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Parametric Lie group structures on the probabilistic simplex and generalized Compositional Data

In this paper we build a set of parametric quotient Lie group structures on the probabilistic simplex that can be extended to real vector space structures. In particular, we rediscover the main mathematical objects generally used when treating compositional data as elements associated to the quotient Lie group with respect to the equivalence relation induced by the scale invariance principle. This perspective facilitates the adaptation of the statistical methods used for classical compositional data to data that follows a more general equivalence relation.


Introduction
The unit simplex can be regarded as the set of discrete probability distributions for random variables with a fixed finite number of possible values. It is also used as a model space for Compositional Data analysis ( [5], [11]), for word frequencies in text classification ( [15]), portfolio selection problems ( [16]) and a myriad of other problems. Using the global diffeomorphism property of Exp and Log, we can introduce a scalar multiplication on R N +1 >0 as follows, c x := Exp(cLog(x)). (1) Writing in coordinates, we have for all c ∈ R. The set R N +1 >0 endowed these two operations becomes a real vector space (R N +1 >0 , ⊕, ). The probabilistic open simplex is a constraint manifold, which also has an Abelian group structure : This group operation, also called perturbation, lays at the core of the Compositional Data analysis methods. It was also used together with the Fisher Riemannian metric on the simplex in the problem of metric learning for text classification in [15].
A first observation is that (∆ 0 N , * ) is not a subgroup of (R N +1 >0 , ⊕). A natural question that arises is how can one geometrically explain the group operation * in relation with the group operation ⊕.
We prove that (∆ 0 N , * ) can be regarded as a quotient Lie group structure of (R N +1 >0 , ⊕) relative to the one-dimensional subgroup H = {Exp(t1) = (. . . , e t , . . . )| t ∈ R}. Furthermore, this structure can be extended to a vector space one. This allows, for the first time to give a Lie group explanation for several of the main objects encountered in Compositional Data analysis. For example, the Centered-Log ratio transformation is non other than the Lie logarithmic map of the quotient structure (∆ 0 N , * ) (for details, see sub-section 3.1). It also facilitates a Lie group interpretation to the famous Boltzmann distribution, also known as softmax function in the machine learning community. In section 2, we give a follow up of the above constructions to the case where the Lie group structure and the associated objects depend on a parameter. In particular we obtain a parametric family of distances on the simplex. Parametric families of distances are an important tool in metric learning problems ( [14], [21]). Our idea is to embed this parameter dependence into the family of one-dimensional subgroups, with respect to which we construct the quotient structure on the probabilistic simplex.
In Section 3, we detail the link between the quotient Lie group structure induced by H and the Aitchison geometry. Sometimes the nature of the data might require a different quotient geometry, like for example data that comes from exponential decay phenomena [20]. In Section 4, we extend the main principles of Compositional Data analysis to this more general framework. Finally, in Section 5, we present the adaptation of some basic objects of descriptive statistics and of the multivariate normal distribution to the probabilistic simplex endowed with this new geometric structures.

The probabilistic simplex as a quotient Lie group
From general theory of Lie groups, we have that any one-dimensional subgroup is of type It is easy to see that those are also all the one-dimensional linear subspaces of (R N +1 >0 , ⊕, ). The one-dimensional subgroup H a is a normal subgroup of the Lie group (R N +1 >0 , ⊕) and consequently, we obtain the quotient Lie group R N +1 >0 /H a , with π a : R N +1 >0 → R N +1 >0 /H a being the projection map. The quotient group structure is defined by [x] a a [y] a := π a (x ⊕ y). and we have the following commutative diagram: >0 /H a is the Lie algebra corresponding to the quotient Lie group R N +1 >0 /H a . The maps Log a and Exp a are the exponential and logarithmic maps associated with the quotient group structure. In this case they are also global diffeomorphisms. Furthermore, the quotient group R N +1 >0 /H a can be endowed with a scalar multiplication as in (1) and the corresponding vector structures are compatible with Log a and Exp a , in the sense that these two become linear maps.
Next, we show that for certain choices of a, the simplex ∆ 0 N can be regarded as a model space for the quotient group (R N +1 >0 /H a , a ). The classes defined by the equivalence relation induced by the subgroup H a are the smooth curves xH a = {x ⊕ Exp(ta) = (. . . , x i e tai , . . . and π a (xH a ) = [x] a . In order to conclude, we need the following result. Lemma 1. Let a ∈ R N +1 and H a the associated subgroup defined in (2). The following statements are equivalent: Assume that all the coordinates a i of the vector a are strictly positive. Then, for any x i e ait is a strictly monotone continuous function whose image is (0, +∞). The same type of reasoning applies for a i < 0, i = 1, N + 1.
To see that this condition is also necessary to ensure that each equivalence class intersects the simplex in a unique point, first observe that if there exists a j = 0, then N +1 i=1 e ait > 1, for all t ∈ R and thus, for x = (1, . . . , 1), we obtain xH a ∩ ∆ 0 N = ∅. Next, suppose that there exists i = j such that a i < 0 and a j > 0. Then for all t ∈ R either e ait > 1 or e aj t > 1 which implies e aj t + e ait > 1, for all t ∈ R. Therefore, we can conclude that if there exist i = j such that a i < 0 and a j > 0 then once more, for x = (1, . . . , 1), we get xH a ∩ ∆ 0 N = ∅.
The above Lemma allows us to define the map C a : R N +1 We endow ∆ 0 N with the group structure that makes the bijection s a a homomorphism, i.e.
where in the right side we regard λ 1 and λ 2 as points in ∆ 0 N ⊂ R N +1 >0 . In other words, to obtain the composition of two elements λ 1 and λ 2 in the open simplex, we regard λ 1 and λ 2 as elements of the ambient group (R N +1 >0 , ⊕) and compute the intersection of the class (λ 1 ⊕ λ 2 )H a with ∆ 0 N . In order to render some of the above constructions in a more explicit form, we need to give the analytical construction of the map C a . By definition, C a can be constructed as where the smooth curve h x,a (t) = (. . . , x i e ait , . . . ) defines the class xH a and t a (x) ∈ R is the solution of the equation x i e ait = 1.
In some particular cases, we can solve the above equation for t but considering a general vector a with strictly positive/negative coordinates, the above equation is impossible to solve analytically.
Nevertheless, in order to compute the exponential and logarithm map for the quotient Lie group (∆ 0 N , ⊕ Ca ), we need only the derivative of the homomorphism C a : (R N +1 >0 , ⊕) −→ (∆ 0 N , ⊕ Ca ) computed in the neutral element 1. More precisely, where e aita(1) is the i th coordinate of the neutral element e a of the quotient Lie group (∆ 0 N , ⊕ Ca ) and ∂t a ∂x j (1) is computed using The Implicit Function Theorem applied to equation (4) as follows, Consequently, for the derivative of the homomorphism C a : R N +1 The homomorphism C a : R N +1 >0 −→ ∆ 0 N makes the following diagram commutative, see Appendix, where the Lie algebras are given by As in the case of the Lie group (R N +1 >0 , ⊕), we obtain for the Lie group (∆ 0 N , ⊕ Ca ) that Exp Ca and Log Ca are also global diffeomorphisms. The commutativity of the above diagram leads directly to an explicit formula for Log Ca but not for Exp Ca . Theorem 1. The logarithm map Log Ca : ∆ 0 N → T ea ∆ 0 N of the quotient Lie group structure on ∆ 0 N induced by the subgroup H a has the formula : where e a = C a (1) is the neutral element of (∆ 0 N , ⊕ Ca ). Proof. Note that, for λ ∈ ∆ 0 N , we have C a (λ) = λ, and consequently, . By a straight forward computation, we obtain the announced formula.

The map Log
Ca can be seen as the representation of Log a taking the open simplex as a model space for the quotient R N +1 >0 /H a and using the identification map C a .
In order to compute Exp Ca (ξ), for ξ ∈ T ea ∆ 0 N , we can proceed as follows: Next, we extend the Abelian Lie group (∆ 0 N , ⊕ Ca ) to a vector space structure by defining the following scalar multiplication. Definition 1. Let c ∈ R and λ ∈ ∆ 0 N . The scalar multiplication Ca of λ by c is defined as: c Ca λ := Exp Ca (cLog Ca λ) An easy consequence of this definition is that : To compute the scalar multiplication Ca , an explicit expression for Exp Ca is not needed. Going back to the definition of Log Ca , we have that for any λ ∈ ∆ 0 N , and thus Proposition 1. The simplex ∆ 0 N endowed with the group composition ⊕ Ca and the scalar multiplication Ca forms a R-vector space. Moreover, Exp Ca and Log Ca are linear isomorphisms between (T ea ∆ 0 Having constructed the logarithm map Log Ca as a linear isomorphism and giving a scalar product ·, · on the Lie algebra T ea ∆ 0 N , we can induce a scalar product on (∆ 0 N , ⊕ Ca , Ca ) : and the associated norm λ Ca = Log Ca (λ), Log Ca (λ) .
Theorem 2. We have the following parameterized induced distance functions on the simplex, d Ca : ii) The maps Log Ca and Exp Ca are linear isometries between iii) Choosing the Euclidean scalar product on T ea ∆ 0 N , we obtain the following formula : The Euclidean scalar product is a natural choice on T ea ∆ 0 N because this is an Abelian Lie algebra. Another choice would be the Killing scalar product but this is zero on an Abelian algebra. Consequently, in what follows, we consider the Euclidean scalar product on T ea ∆ 0 N .
Putting together the previous diagrams, we obtain the following depiction of the constructions we have made so far: , it is not a subgroup of (R N +1 >0 , ⊕) and thus the quotient operation ⊕ Ca is not simply the restriction of ⊕ to ∆ 0 N . This further implies that ∆ 0 N is non-linear (not a linear subspace) in the ambient vector space (R N +1 >0 , ⊕, ). As we have seen so far, the quotient group R N +1 >0 /H a exists for any choice of a ∈ R N +1 and we can regard a as a parameter for the quotient structures. Nevertheless, only the choices with a i > 0 (or a i < 0), for all i ∈ 1, N + 1 can accommodate the simplex ∆ 0 N as a model space. The simple observation that H a = H ca , for any scalar c ∈ R shows that the quotient structure depends only on the line [a] 1 and consequently, the quotient structures are parametrized by the real projective space RP N . The simplex ∆ 0 N can be a model space for those quotient structures parameterized by [a] 1 ∈ RP N with a i > 0 (or a i < 0), for all i ∈ 1, N + 1. Everything can be organized in a bundle of groups (Lie groupoid) (see, [7]) with base RP N but this is beyond the purpose of this paper.
A difficulty that we encounter is the computation of the neutral element e a ∈ ∆ 0 N for a given quotient structure parametrized by [a] 1 . Nevertheless, we can solve the converse problem noticing that for any λ ∈ ∆ 0 N , the class [(· · · , lnλ i , · · · )] 1 generates the quotient Lie group structure where λ is the neutral element. We have the following result.
Lemma 2. The relation between the parameter [a] 1 with a i > 0 (or a i < 0), for all i ∈ 1, N + 1 and the neutral element of the quotient Lie group structure it generates on ∆ 0 N is given by the equality

Particular cases
We illustrate the above constructions for some basic choices of a that allow for explicit computations. We start with the basic case a = (1, . . . , 1), that gives a new perspective from a Lie group theory point of view on several objects encountered in the framework of compositional data. a = (1, . . . , 1) i.e., Compositional Data analysis Consider the case a = 1, i.e. a i = 1, i = 1, N + 1. Then

Case
In one of his seminal works [2], Aitchison says that "When we say that a problem is compositional we are recognizing that the sizes of our specimens are irrelevant." This is known as the scale invariance principle. When dealing with classical Compositional Data analysis, this comes down to saying that two specimen vectors v, w ∈ R N +1

>0
are equivalent from a compositional point of view iff v = αw, with α ∈ R + >0 . The above equivalence can be rewritten in terms of proportions as v i v j = w i w j , for all i, j ∈ 1, N + 1. This, of course, corresponds to considering equivalence classes generated by H 1 and consequently, the quotient with respect to H 1 .
In this case it is easy to see that for all x ∈ R N +1 >0 , where · denotes the L 1 norm and t 1 (x) is the solution of equation (4). The map C 1 is known in the Compositional Data analysis literature as the closure operation.
The corresponding quotient group structure can be easily deduced: for all λ, µ ∈ ∆ 0 N , This is what Aitchison has defined as the perturbation operation in [3].
Furthermore, the neutral element e 1 is given by . . . , 1 N + 1 , . . . . Thus, the derivative of C 1 given by (5) becomes: From Theorem 1 it follows that, for all λ ∈ ∆ 0 N : In particular, this formula implies that, up to a constant, the well-known centered log-ratio transformation (see for example: [4], [11]) can actually be seen as the logarithmic map Log C 1 of the quotient Lie group (∆ 0 can be written as: This distance, sometimes called Aitchison distance, first appeared in [5] and [1].
For this particular case of a, we can give an explicit formula for Exp C 1 . This is facilitated by the property that for all ξ ∈ T e1 ∆ 0 DC 1 (1)ξ = 1 N +1 ξ and thus, the commutativity of the diagram (6) implies that : This exponential map is also known as the softmax function encountered in logistic regression and neural networks. To summarise, we give a dictionary between the constructions resulting from the Lie group theory and the ones encountered in compositional data analysis.

Compositional Data
Lie Group Theory closure operation

Case a = (1, . . . , 1, 2)
To highlight the impact of the reference subgroup H a on the quotient group structure and the afferent objects, let us consider a second example. Set a = (1, . . . , 1, 2) and consider the associate subgroup: H a = exp(ta) = (e t , . . . , e t , e 2t )| t ∈ R .
Applying (3) implies, for all x ∈ R N +1 >0 , that the intersection of the equivalence class of x with the simplex is given by : x i e t + x N +1 e 2t = 1.
Thus C a can be explicitly computed. (7) can be written as: x Hence a solution of (7) can be obtained from the positive solution of (8), implying that t a (x) = ln(y + ) and consequently : In this case, the neutral element can be explicitly computed and we have that : where · ea denotes the L 1 norm weighted by e a .
Finally, the associated distance is given by :

Generalised compositional data
As we have seen before, the main concepts and mathematical objects used in compositional data arise naturally in the Lie group theory, when considering the simplex as a space model for the quotient group R N +1 >0 /H 1 . If the nature of the data implies a different link between the components, considering a different equivalence relation, determined by a general H a , might be a more appropriate choice. There are several principles, introduced by Aitchison, that are considered being fundamental in an large part of the compositional data literature. A detailed historical account of these principles and an analysis of their usefulness when studying data is given in [19]. In what follows, we explain what these principles become in the more general framework when the equivalence relation is generated by H a .

Scale invariance
The scale invariance principle used in compositional data corresponds to the equivalence relation determined by H 1 . We give a straightforward generalisation of this principle when the equivalence class is determined by H a . In this case, two specimens v and w belong to the same class determined by H a , i.e. v ∼ w, iff there exists α ∈ R >0 such that v = (w 1 α a1 , . . . , w N +1 α a N +1 ) , for all i, j ∈ 1, N + 1.
If the two specimens are two different instances of a variable X : R → R N +1 >0 , i.e. v = X(τ 1 ) and w = X(τ 2 ), where τ is some parameter describing mass, or volume, time, concentration, etc., then the equivalence relation above can be restated as δ δτ X(τ ) := X(τ ) −1Ẋ (τ ) = a. The operation δ δτ is the logarithmic derivative in the Abelian group (R N +1 >0 , ⊕). This leads to considering the quotient structure with respect to H a . Such situations are encountered in practice, for example, when analyzing the mass evolution of components in a system, see [20]. For instances, when the data comes from an exponential decay dynamic, a situation also described in [20], the Aitchison geometry (case a = 1) might not be the best approach when dealing with general equivalence classes. Another case when a more general geometry is better adaptive is for data in which some components are measured in unit and other, for example, in unit 2 or unit 3 (for instance one component that represents a length and another an area or a volume).
Permutation invariance This principle refers to the fact that the results of an analysis of the data should be invariant under any permutation of the components, as long as the same permutation is applied to the whole data set. This property still holds for the generalized framework corresponding to any a ∈ R N +1 >0 , as soon as the quotient structure is adapted accordingly : if a permutation σ is used on the components of the data, then the proper quotient structure is the one with respect to H σ(a) .

Subcompositional coherence
In what follows, we define a subcomposition procedure in the previously proposed general framework and show that the subcomposition procedure is a linear map between two vector spaces.
We denote by A k : R N +1

>0
→ R k+1 >0 the map that assigns to each x = (x 1 , . . . , x N +1 ) ∈ R N +1 >0 a sub-part of k + 1 elements (x σ1 , . . . , x σ k+1 ) (not necessarily in the initial order) and set a := A k (a). Then, for λ ∈ ∆ 0 N we can define the subcomposition corresponding to the k + 1 elements chosen by A k as : The well definiteness of the operator is a consequence of the fact that ∀x ∈ R N +1 >0 , C a (A k (x)) = C a (A k (C a (x))). Indeed, let x ∈ R N +1 >0 . Then C a (x) = (x 1 e ta(x)a1 , . . . , x N +1 e ta(x)a N +1 ), where t a (x) is the solution of the equation (4) and In other words, we have that A k (C a (x)) is in the same equivalence class as A k (x) with respect to H a and thus by definition C a (A k (C a (x))) = C a (A k (x)).
The fact that Sub is a linear map is now an immediate consequence of this last equality and the fact that C a and A k are themselves linear maps.
Subcompositional coherence principle is generally described as the fact that the results obtained for a sub-part of a compositional data (studied after closure) should remain coherent with the results obtained for the full composition. A classical example of subcompositional incoherence, shows that when compositional data framework is not taken into account, we can obtain a positive correlation between two variables when they are studied as composites of a specimen with n components and a negative correlation when the same variables are studied on a (reclosed) subcomposition of the same original specimen (see for example [5]). Of course, not all results obtained on a compositional data are preserved by the Sub operator. For example, the arithmetic mean on the simplex is subcompositional coherent in the sense that :

Elements of descriptive statistics on the simplex and normal distribution
Centrality for generalized compositional data Once we endow the simplex with a distance d Ca , a natural first notion of centrality that can be considered is the corresponding Fréchet mean. Fréchet means were introduced by M Fréchet in [10] and represent a generalization of the classical expected value to arbitrary metric spaces. For a random variable X, defined on a metric space (M, d), a Fréchet mean is a minimizer of y → E d 2 (y, X) . When there exists a unique minimizer m(X), we say that m(X) is the Fréchet mean of X with respect to d. Aitchison used this notion to define the geometric mean as a measure of central tendency (see for example [2]) for compositional data. A similar result can be obtained in a straightforward way for a more general distance d Ca . Let X be a random variable on ∆ 0 N . Then a Fréchet mean of X with respect to d Ca is a minimizer of the real valued functional: Now, for any λ, µ ∈ ∆ 0 N , the definition of the distance d Ca implies that d Ca (λ, µ) = d(Log Ca (λ), Log Ca (µ)). We also have that, for all λ ∈ ∆ 0 , thus a minimizer of (9) is a Fréchet mean of Log a (X) and therefore, as soon as Log Ca (X) is integrable, we have that m a (X), the Fréchet mean of X with respect to d Ca , is unique and that : This last result can also be seen as a consequence of Proposition 1 of [9] since Log Ca is an isometry. Since the simplex is endowed with additional mathematical structure we can also construct the expected value of a random variable. Let a ∈ R N +1 >0 and denote B a = Exp Ca (B), where B is the Borel σ-algebra of T ea ∆ 0 N . B a is also the smallest σ-algebra containing all the open balls associated to the distance d Ca . Following the same procedure as [9], for each random variable X : (Ω, F, P) → (∆ 0 N , B a ), we can consider the real valued random vector Log Ca (X) : (Ω, F, P) → (T ea ∆ 0 N , B) and obtain that : It is well known that in R N , any integrable random variable Z (ie for which E[Z] exists and is well defined) has a unique Fréchet mean with respect to the euclidean distance d and that m(Z) = E[Z]. This property holds here too, in the sense that when E a [X] is well defined, m a (X) = E a [X].
PCA Proposition 1 implies in particular that Log Ca can be seen as the equivalent of crl map for generalized compositional data. With this is mind, all the statistical methods based on log-ratios used for compositional data can be extended in a straightforward way to generalized compositional data. Take for example principal component analysis (PCA). This is a well known multivariate statistical analysis method, often used for dimensionality reduction. The main idea of PCA is to synthesize a data set consisting in a large number of correlated variables, while retaining as much as possible the variation present in it. To achieve this, PCA computes new uncorrelated variables, called principal components. For more details see [13].
In an Euclidean framework, PCA can be reduced to the study of eigenvalues and eigenvectors of a positive-semidefinite symmetric matrix, and is generally treated in this manner. In the context of generalized compositional data, principal component analysis can be applied on the matrix obtained through a Log Ca transformation of the data set. This is the equivalent of logratio analysis used for classical compositional data.
To highlight the link between the PCA on the image of a data set through Log Ca and an analysis directly on the simplex, it is convenient to look at the variational formulation of the PCA.
Indeed, principal components can be described as a sequence of nested affine sub-spaces of increasing dimension that maximize the variance of the projections or minimize the sum of norms of projection residuals. For example, consider the case of m points x 1 , . . . x m , in R n and for all v ∈ R n denote: A first principal component of the data can be described as S v1 , where v 1 is such that: This formulation is at the root of several generalizations of PCA to non-euclidean framework (like on the Wasserstein space of probability measures over R in [6], for a survey of methods employed in Riemannian settings see [12]). Now consider the equivalent objects on (∆ 0 N , ⊕ Ca , Ca ). Suppose we have λ 1 , . . . λ m ∈ ∆ 0 N and for all µ ∈ ∆ 0 N denote : A first principal component becomes S µ 1 , with µ 1 such that : For i ∈ {1, . . . m}, set x i = Log Ca (λ i ). Let µ ∈ ∆ 0 N and denote v = Log Ca (µ). Using the previous notations and the definition of the distance given by Theorem 2 we have that : . Keeping in mind that µ Ca = Log Ca (µ) = v , we have that And thus, S µ is a first principal component for . This argument can be easily extended to principal components of higher order, showing thus that applying PCA on the data transformed through Log Ca is the same as using it directly on the simplex with the appropriate operations.
Normal distribution on the simplex The vector structure also facilitates the definition of a Gaussian vector on the simplex. We can apply for example the definition of a Gaussian vector on a general vector space given by [8] to define a normal law on the simplex with respect with the structure corresponding to H a . Definition 2. We say that a random vector X : (Ω, F, P) → (∆ 0 N , B a , ⊕ Ca , Ca ) has a normal distribution on the simplex iff, for all λ ∈ ∆ 0 N , λ, X Ca = Log Ca (λ), Log Ca (X) has a normal distribution on R, i.e. λ, X Ca ∼ N (µ λ , σ 2 λ ) for some µ λ , σ λ ∈ R.
Let B = (b 1 , . . . , b N ) be an orthonormal basis of (∆ 0 N , ⊕ Ca , Ca ). Then X is a normal random vector iff there exists a real multivariate normally distributed vector Z = (Z 1 , . . . , Z N ) such that To see this, first observe that Since B is an orthonormal basis, we have that X, λ Ca = N i=1 c i Z i . According to Definition 2, this last term is Gaussian. Therefore we can conclude that any linear combination of components of Z is Gaussian and thus Z is a multivariate normal vector.
Conversely if equality (10) holds, then for all λ ∈ ∆ 0 N , This shows that λ, X Ca is a linear combination of the components of a real normal vector and thus has a normal distribution. When X is defined by (10), its expected value can be written as:   Let µ B , Σ B be the parameters of a normal random vector X with respect to a fixed orthonormal basis B and suppose that Σ B is non-degenerate (of rank N ). Denote T : ∆ 0 N → R N the bijective linear map that associates to each λ ∈ ∆ 0 N its coordinates in the basis B. Define the measure ν T on the Borel sets A ∈ B a , by ν T (A) = ν(T (A)), where ν denotes the Lebesgue measure on R N .
Then the density function of X with respect to ν T is given by: One can see that f X (λ) = f Z (T (λ)), where Z ∼ N (µ B , Σ B ) is a real valued random vector and f Z is its density with respect to the Lebesgue measure on R N . For more details about the density function of a normal random vector on a general vector space, see [8].
For a = 1, setting T as the ilr transformation defined in [18], we obtain the same density function as for the normal distribution defined in [18]. The authors of the same paper also highlight that the normal law on ∆ 0 N , corresponding to this density function is equivalent in terms of probabilities to the additive logistic normal law given by Aitchison in [4].

Appendix
We give a proof for the commutativity of Diagram (6). First, we recall the necessary theoretical elements from Marsden and Ratiu [17].
Let (G, ·) be a Lie group and G its Lie algebra. One can regard the group multiplication as the left action of the group to itself, g · h = L g (h). The corresponding infinitesimal action is defined by (ξ, g) → ξ L (g) := DL g (e G ) ξ, where DL g (e G ) is the differential of the left translation computed at the neutral element e G ∈ G.
The exponential map Exp G : G → G is defined by where γ ξ (1) is the solution at time t = 1 for the system of differential equationṡ γ ξ (t) = ξ L (γ ξ (t)) with initial condition γ ξ (0) = e G . For the case of Lie group (∆ 0 N , ⊕ Ca ), the left translation is defined by the maps composition By differentiation, we obtain the operator equality, and consequently, for any ξ ∈ T ea ∆ 0 N , we have the following computation where v ξ ∈ T 1 R N +1 >0 such that ξ = DC a (1)v ξ .This shows that the left invariant vector fields on (∆ 0 N , ⊕ Ca ) are the push forward of the left invariant vector fields on (R N +1 >0 , ⊕). Using the definition of the exponential map, we obtain the commutativity of Diagram 6.