Zassenhaus formula

Let be the (graded) free Lie algebra on two generators over , the graded free associative algebra on two generators over and its completion with respect to the augmentation ideal (where both and have degree ). For each without constant term, let denote the element

of . Then there exist elements , homogeneous of degree , and , homogeneous of degree in and of degree in , in which are Lie elements, i.e. they are in , and which are such that

(a1)

(a2)

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 and then over . The are recursively defined by:

These formulas find application in (combinatorial) group theory, algebraic topology and quantum theory, cf., e.g., [a2]–[a4]. For convergence results (for and 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)