˆ q ö ` ` p ˆ , ˆ q ö ` p ˆ , ˆ q ö ` p ˆ q ö p ˆ q ö p ˆ q ö ` p ˆ q ö ` p ˆ q ö p ˆ q ö p ˆ , ˆ q ö ` p ˆ q ö p ˆ q ö ` p ˆ q ö p ˆ q ö p ˆ q ö ` p ˆ q ö ` p ˆ q ö p ˆ q ö p ˆ q ö ` p ˆ q ö ` p ˆ q ö p ˆ q ö p ˆ q ö ` p ˆ q ö ` p ˆ q ö p ˆ q ö p ˆ q ö ` p ˆ q ö ` p ˆ q ö p ˆ , ˆ q ö p ˆ , ˆ q ö ` 2 p ˆ , ˆ q ö p ˆ , ˆ q ö p ˆ q ö ` 2 p ˆ , ˆ q ö p ˆ q ö p ˆ q ö p ˆ q ö p ˆ q ö ` 2 p ˆ q ö p ˆ q ö p ˆ q ö p ˆ q ö p ˆ q ö ` 2 p ˆ q ö p ˆ q ö r , s r , s ` r , s ` r , s p ˆ , ˆ q ö ´ p ˆ , ˆ q ö ` p ˆ q ö ´p ˆ q ö ` p ˆ q ö ´ p ˆ q ö T able 1: Divergence of elements of Lie A p AT q of order up to three (see also App endix A). wher e p R , L, r´ , ´s , ρ, ▷ , τ q is a k -tr acial p ost-Lie-Rinehart algebr a, ther e exists a unique (epi- )morphism of k -tr acial p ost-Lie-Rinehart algebr a F : p A C , Lie A C p AT C qq Ý Ñ p R , L q extending the map F . C p R, L, r´ , ´s , ρ, ▷ , τ q p A C , Lie A C p AT C q , r´ , ´s , ρ, ñ , τ q F ι F Let us now presen t a concise pro of of Theorem 5.10. W e refer to [53] for a fully detailed pro of of this result. Pr o of. As Lie p T C q is the free p ost-Lie algebra, there exists a unique p ost-Lie algebra morphism F : Lie p T C q Ñ L (see Prop osition 5.2). F ollowing the pro of of [44], it also naturally yields a unique algebra morphism ψ : El k p Lie p T C qq Ñ El R p L q , whic h naturally induces the following map (analogously to the pre-Lie-Rinehart case [20]): ˆ ψ : El k p Lie p T C qq{r T 0 C , T 0 C s ˝ Ñ El R p L q{r El R p L q , El R p L qs ˝ , where r u, v s ˝ “ u ˝ v ´ v ˝ u . W e th us obtain F : T 0 C {r T 0 C , T 0 C s ˝ Ñ R, F “ ˆ τ ˝ ˆ ψ , where we factor the trace ˆ τ : El R p L q{r El R p L q , El R p L qs ˝ Ñ R . As T 0 C {r T 0 C , T 0 C s ˝ is exactly the space of planar aromas, this defines F on aromas, and one thus extends F to the symmetric 16 algebra of multiaromas A C as an algebra morphism F : A C Ñ R . Then, we obtain the unique k -tracial p ost-Lie-Rinehart algebra F b y F : p A C , Lie A C p AT C qq Ñ p R, L q , F p at q “ F p a q F p t q . Hence the result. 6 Numerical preserv ation of div ergence-free features on mani- folds 6.1 P ost-Lie-Rinehart structure of the connection algebra Consider a manifold M on k equipp ed with a frame basis p E d q , defined globally for simplicit y 1 . The manifold is naturally equipp ed with the W eitzenbö c k connection, that we write as a pro duct on the v ector fields X p M q : Y Ź X “ Y r x i s E i , X “ x i E i , where we use the Einstein summation notation. The connection Ź is naturally curv ature-free R “ 0 , but do es not hav e constant torsion in general. Thus, w e further assume that the p E d q span a Lie algebra, or equiv alently that M is lo cally a Lie group, or equiv alently that w e hav e constan t torsion ∇ T “ 0 (see [47]). Let the Jacobi brac ket J ´ , ´ K J and the brack et derived from the torsion r´ , ´s “ ´ T , giv en explicitly for the W eitzenbö c k connection by r X , Y s “ x i y j J E i , E j K J , X “ x i E i , Y “ y j E j . Note that the tw o brack ets are link ed b y the identit y J X , Y K J “ r X , Y s ` X Ź Y ´ Y Ź X . Then, the connection algebra p X p M q , r´ , ´s , Źq is a p ost-Lie algebra [44]. A p ost-Lie-Rinehart structure on the C 8 p M q -mo dule X p M q is con venien tly defined when using the W eitzenbö ck connection. Let the anchor map ρ p X qp ϕ q “ X r ϕ s , X “ x i E i , ϕ P C 8 p M q , the trace τ p u q “ u i i , u P End C 8 p M q p X p M qq , u p E j q “ u i j E i , and the div ergence div p X q “ E i r x i s , X “ x i E i . Prop osition 6.1. A ssume that Ź has c onstant torsion, then p C 8 p M q , X p M q , r´ , ´s , ρ , Ź , τ q is a k -tr acial p ost-Lie-R inehart algebr a. Remark 6.2. The T aylor exp ansions of numeric al metho ds ar e not expr esse d dir e ctly in X p M q but r ather in the enveloping algebr a of X p M q , e quipp e d with the fr ozen pr o duct ¨ . F our our choic e of c onne ction, the fr ozen pr o duct ¨ acts on ve ctor fields and functions as the differ ential op er ator p X 1 ¨ ¨ ¨ X p q Ź Y “ pp X 1 ¨ ¨ ¨ X p q Ź y j q E j , p X 1 ¨ ¨ ¨ X p q Ź ϕ “ x i 1 1 . . . x i p p E i 1 r . . . E i p r ϕ s . . . s . It is showe d in p articular in [18, 7] that the universal enveloping algebr a of p ost-Lie algebr as (r esp e ctively p ost-Lie-Rinehart algebr as) ar e, under te chnic al assumptions, p ost-Hopf algebr as (r esp e ctively p ost-Hopf algebr oids). 1 The op erations used in Lie-group metho ds only require a lo cal frame family (see [49, 6]). 17 6.2 Div ergence-free vector fields, volume-preserv ation, and planar aromatic trees Let a smo oth ordinary differential equation on M led by a divergence-free v ector field y 1 p t q “ F p y p t qq , y p 0 q “ y 0 , F “ x i E i , div p F q “ 0 , (6.1) where F is a smo oth Lipschitz vector field. The divergence characterises geometric prop erties in sp ecific cases. Prop osition 6.3. A ssume that M is c omp act and e quipp e d with a Riemannian bi-invariant metric and the c orr esp onding volume form d vol . L et the volume of a me asur able set Ω of M b e V ol p Ω q “ ş Ω d vol . W e say that a flow φ t is volume-pr eserving if for al l me asur able Ω and al l t ą 0 , we have V ol p φ t p Ω qq “ V ol p Ω q . Then, the flow of (6.1) is volume-pr eserving if and only if div p F q “ 0 . Pr o of. The preserv ation of volume is characterised by the divergence for the Levi-Civita con- nection [32, Liouville’s theorem], which coincides with the W eitzenbö c k divergence div in the case of a bi-inv ariant metric. Example 6.4. Consider the differ ential e quation on the sp e cial ortho gonal gr oup SO d p R q : y 1 p t q “ A p y p t qq y p t q , A : SO d p R q Ñ so d p R q . Given a fr ame b asis of right-invariant ve ctor fields E i p y q “ A i y and p A i q b asis of so d p R q , the W eitzenb ö ck c onne ction b e c omes the standar d right-invariant c onne ction on SO d p R q . If A p y q “ f i p y q A i and E i r f i s “ 0 , then the flow pr eserves volume. Remark 6.5. Finding inte gr ators that pr eserve volume in a gener al ge ometric c ontext is a chal lenging op en pr oblem, for which this work is a first step. Inde e d, this r e quir es the design of a high-or der analysis and the study of algebr aic structur es for r epr esenting the c onne ction algebr a for a gener al c onne ction. W e cite in p articular the r e c ent works [55, 56, 41] that intr o duc e inte gr ators and algebr aic to ols for c onne ctions with vanishing torsion and c onstant curvatur e. The gener al c ase wil l b e studie d in futur e works. Planar aromatic trees represent sp ecific vector fields and functions through the application of the elemen tary differen tial map. Definition 6.6. Given a smo oth ve ctor field F P X p M q , the elementary differ ential map is define d on Lie k p T q by F F p q “ F , F F pp t 1 ¨ ¨ ¨ t p q ñ q “ p F F p t 1 q ¨ ¨ ¨ F F p t p qq Ź F , F F pr t 1 , t 2 sq “ r F F p t 1 q , F F p t 2 qs , wher e we use the fr ozen pr o duct of R emark 6.2. The map extends to T 0 by F F p ˆ qp X q “ X , F F pp t 1 ¨ ¨ ¨ t p q ñ qp X q “ p F F p t 1 qp X q ¨ ¨ ¨ F F p t p qp X qq Ź F , and on ar omas by F F p 1 q “ 1 , F F pp t 1 , . . . , t n q ö q “ F F p t 1 q i 1 p E i 2 q ¨ ¨ ¨ F F p t n q i n p E i 1 q . Final ly, F F extends into F F : Lie A p AT q Ñ X p M q as a morphism F F p a 1 . . . a n t q “ F F p a 1 q . . . F F p a n q F F p t q . 18 Prop osition 6.7. The elementary differ ential F F is a morphism of p ost-Lie-Rinehart algebr as: F F : p A , Lie A p AT q , r´ , ´s , ρ, ñ , τ q Ñ p C 8 p M q , X p M q , r´ , ´s , ρ, Ź , τ q . In addition, F F c ommutes with the diver genc e: div ˝ F F “ F F ˝ div . The divergence-free assumption of F yields degeneracies. Lemma 6.8. A ssume that div p F q “ 0 . Then, the differ ential asso ciate d to the fol lowing ar omas vanishes: F F pp t 1 ¨ ¨ ¨ t p ¨ ˆq ñ q ö q “ 0 . The degeneracies of Lemma 6.8 prov e useful in the design of efficient pseudo-divergence-free metho ds as it allo ws to use only first deriv atives to represen t some differen tial op erators of higher order in general. In particular, one obtains with div p F q “ 0 , F F ppˆ ¨ t q ñ q ö q “ J E i , F F p t q K J r f i s , where we recall that J E i , F F p t q K J is a v ector field and th us acts as a first order differen tial op erator. 6.3 Div ergence-free Lie-group metho ds with aromatic Lie-Butc her series Our aim is to create Lie-group metho ds [24] that satisfy similar prop erties as the flow of (6.1). Exact volume-preserv ation is an op en problem already in the Euclidean setting, so that we prop ose a metho dology for the design of pseudo-divergence-free metho ds. Let the exp onen tial exp p tX q p for X P X p M q b e the solution of the ODE y 1 p t q “ X p y p t qq , y p 0 q “ p. F reezing the comp onen ts at a p oin t q P M yields the frozen exp onen tial exp p tx d p q q E d q p , which giv es the geo desics for the connection Ź . W e consider Lie-Runge-Kutta metho ds of the form Y i “ exp p h ÿ j a ij f d p Y j q E d q y n , y n ` 1 “ exp p h ÿ i b i f d p Y i q E d q y n , (6.2) where A P R s ˆ s and b P R s are the co efficien ts of the metho ds, h is the timestep of the metho d, and s is the num ber of stages. Our approach extends straightforw ardly to RKMK metho ds [38, 43, 39, 40] and frozen-flo w metho ds [16, 50, 12, 49], and we restrict our approac h to the metho ds (6.2) for simplicity . As exact volume preserv ation is imp ossible for metho ds of the form (6.2) (see [15, 25]), we design prepro cessors to hav e a high-ord er of volume-preserv ation with a lo w order of conv ergence, similarly to the metho d prop osed in [3]. W e recall in particular that the metho ds (6.2) cannot b e of order more than tw o in general. Lie-group metho ds are naturally describ ed by planar Butcher trees and forests. 19 Definition 6.9. A Lie-Butcher (LB) series is a formal series indexe d by planar tr e es B F p a q “ ÿ t P T a p t q F F p t q , a P T ˚ . A nalo gously, an ar omatic LB series is a formal series indexe d by planar ar omatic tr e es B F p a q “ ÿ t P AT a p t q F F p t q , a P AT ˚ . Thanks to Prop osition 6.3, the backw ard error analysis of Lie-group metho ds allows us to c haracterise metho ds that preserve the features of the system (6.1) (see also [23, 14, 11]). Prop osition 6.10 ([35, 52]) . Consider a metho d of the form (6.2) . Then, its T aylor exp ansion c oincides with the one of the exact flow of the mo difie d ODE y 1 p t q “ ˜ F h p y p t qq , h ˜ F h “ B hF p b q , wher e the mo difie d ve ctor field ˜ F h is given by a LB series. Mor e gener al ly, if the metho d (6.2) is applie d to a pr epr o c esse d ve ctor field h ˆ F h “ B hF p b q given by an ar omatic LB series, then the mo difie d ve ctor field of the r esulting metho d ˜ F h is given by a LB series. The numeric al metho d is c al le d diver genc e-fr e e if its mo difie d ve ctor field satisfies div p ˜ F h q “ 0 and is c al le d pseudo-diver genc e-fr e e of or der p if div p ˜ F h q “ O p h p q . Example 6.11. Consider the Lie-Euler metho d y n ` 1 “ exp p hf d p y n q E d q y n . (6.3) Then, the first terms of its mo difie d ve ctor field ar e h ˜ F h “ F hF ´ ´ 1 2 ` 1 3 ` 1 12 ´ 1 12 r , s ` . . . ¯ . The combinations of planar aromatic forests of order up to four of v anishing divergence div p Ψ k q “ 0 are listed b elo w. Note that we reco ver the solenoidal forms from [28, 27] when remo ving the commutators and using non-planar aromatic trees. Ψ 1 “ ` p ˆ q ö ´ p ˆ q ö ´ p ˆ , ˆ q ö ` r , s , Ψ 2 “ ` ` p ˆ q ö ´ p ˆ q ö ´ p ˆ , ˆ q ö ´ p ˆ , ˆ , ˆ q ö ` r , s , Ψ 3 “ ` ` ` p ˆ q ö ´ ´ p ˆ q ö ´ p ˆ , ˆ q ö ´ p ˆ , ˆ q ö ` r , s , Ψ 4 “ p ˆ q ö ` p ˆ q ö ` p ˆ q ö p ˆ q ö ´ p ˆ q ö ´ p ˆ q ö p ˆ q ö ´ p ˆ q ö p ˆ , ˆ q ö ` p ˆ q ö r , s , Ψ 5 “ ` p ˆ q ö ` p ˆ q ö ` p ˆ , ˆ q ö ´ ´ p ˆ q ö ´ p ˆ q ö ´ p ˆ , ˆ q ö ` r , s ` r , s ` p ˆ q ö r , s ` r , r , ss . Our analysis with planar aromatic trees allo ws us to tac kle the tedious calculations for deriving pseudo-divergence-free metho ds of high order. In particular, we prop ose the following Lie-R unge-Kutta metho ds, where we use prepro cessed aromatic vector fields that only require the first deriv ative of the f d . 20 Prop osition 6.12. Consider the Lie-Euler metho d (6.3) for solving (6.1) with div p F q “ 0 . A pply the metho d with the pr epr o c esse d ve ctor field h ˆ F h “ F hF ´ ` 1 2 ´ 1 3 ´ 1 12 p ˆ q ö ´ 1 12 p ˆ , ˆ q ö ` 1 6 r , s ¯ “ hf i E i ` h 2 2 f j E j r f i s E i ´ h 3 3 f k E k r f j s E j r f i s E i ´ h 3 12 f k J E j , E k K J r f j s f i E i ´ h 3 12 E j r f k s E k r f j s f i E i ` h 3 6 f k f j E j r f i s J E k , E i K J . Then, the r esulting metho d has or der two of c onver genc e and is pseudo-diver genc e-fr e e of thir d or der. Prop osition 6.13. Consider the Lie-Runge-K utta metho d (6.2) with the fol lowing Butcher table au for solving (6.1) with div p F q “ 0 . 0 0 0 0 ´ 1 ` ? 5 12 ´ 1 ` ? 5 12 0 0 3 ` ? 5 12 ´ 9 ´ 5 ? 5 12 2 ` ? 5 2 0 ´ 7 ` 3 ? 5 2 0 9 ´ 3 ? 5 2 A pply the inte gr ator to the pr epr o c esse d ve ctor field h ˆ F h “ F hF ´ ´ 1 12 ` 1 ´ ? 5 24 p ˆ q ö ` 1 ´ ? 5 24 p ˆ , ˆ q ö ` 1 ` ? 5 24 r , s ` 1 8 ´ 1 12 ` ´ ˘ ` 1 36 ` p ˆ q ö ´ 2 p ˆ q ö ˘ ` 1 12 ` p ˆ , ˆ q ö ´ p ˆ , ˆ q ö ˘ ` 1 18 p ˆ , ˆ , ˆ q ö ´ 1 12 r , s ` 1 18 r , r , ss ¯ “ hF ` h 3 ´ ´ 1 12 f k E k r f j s E j r f i s E i ` 1 ´ ? 5 24 f k J E j , E k K J r f j s f i E i ` 1 ´ ? 5 24 E j r f k s E k r f j s f i E i ` 1 ` ? 5 24 f k f j E j r f i s J E k , E i K J ¯ ` h 4 ´ 1 8 f l E l r f k s E k r f j s E j r f i s E i ´ 1 12 f l f k E k r f j s J E l , E j K J r f i s E i ` 1 36 f l f k JJ E j , E l K J , E k K J r f j s f i E i ` 1 12 f l E j r f k s J E k , E l K J r f j s f i E i ` 1 18 E j r f l s E l r f k s E k r f j s f i E i ´ 1 12 f l f k E k r f j s E j r f i s J E l , E i K J ` 1 18 f l f k f j E j r f i s E i J E l , J E k , E i K J K J ¯ . Then, the r esulting ar omatic metho d has or der two of c onver genc e, but is pseudo-diver genc e-fr e e of fourth or der. 7 Conclusion In this pap er, we defined planar aromatic trees and show ed that they are the free tracial p ost-Lie- Rinehart algebra. This algebraic b ject finds concrete applications in the numerical integration of ODEs on manifolds for the design of pseudo-divergence-free maps via backw ard error analysis. 21 This work op ens sev eral research a ven ues that we will explore in future works. First, the c haracterisation and existence of divergence-free in tegrators is an imp ortant open problem of geometric integration, even for Euclidean ODEs . This calls for the generalisation of the works [4, 27, 28, 17] to the manifold case. The understanding of divergence-free maps for ODEs shows strong links with the discretisations of ergodic sto c hastic differen tial equations that sample the in v arian t measure exactly , with numerous applications in molecular dynamics, stochastic optimisation, and mac hine learning. This link is uncov ered in [30, 5] in R D and in [6] through the use of exotic Butc her series and extensions, so that generalising the present w ork to the stochastic con text is natural. On the geometric side, it w ould b e interesting to characterise planar aromatic B-series with univ ersal geometric prop erties in the spirit of [42, 36, 29]. On the algebraic side, the structures asso ciated to aromas were recently studied from v arious p oin t of views in [28, 7, 58] with p ossible applications b ey ond numerical analysis and combinatorial algebra. Moreov er, w e recall from [34, 21] that post-Lie algebras are asso ciated to the connection algebra in a context of constan t torsion and v anishing curv ature, which is not the natural framework for working with v olume forms. It is imp ortan t to generalise the concept of aromas to more general connection algebras in order to obtain general statemen ts for v olume-preserv ation on Riemannian manifolds. A ckno wledgemen ts. The authors ackno wledge the supp ort of the F renc h program ANR- 25-CE40-2862-01 (MaStoC - Manifolds and Sto c hastic Computations), the Researc h Council of Norw ay through pro ject 302831 “Computational Dynamics and Sto c hastics on Manifolds” (COD YSMA), and the UiT–MaSCoT pro ject at UiT, T romsø. The first author w ould like to thank Pauline Baudat for an enlightening discussion that inspired our notation of planar aro- mas. 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Aromatic and clump ed m ulti-indices: algebraic structure and Hopf embeddings. arXiv pr eprint arXiv:2603.13105 , 2026. 25 App endices A Computations on planar aromatic trees of order four t P Lie A p AT q div p t q p ˆ , ˆ , ˆ , ˆ q ö ` p ˆ , ˆ , ˆ q ö ` p ˆ , ˆ q ö ` p ˆ q ö p ˆ , ˆ , ˆ q ö ` p ˆ , ˆ , ˆ q ö ` p ˆ , ˆ q ö ` p ˆ q ö p ˆ , ˆ , ˆ q ö ` p ˆ , ˆ q ö ` p ˆ , ˆ q ö ` p ˆ q ö p ˆ , ˆ q ö ` p ˆ , ˆ , ˆ q ö ` p ˆ , ˆ q ö ` p ˆ q ö p ˆ , ˆ q ö ` p ˆ , ˆ q ö ` p ˆ , ˆ q ö ` p ˆ q ö p ˆ q ö p ˆ q ö p ˆ , ˆ , ˆ q ö ` p ˆ q ö p ˆ , ˆ q ö ` p ˆ q ö p ˆ q ö ` p ˆ q ö p ˆ q ö p ˆ q ö p ˆ , ˆ q ö ` p ˆ q ö p ˆ , ˆ q ö ` p ˆ q ö p ˆ q ö ` p ˆ q ö p ˆ q ö p ˆ q ö p ˆ , ˆ q ö ` p ˆ q ö p ˆ q ö ` p ˆ q ö ` p ˆ q ö p ˆ q ö p ˆ q ö p ˆ , ˆ q ö ` p ˆ q ö p ˆ q ö ` p ˆ q ö ` p ˆ q ö p ˆ , ˆ q ö p ˆ , ˆ q ö p ˆ , ˆ q ö ` p ˆ , ˆ q ö p ˆ q ö ` 2 p ˆ , ˆ q ö p ˆ q ö p ˆ q ö p ˆ q ö p ˆ q ö p ˆ , ˆ q ö ` p ˆ q ö p ˆ q ö p ˆ q ö ` 2 p ˆ q ö p ˆ q ö p ˆ q ö p ˆ q ö p ˆ q ö ` p ˆ q ö ` p ˆ q ö ` p ˆ q ö p ˆ q ö p ˆ q ö p ˆ q ö ` p ˆ q ö ` p ˆ q ö ` p ˆ q ö p ˆ q ö p ˆ q ö p ˆ q ö ` p ˆ q ö ` p ˆ q ö ` p ˆ q ö p ˆ q ö p ˆ q ö p ˆ q ö ` p ˆ q ö ` p ˆ q ö ` p ˆ q ö p ˆ q ö p ˆ q ö p ˆ q ö ` p ˆ q ö ` p ˆ q ö ` p ˆ q ö p ˆ , ˆ q ö p ˆ , ˆ q ö p ˆ q ö ` p ˆ , ˆ q ö ` p ˆ , ˆ q ö ` p ˆ , ˆ q ö p ˆ , ˆ q ö p ˆ , ˆ q ö p ˆ q ö ` p ˆ , ˆ q ö ` p ˆ , ˆ q ö ` p ˆ , ˆ q ö p ˆ , ˆ , ˆ q ö p ˆ , ˆ , ˆ q ö p ˆ q ö ` 3 p ˆ , ˆ , ˆ q ö p ˆ q ö p ˆ q ö p ˆ q ö p ˆ q ö p ˆ q ö ` p ˆ q ö p ˆ q ö ` p ˆ q ö p ˆ q ö ` p ˆ q ö p ˆ q ö p ˆ q ö p ˆ q ö p ˆ q ö p ˆ q ö p ˆ q ö ` p ˆ q ö p ˆ q ö ` p ˆ q ö p ˆ q ö ` p ˆ q ö p ˆ q ö p ˆ , ˆ q ö p ˆ q ö p ˆ , ˆ q ö p ˆ q ö p ˆ q ö ` p ˆ , ˆ q ö p ˆ q ö ` 2 p ˆ , ˆ q ö p ˆ q ö p ˆ q ö p ˆ q ö p ˆ q ö p ˆ q ö p ˆ q ö p ˆ q ö p ˆ q ö ` 3 p ˆ q ö p ˆ q ö p ˆ q ö 26 t P Lie A p AT q div p t q r , s p ˆ , ˆ , ˆ q ö ´ p ˆ , ˆ , ˆ q ö ` p ˆ , ˆ q ö ´ p ˆ , ˆ q ö `p ˆ q ö ´ p ˆ q ö ` p ˆ q ö ´ p ˆ q ö r , s p ˆ , ˆ q ö ´ p ˆ , ˆ q ö ` p ˆ q ö ´p ˆ q ö ` p ˆ q ö ´ p ˆ q ö p ˆ q ö r , s p ˆ q ö p ˆ , ˆ q ö ´ p ˆ q ö p ˆ , ˆ q ö ` p ˆ q ö p ˆ q ö ´ p ˆ q ö p ˆ q ö `p ˆ q ö p ˆ q ö ´ p ˆ q ö p ˆ q ö ` p ˆ q ö ´ p ˆ q ö r , r , ss p ˆ q ö ` p ˆ q ö ` p ˆ q ö ` p ˆ q ö ´ 2 p ˆ q ö ´ 2 p ˆ q ö `p ˆ q ö ` p ˆ q ö ´ 2 p ˆ q ö ` p ˆ q ö ` p ˆ q ö ´ 2 p ˆ q ö 27