Formal Lie groupoids
Magnus algebras and Magnus group
Let be a set of symbols. The free associative k-algebra on the set where is a commutative unital ring, will be denoted . It is clearly graded (by the length of the word) as . The product of -modules has a natural multiplication
where and . Furthermore, has the topology of the product of discrete topological spaces. This makes a Hausdorff topological algebra, where the ground field is considered discrete and is dense in . We say that is the Magnus algebra with coefficients in . (Bourbaki-Lie gr. II.5).
An element in is invertible (under multiplication) iff it’s free term is invertible in .
The Magnus group is the (multiplicative) subgroup of the Magnus algebra consisting of all elements in the Magnus algebra with free term .
Hausdorff group and Hausdorff series
The free Lie algebra naturally embeds in (the Lie algebra corresponding to the associative algebra) ; one defines as the closure of in . The exponential series and the makes sense in ; when restricted to it gives a bijection between and a closed subgroup of the Magnus group which is sometimes called the Hausdorff group .
Hausdorff series is an element in .
The formula implies the basic symmetry of the Hausdorff series: .
The specializations of the Hausdorff series in Lie algebras which are not necessarily free are known as the Baker-Campbell-Hausdorff series and play the role in the corresponding BCH formula .
The BCH formula can be written in many ways, the most important which belong to Dynkin. The part which is linear in one of the variables involves Bernoulli numbers.
There is a decomposition where Dynkin’s Lie polynomials are defined recursively by and
where the sum over is the sum over all -tuples of strictly positive integers whose sum .
Hausdorff series satisfies the symmetry .
First few terms of Hausdorff series are
- N. Bourbaki, Lie groups and algebras, chapter II
- M M Postnikov, Lectures on geometry, Semester V, Lie groups and algebras
- E. B. Dynkin, Calculation of the coefficents in the Campbell-Hausdorff formula, Doklady Akad. Nauk SSSR (N.S.) 57, 323-326, (1947).
cf. Malcev completion
- Anton Alekseev, Charles Torossian, The Kashiwara-Vergne conjecture and Drinfeld’s associators, pdf