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The Baker-Campbell-Hausdorff formula and the Zassenhaus formula in synthetic differential geometry

After the torch of Anders Kock [Taylor series calculus for ring objects of line type, Journal of Pure and Applied Algebra, 12 (1978), 271-293], we will establish the Baker-Campbell-Hausdorff formula as well as the Zassenhaus formula in the theory of Lie groups.


Introduction
The Baker-Campbell-Hausdorff formula (the BCH formula for short) was first discovered by Campbell ([2] and [3]) on the closing days of the 19th century so as to construct a Lie group directly from a given Lie algebra (i.e., Lie's third fundamental theorem !). However, his investigation failed in convergence problems, let alone dealing only with matrix Lie algebras. The BCH formula was finally established by Baker [1] and Hausdorff [6] independently within a somewhat more abstract framework of formal power series on the dawning days of the 20th century, getting rid of convergence problems completely while losing touch with the theory of Lie groups. The BCH formula resurrected its touch with the theory of Lie groups thanks to Magnus [10] in the middle of the 20th century.
The BCH formula claims, roughly speaking, that the multiplication in a Lie group is already encoded in its Lie algebra. More precisely, the multiplication in a Lie group is expressible in terms of Lie brackets in its Lie algebra, which readily gives rise to Lie's second fundamental theorem in the theory of finite-dimensional Lie groups, though the modern treatment of the theory of finite-dimensional Lie groups is liable to base Lie's second fundamental theorem somewhat opaquely upon the Frobenius theorem.
The so-called Taylor formula was introduced by the English mathematician called Brook Taylor in the early 18th century, though its pedigree can be traced back even to Zeno in ancient Greece. Kock [7] has shown that the nature of the Taylor formula in differential calculus is more combinatorial or algebraic than analytical, dodging convergence problems completely, as far as we are admitted to speak on the infinitesimal level, where nilpotent infinitesimals are available in plenty. The principal objective in this paper is to do the same thing to the BCH formula and its inverse companion called the Zassenhaus formula in the theory of Lie groups, though we must confront the noncommutative world in sharp contrast to the Taylor formula living a commutative life. We have found out that the Zassenhaus formula is much easier to deal with than the BCH formula itself, albeit, historically speaking, the former having been found out by Zassenhaus [20] within an abstract framework of formal power series more than three decades later than the latter and its continuous counterpart having been established by Fer [4] four years later than [10]. Quirky enough, our BCH formula diverges from the usual one in the 4-th order. The BCH formula will be dealt with in §7 and §8 by two different methods, while we will be engaged upon the Zassenhaus formula in §6. We approach the BCH formula in anticipation of its validity in §7 by using only the left logarithmic derivative of the exponential mapping, while we will do so from scratch in §8 by using both of the left and right logarithmic derivatives of the exponential mapping. As is expected, the latter proofs are longer than the former ones.
We will work within the framework of synthetic differential geometry as in [9]. We assume the reader to be familiar with Chapters 1-3 of [9]. Now we fix our terminology and notation. Given a microlinear space M , we denote M D by TM , while we denote the tangent space of M at x ∈ M by T x M = {γ ∈ TM | γ (0) = x}. Given a mapping f : M → N of microlinear spaces, its differential is denoted by df , which is a mapping from TM to TN , assigning f • γ ∈ TN to each γ ∈ TM . We denote the identity mapping of M by id M . The unit element of a group G is usually denoted by e. In the proof of a theorem or the like, we insert some comment surrounded with parentheses )(. θ (0) = e and θ(t + d) = θ (t) .ς (t) d for any t ∈ R and any d ∈ D.
From now on, we will assume the Lie group G to be regular, so that exp G : g → G is indeed a total function. Notation 18 Given ξ ∈ gl (V ) with ξ n+1 vanishing for some natural number n, we write It is easy to see that Lemma 19 Given ξ ∈ gl (V ) with ξ n+1 vanishing for some natural number n, we have exp GL(V ) ξ = e ξ Proposition 20 Given a homomorphism ϕ : G → H of Lie groups and X ∈ g, exp H ϕ ′ (X) is defined, and we have

Remark 21
The Lie group G is assumed to be regular, as we have said before, but the Lie group H is not assumed to be regular, so that exp H is not necessarily a total function.
Proof. It suffices to note that, given a one-parameter subgroup θ : Proposition 22 Given X ∈ g with (ad X) n+1 vanishing for some natural number n, we have Ad (exp X) = e ad X Proof. We have )By Proposition 20( = e ad X )By Lemma 19( We conclude this section by the following simple but significant proposition.

Proposition 23
We have exp t (dX) = X td for any t ∈ R. In particular, we have Proof. For any d ′ ∈ D, we have so that we have the desired conclusion. 7

Logarithmic Derivatives
In this section we deal with the left and right derivations. First we deal with the left derivation.
Definition 24 Given a microlinear space M and a function f : M → G, the function δ left f : TM → g is defined to be such that The following is the Leibniz rule for the left logarithmic derivation.
Proposition 25 Let M be a microlinear space. Given two functions Proof. For any d ∈ D, we have so that we get the desired formula.
Theorem 26 Given X ∈ g with (ad X) n+1 vanishing for some natural number n, we have Proof. The proof is essentially on the lines of Lemma 4.27 of [12]. We have [By the chain rule of differentiation] so as to introduce a function by fixing s and differentiaing with respect to t at t = 0. Since we have also we get F ′ (0) = id g , by letting s = 0, so that the formula (1) is transmogrified into the ordinary differential equation F ′ (s) = id g − (ad X) (F (s)) on L (g, g). Its unique solution with the initial condition of F (0)'s vanishing is which results in the desired formula by letting s = 1.
In particular, we have Letting so as to get a function H : R → G, we have H (0) = e evidently. By differentiating H logarithmically, we have so that we have the desired formula.
Proposition 28 Given X, Y ∈ g and d 1 , d 2 ∈ D, we have )By Theorem 26( )By Proposition 23( , while we have by the same token. Therefore we have By multiplying from the right and making use of Proposition 27, we get the desired formula. Now we deal with the right derivation.
Definition 29 Given a microlinear space M and a function f : M → G, the function δ right f : TM → g is defined to be such that for any x ∈ M , any X ∈ T x M and any d ∈ D. It is called the right logarithmic derivative of f . The restriction of δ right f to T x M is denoted by δ right f (x).
Proposition 30 Let M be a microlinear space. Given two functions f, g : M → G together with X ∈ TM , we have Theorem 31 Given X ∈ g with (ad X) n+1 vanishing for some natural number n, we have

The Zassenhaus Formula
Lemma 32 Given d 1 , ...d n ∈ D, we have for any natural number m with m ≤ n.
Theorem 33 Given X, Y ∈ g and d 1 ∈ D, we have )By Proposition 23( so that we have got to the desired formula.
Theorem 34 Given X, Y ∈ g and d 1 , d 2 ∈ D, we have Proof. We have )By Theorem 26( so that we have got to the desired formula.
Theorem 35 Given X, Y ∈ g and d 1 , d 2 , d 3 ∈ D, we have Proof. We have )left logarithmic derivation( )By Theorem 26( )By Theorems 33 and 34( By moving exp d 3 X left towards exp (d 1 + d 2 ) X via Propositions 27 and 28 By moving exp d 3 Y left towards exp (d 1 + d 2 ) Y via Propositions 27 and 28 so that we have got to the desired formula.
14 Theorem 36 Given X, Y ∈ g and d 1 , Proof. We have By Theorems 33 and 35( via Propositions 27 and 28 via Propositions 27 and 28 16 We keep on: via Propositions 27 and 28 so that we have got to the desired formula. We could keep on, but the complexity of computation increases rapidly.

The First Approach to the Baker-Campbell-Hausdorff Formula
The following result is no other than Theorem 33 itself.
Theorem 37 Given X, Y ∈ g and d 1 ∈ D, we have Corollary 38 Given X 1 , ..., X n ∈ g and d 1 ∈ D, we have Proof. By simple induction on n.
Theorem 39 Given X, Y ∈ g and d 1 , d 2 ∈ D, we have Proof. We have )By Theorem 26( )By Proposition 27( so that we get the desired formula by multiplying from the right and making use of Proposition 27. Corollary 40 (cf. Theorem 2.12.4 of [19]). Given X 1 , ..., X n ∈ g and d 1 , d 2 ∈ D, we have Proof. Here we deal only with the case of n = 3, leaving the general treatment by induction on n to the reader. We note in passing that the case of n = 2 is no other than Theorem 39 itself. We have so that we are done.
Theorem 41 Given X, Y ∈ g and d 1 , Proof. We have .
)By Proposition 28( )By repeated use of Proposition 27( so that we get the desired formula by multiplying from the right and making use of Proposition 27.
Theorem 42 Given X, Y ∈ g and d 1 , Proof. We have . .

22
We keep on By moving exp d 4 X left towards exp (d 1 + d 2 + d 3 ) X via Propositions 27 and 28 By exchanging exp d 1 d 4 [Y, X] and exp d 2 Y via Proposition 28 and using Proposition 27 By moving exp d 3 Y left towards exp (d 1 + d 2 ) Y via Propositions 27 and 28 )By Lemma 10( so that we get the desired formula by multiplying from the right and making use of Proposition 27. We could keep on, but the complexity of computation increases rapidly.

The Second Approach to the Baker-Campbell-Hausdorff Formula
Theorem 43 Given X, Y ∈ g and d 1 ∈ D, we have Proof. By Proposition 27.
Theorem 44 Given X, Y ∈ g and d 1 , d 2 ∈ D, we have Proof. We have Theorem 45 Given X, Y ∈ g and d 1 , d 2 , d 3 ∈ D, we have Proof. We have .
Theorem 46 Given X, Y ∈ g and d 1 , d 2 , d 3 , d 4 ∈ D, we have Proof. We have We keep on.
We keep on again.
We keep on once more.
We keep on once more.
We keep on once more.