Zassenhaus formula

Let $ L( X, Y) $ be the (graded) free Lie algebra on two generators over $ \mathbf Z $, $ \mathop{\rm Ass} ( X, Y) $ the graded free associative algebra on two generators over $ \mathbf Z $ and $ { \mathop{\rm Ass} } hat ( X, Y) $ its completion with respect to the augmentation ideal (where both $ X $ and $ Y $ have degree $ 1 $). For each $ z \in \mathop{\rm Ass} ( X, Y) $ without constant term, let $ e ^ {z} $ denote the element

$$ e ^ {z} = 1 + z + \frac{z ^ {2} }{2!} + \frac{z ^ {3} }{3!} + \dots $$

of $ { \mathop{\rm Ass} } hat ( X, Y) $. Then there exist elements $ c _ {n} ( X, Y) $, homogeneous of degree $ n $, and $ R _ {m} ( X, Y) $, homogeneous of degree $ m $ in $ X $ and of degree $ n $ in $ Y $, in $ \mathop{\rm Ass} ( X, Y) $ which are Lie elements, i.e. they are in $ L( X, Y) \subset \mathop{\rm Ass} ( X, Y) $, and which are such that

$$ \tag{a1 } e ^ {X} e ^ {Y} = \prod _ {n \geq 1 } e ^ {c _ {n} ( X, Y) / n! } , $$

$$ \tag{a2 } e ^ {- X} e ^ {- Y} e ^ {X} e ^ {Y} = \prod _ { n= 1} ^ \infty \prod _ { m= 1} ^ \infty e ^ {R _ {m,n } ( X, Y)/m!n! } . $$

Here the factors on the right-hand side are to be taken in the natural order for (a1), while in the case of (a2) the product is first taken over $ m $ and then over $ n $. The $ c _ {n} ( X, Y) $ are recursively defined by:

$$ c _ {n} ( X, Y) = $$

$$ = \ \left . \frac{\partial ^ {n} }{\partial t ^ {n} } \left ( e ^ {- t ^ {n- 1} c _ {n- 1} / ( n- 1)! } \dots e ^ {- t ^ {2} c _ {2} / 2! } e ^ {- t c _ {1} } e ^ {tX} e ^ {tY} \right ) \right | _ {t= 0 } . $$

These formulas find application in (combinatorial) group theory, algebraic topology and quantum theory, cf., e.g., [a2]–[a4]. For convergence results (for $ X $ and $ Y $ elements of a Banach algebra) concerning formula (a1) and for more general formulas cf., e.g., [a2].

References

[a1]	H. Zassenhaus, "Über Lie'schen Ringe mit Primzahlcharakteristik" Abh. Math. Sem. Univ. Hamburg , 13 (1940) pp. 1–100
[a2]	M. Suzuki, "On the convergence of exponential operators - the Zassenhaus formula, BCH formula and systematic approximants" Comm. Math. Phys. , 57 (1977) pp. 193–200
[a3]	W. Magnus, A. Karrass, B. Solitar, "Combinatorial group theory: presentations in terms of generators and relations" , Wiley (Interscience) (1966) pp. 412
[a4]	H.J. Baues, "Commutator calculus and groups of homotopy classes" , Cambridge Univ. Press (1981)