nLab Hausdorff series

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Context

$\infty$ -Lie theory

Magnus algebras and Magnus group
Hausdorff group and Hausdorff series
Related entries
Literature

Magnus algebras and Magnus group

Let $X$ be a set of symbols and let $k$ denote the ground field.

The free associative k-algebra $k\langle X \rangle$ on the set $X$ where $k$ is a commutative unital ring, will be denoted $A(X)$ . It is clearly graded (by the length of the word) as $A(X) = \oplus_n A^n(X)$ . The product of $k$ -modules $\hat{A}(X) = \prod_n A^n(X)$ has a natural multiplication

(ab)_n = \sum_{i = 0}^n a_i b_{n-i}

where $a = (a_n)_n$ and $b = (b_n)_n$ . Furthermore, $\hat{A}(X)$ has the topology of the product of discrete topological spaces. This makes $\hat{A}(X)$ a Hausdorff topological algebra, where the ground field is considered discrete and $A(X)$ is dense in $\hat{A}(X)$ . We say that $\hat{A}(X)$ is the Magnus algebra with coefficients in $k$ . (Bourbaki-Lie gr. II.5).

An element in $\hat{A}(X)$ is invertible (under multiplication) iff it’s free term is invertible in $k$ .

The Magnus group is the (multiplicative) subgroup of the Magnus algebra consisting of all elements in the Magnus algebra with free term $1$ .

Hausdorff group and Hausdorff series

The free Lie algebra $L(X)$ naturally embeds in (the Lie algebra corresponding to the associative algebra) $A(X)\hookrightarrow \hat{A}(X)$ ; one defines $\hat{L}(X)$ as the closure of $L(X)$ in $\hat{A}(X)$ . The exponential series and the makes sense in $\hat{A}(X)$ ; when restricted to $\hat{L}(X)$ it gives a bijection between $\hat{L}(X)$ and a closed subgroup of the Magnus group which is sometimes called the Hausdorff group $exp(\hat{L}(X))$ .

Hausdorff series $H(U,V)$ is an element $log(exp(U)exp(V))$ in $\hat{L}(\{U,V\})$ .

The formula $exp(X)exp(Y) = (exp(Y)exp(X))^{-1}$ implies the basic symmetry of the Hausdorff series: $H(-Y,-X) = -H(X,Y)$ .

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 $exp(U)exp(V) = exp(H(U,V))$ .

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 $H(X,Y) = \sum_{N=0}^\infty H_N(X,Y)$ where Dynkin’s Lie polynomials $H_N = H_N(X,Y)$ are defined recursively by $H_1 = X+Y$ and

(N+1)H_{N+1} = \frac{1}{2}[X-Y,H_N] + \sum_{r = 0}^{\lfloor N/2 -1\rfloor} \frac{B_{2r}}{(2r)!}\sum_s [H_{s_1},[H_{s_2},[ \ldots, [H_{s_{2r}},X+Y]\ldots]]]

where the sum over $s$ is the sum over all $2r$ -tuples $s = (s_1,\ldots,s_{2r})$ of strictly positive integers whose sum $s_1 +\ldots+s_{2r} = N$ .

The Hausdorff series satisfies the symmetry $H(-Y,-X) = -H(X,Y)$ .

First few terms of Hausdorff series are:

H(X,Y) \;=\; X + Y + \tfrac{1}{2}[X,Y] + \tfrac{1}{12} \Big( \big[X,[X,Y]\big] + \big[Y,[Y,X]\big] \Big) + \tfrac{1}{24} \Big[ Y, \big [X,[Y,X] \big] \Big] + \ldots \,.

Related entries

Literature

The following references concern the classical part of the subject. For the references connecting Hausdorff series to Drinfeld associators, the Grothendieck-Teichmueller group and the Kashiwara-Vergne conjecture see there.

Early discussion:

Rüdiger Achilles, Andrea Bonfiglioli: The early proofs of the theorem of Campbell, Baker, Hausdorff, and Dynkin, Arch. Hist. Exact Sci. 66 (2012) 295–358 [doi:10.1007/s00407-012-0095-8]
Eugene B. Dynkin, Calculation of the coefficents in the Campbell-Hausdorff formula, Doklady Akad. Nauk SSSR (N.S.) 57 (1947) 323-326
Wilhelm Magnus, A connection between the Baker-Hausdorff formula and a problem of Burnside, Ann. of Math. 52 (1950) 111-126
Kuo-Tsai Chen, Integration of paths, geometric invariants and a generalized Baker–Hausdorff formula, Annals of Mathematics 65:1 (1957) 163–178 doi jstor

Most monographs on Lie algebras discuss the Hausdorff series, for instance:

Jean-Pierre Serre, Section IV.7 in: Lie Algebras and Lie Groups – 1964 Lectures given at Harvard University, Lecture Notes in Mathematics 1500, Springer (1992) [doi:10.1007/978-3-540-70634-2]
Nicolas Bourbaki, Chapter 2 of: Lie groups and Lie algebras, Springer (1975, 1989) [ISBN:9783540642428]
Gerhard P. Hochschild, p. 226 in: Basic theory of algebraic groups and Lie algebras, Graduate Texts in Mathematics 75, Springer (1981) [doi:10.1007/978-1-4613-8114-3_16]
M. M. Postnikov, Lectures on geometry: Semester V, Lie groups and algebras (1986) [ark:/13960/t4cp9jn4p]
Hans Duistermaat, Johan A. C. Kolk, Dynkin’s formula, Section 1.7 in: Lie groups, Springer (2000) [doi:10.1007/978-3-642-56936-4]
Shlomo Sternberg: The Campbell Baker Hausdorff Formula, Section 1 of: Lie Algebras (2004) [pdf, pdf]
James Milne, p. 260 in: Algebraic Groups – The theory of group schemes of finite type over a field, Cambridge University Press (2017) [doi:10.1017/9781316711736, webpage, pdf]
Richard Borcherds, Mark Haiman, Theo Johnson-Freyd, Nicolai Reshetikhin, Vera Serganova: §3.3 in: Berkeley Lectures on Lie Groups and Quantum Groups (2020-2024) [pdf]