[Paper Review] A Non-associative Baker-Campbell-Hausdorff formula

Jacob Mostovoy, José M. Pérez-Izquierdo+1 more|arXiv (Cornell University)|May 3, 2016

Advanced Topics in Algebra10 references3 citations

TL;DR

This paper develops a non-associative version of the Baker-Campbell-Hausdorff (BCH) formula using Shestakov-Umirbaev primitive operations in non-associative algebras. It introduces a non-associative Dynkin-Specht-Wever Lemma and a recursive Magnus expansion to compute the BCH series up to degree 4, recovering the classical formula when associativity is imposed.

ABSTRACT

We address the problem of constructing the non-associative version of the Dynkin form of the Baker-Campbell-Hausdorff formula; that is, expressing $\log (\exp (x)\exp(y))$, where $x$ and $y$ are non-associative variables, in terms of the Shestakov-Umirbaev primitive operations. In particular, we obtain a recursive expression for the Magnus expansion of the Baker-Campbell-Hausdorff series and an explicit formula in degrees smaller than 5. Our main tool is a non-associative version of the Dynkin-Specht-Wever Lemma. A construction of Bernouilli numbers in terms of binary trees is also recovered.

Motivation & Objective

• To extend the classical Baker-Campbell-Hausdorff formula to non-associative algebras by generalizing the Dynkin-Specht-Wever Lemma.
• To define and compute the non-associative logarithm of the product of two non-associative exponentials using primitive operations.
• To provide a recursive method for computing the non-associative BCH series using a generalized Magnus expansion.
• To recover known structures such as Bernoulli numbers via binary trees in the non-associative setting.
• To establish a framework for non-associative Lie theory based on Sabinin algebras and universal enveloping algebras.

Proposed method

• Introduce a non-associative exponential map $\exp_l(x) = \sum_{n \geq 0} \frac{1}{n!} (\cdots((xx)x)\cdots)x $ in a free non-associative algebra on $x$ and $y$.
• Define the non-associative logarithm $\log_l(\exp_l(x)\exp_l(y))$ as a series in Shestakov-Umirbaev primitive operations.
• Use a non-associative version of the Dynkin-Specht-Wever Lemma to express the logarithm in terms of cuts of monomials.
• Apply a generalized Magnus expansion to derive a differential equation for $\Omega(t)$ satisfying $X'(t) = X(t)A(t)$, leading to a recursive formula.
• Characterize BCH-cuts as decompositions of monomials into branches of the form $x^i y^j$, enabling coefficient computation via combinatorial sums.
• Compute coefficients using $c_\tau = \frac{B_\tau}{\tau!}$, where $B_\tau$ are coefficients from $\log_l(1+x)$, and sum over all valid BCH-cuts.

Experimental results

Research questions

• RQ1 How can the Baker-Campbell-Hausdorff formula be generalized to non-associative algebras where associativity fails?
• RQ2 What is the non-associative analog of the Dynkin-Specht-Wever Lemma, and how does it facilitate the expansion of $\log_l(\exp_l(x)\exp_l(y))$?
• RQ3 Can the Magnus expansion be adapted to non-associative settings to recursively compute the BCH series?
• RQ4 What is the explicit form of the BCH series up to degree 4 in non-associative variables?
• RQ5 How do Bernoulli numbers emerge in the context of non-associative algebras and binary trees?

Key findings

• The non-associative BCH formula is explicitly computed up to degree 4, yielding $\mathrm{BCH}_l(x,y) = x + y + \frac{1}{2}[x,y] + \frac{1}{12}[x,[x,y]] - \frac{1}{3}\langle x;x,y\rangle - \cdots$.
• When all non-associative operations vanish, the formula reduces exactly to the classical Baker-Campbell-Hausdorff series.
• The coefficient of $x^m y^n$ in $\log_l(\exp_l(x)\exp_l(y))$ is $\frac{1}{m!n!}$ for $n \geq 2$, and $\frac{m}{(m+1)!}$ for $n = 1$.
• The coefficient of $x^2y$ is $\frac{1}{3}$, computed as $\frac{c_x}{2} + \frac{c_{x^2}}{2} + c_{x^2x} = \frac{1}{2} - \frac{1}{4} + \frac{1}{12}$.
• The coefficient of $x(xy)$ is $-\frac{1}{4}$, derived from $c_{x^2} + c_{xx^2} = -\frac{1}{2} + \frac{1}{4}$.
• The paper recovers a construction of Bernoulli numbers via binary planar rooted trees in the non-associative setting.