2-division-free Baker-Campbell-Hausdorff formula

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Definition

The 2-division-free Baker-Campbell-Hausdorff formula is a variant of the Baker-Campbell-Hausdorff formula where the Lie brackets are replaced by an arbitrary binary operation and the coefficient for any product of n terms is 2^{n-1} times its coefficient in the original Baker-Campbell-Hausdorff formula. In particular, it turns out, due to an observation of Lazard about denominators of these formulas, that the new coefficients do not have any power of 2 in their denominators.

Particular cases

In the case that we have a nilpotent group or equivalently a nilpotent Lie ring, the Baker-Campbell-Hausdorff formula terminates in finitely many steps, because all terms that involve more than a given number of Lie bracket iterations vanish. Below are the formulas for small values of nilpotency class.

For class c, the formula makes sense over any field or ring where all the primes (other than 2) less than or equal to c are invertible.

Note that in the abelian case, the exponential map is a homomorphism, and hence the Baker-Campbell-Hausdorff formula just gives X + Y.

Nilpotency class Original formula in terms of Lie bracket New formula in terms of * Primes required to be invertible
1 X + Y \! X + Y no condition
2 X + Y + \frac{1}{2}[X,Y] \! X + Y + (X * Y) no condition
3 X + Y + \frac{1}{2}[X,Y] + \frac{1}{12}[X,[X,Y]] - \frac{1}{12}[Y,[X,Y]] \! X + Y + (X * Y) + \frac{1}{3}((X * (X * Y)) - (Y * (X * Y))) 3 only
4 X + Y + \frac{1}{2}[X,Y] + \frac{1}{12}[X,[X,Y]] - \frac{1}{12}[Y,[X,Y]] - \frac{1}{24}[Y,[X,[X,Y]]] \! X + Y + (X * Y) + \frac{1}{3}((X * (X * Y)) - (Y * (X * Y))) - \frac{1}{3}(Y * (X * (X * Y))) 3 only