Baker-Campbell-Hausdorff polynomials and Menelaus theorem

This is a continuation of the previous post on the noncommutative BCH formula. For the “Menelaus theorem” part see this post.

Everything is related to “noncommutative techniques” for approximate groups, which hopefully will apply sometimes in the future to real combinatorial problems, like the Tao’ project presented here, and also to the problem of understanding curvature (in non-riemannian settings), see a hint here, and finally to the problem of higher order differential calculus in sub-riemannian geometry, see this for a comment on this blog.

Remark: as everything this days can be retrieved on the net, if you find in this blog something worthy to include in a published paper, then don’t be shy and mention this. I believe strongly in fair practices relating to this new age of scientific collaboration opened by the www, even if in the past too often ideas which I communicated freely were taken in published papers without attribution. Hey, I am happy to help! but unfortunately I have an ego too (not only an ergobrain, as any living creature).

For the moment we stay in a Lie group , with the convention to take the exponential equal to identity, i.e. to consider that the group operation can be written in terms of Lie brackets according to the BCH formula:

x y = x + y + \frac{1}{2} [x,y] + \frac{1}{12}[x,[x,y]] - \frac{1}{12}[y,[y,x]]+...

For any \varepsilon \in (0,1] we define

x \cdot_{\varepsilon} y = \varepsilon^{-1} ((\varepsilon x) (\varepsilon y))

and we remark that x \cdot_{\varepsilon} y \rightarrow x+y uniformly with respect to x,y in a compact neighbourhood of the neutral element e=0. The BCH formula for the operation labeled with \varepsilon is the following

x \cdot_{\varepsilon} y = x + y + \frac{\varepsilon}{2} [x,y] + \frac{\varepsilon^{2}}{12}[x,[x,y]] - \frac{\varepsilon^{2}}{12}[y,[y,x]]+...

Let us define the BCH functions. We start with

BCH^{0}_{\varepsilon} (x,y) = x \cdot_{\varepsilon} y

and BCH^{0}_{0}(x,y) = \lim_{\varepsilon \rightarrow 0} BCH^{0}_{\varepsilon}(x,y) = x + y.

Define the “linearized dilation\delta^{x}_{\varepsilon} y = x + \varepsilon (-x+y) (written like this on purpose, without using the commutativity of the “+” operation; due to limitations of my knowledge to use latex in this environment, I am shying away to put a bar over this dilation, to emphasize that it is different from the “group dilation”, equal to x (\varepsilon(x^{-1}y))).

Consider the family of \beta > 0 such that there is an uniform limit w.r.t. x,y in compact set of the expression

\delta_{\varepsilon^{-\beta}}^{BCH^{0}_{\varepsilon}(x,y)}  BCH^{0}_{0}(x,y)

and remark that this family has a maximum \beta = 1. Call this maximum \alpha_{0} and define

BCH^{1}_{\varepsilon}(x,y) = \delta_{\varepsilon^{-\alpha_{1}}}^{BCH^{0}_{\varepsilon}(x,y)}  BCH^{0}_{0}(x,y)

and BCH^{1}_{0}(x,y) = \lim_{\varepsilon \rightarrow 0} BCH^{1}_{\varepsilon}(x,y).

Let us compute BCH^{1}_{0}(x,y):

BCH^{1}_{0}(x,y) = x + y + \frac{1}{2}[x,y]

and also remark that

BCH^{1}_{\varepsilon}(x,y) = x+y + \varepsilon^{-1} ( -(x+y) + (x \cdot_{\varepsilon} y)).

We recognize in the right hand side an expression which is a relative of what I have called in the previous post an “approximate bracket”, relations (2) and (3). A better name for it is a halfbracket.

We may continue indefinitely this recipe. Namely for any natural number i\geq 1 we first define the maximal number \alpha_{i} among all \beta > 0 with the property that the (uniform) limit exists

\lim_{\varepsilon \rightarrow 0} \delta_{\varepsilon^{-\beta}}^{BCH^{i}_{\varepsilon}(x,y)}  BCH^{i}_{0}(x,y)

Generically we shall find \alpha_{i} = 1. We define then

BCH^{i+1}_{\varepsilon}(x,y) = \delta_{\varepsilon^{-\alpha_{i}}}^{BCH^{i}_{\varepsilon}(x,y)}  BCH^{i}_{0}(x,y)

and BCH^{i+1}_{0}(x,y) = \lim_{\varepsilon \rightarrow 0} BCH^{i+1}_{\varepsilon}(x,y).

It is time to use Menelaus theorem. Take a natural number N > 0. We may write (pretending we don’t know that all \alpha_{i} = 1, for i = 0, ... N):

x \cdot_{\varepsilon} y = BCH^{0}_{\varepsilon}(x,y) = \delta^{BCH^{0}_{0}(x,y)}_{\varepsilon^{\alpha_{0}}} \delta^{BCH^{1}_{0}(x,y)}_{\varepsilon^{\alpha_{1}}} ... \delta^{BCH^{N}_{0}(x,y)}_{\varepsilon^{\alpha_{N}}} BCH^{N+1}_{\varepsilon}(x,y)

Let us denote \alpha_{0} + ... + \alpha_{N} = \gamma_{N} and introduce the BCH polynomial PBCH^{N}(x,y)(\mu) (the variable of the polynomial is \mu), defined by: PBCH^{N}(x,y)(\mu) is the unique element of the group with the property that for any other element z (close enough to the neutral element) we have

\delta^{BCH^{0}_{0}(x,y)}_{\mu^{\alpha_{0}}} \delta^{BCH^{1}_{0}(x,y)}_{\mu^{\alpha_{1}}} ... \delta^{BCH^{N}_{0}(x,y)}_{\mu^{\alpha_{N}}} z = \delta^{PBCH^{N}(x,y)(\mu)}_{\mu^{\gamma_{N}}} z

Such an element exists and it is unique due to (Artin’ version of the) Menelaus theorem.

Remark that PBCH^{N}(x,y)(\mu) is not a true polynomial in \mu, but it is a rational function of \mu which is a polynomial up to terms of order \mu^{\gamma_{N}}. A straightforward computation shows that the BCH polynomial (up to terms of the mentioned order) is a truncation of the BCH formula up to terms containing N-1 brackets, when we take \mu =1.

It looks contorted, but written this way it works verbatim for normed groups with dilations! There are several things which are different in detail. These are:

1. the coefficients \alpha_{i} are not equal to 1, in general. Moreover, I can prove that the \alpha_{i} exist (as a maximum of numbers \beta such that …) for a sub-riemannian Lie group, that is for a Lie group endowed with a left-invariant dilation structure, by using the classical BCH formula, but I don’t think that one can prove the existence of these numbers for a general group with dilations! Remark that the numbers \alpha_{i} are defined in a similar way as Hausdorff dimension is!

2. one has to define noncommutative polynomials, i.e. polynomials in the frame of Carnot groups (at least). This can be done, it has been sketched in a previous paper of mine, Tangent bundles to sub-riemannian groups, section 6.

UPDATE: (30.10.2011) See the post of Tao

Associativity of the Baker-Campbell-Hausdorff formula

where a (trained) eye may see the appearance of several ingredients, in the particular commutative case, of the mechanism of definition of the BCH formula.

The associativity is rephrased, in a well known way,  in proposition 2 as a commutativity of say left and  right actions. From there signs of commutativity (unconsciously assumed) appear:  the obvious first are the “radial  homogeneity  identities”, but already at this stage a lot of familiar  machinery is put in place and the following is more and more heavy of  the same. I can only wonder:  is this  all necessary? My guess is: not. Because for starters, as explained here and in previous posts, Lie algebras are of a commutative blend, like the BCH formula. And (local, well known from the beginning) groups are not.

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