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
![](https://www.encyclopediaofmath.org/legacyimages/z/z099/z099150/z09915022.png) | (a1) |
![](https://www.encyclopediaofmath.org/legacyimages/z/z099/z099150/z09915023.png) | (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) |
How to Cite This Entry:
Zassenhaus formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Zassenhaus_formula&oldid=13224