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\begin{document}

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\section*{Hausdorff series}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{lie_theory}{}\paragraph*{{$\infty$-Lie theory}}\label{lie_theory}

[[!include infinity-Lie theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{magnus_algebras_and_magnus_group}{Magnus algebras and Magnus group}\dotfill \pageref*{magnus_algebras_and_magnus_group} \linebreak
\noindent\hyperlink{hausdorff_group_and_hausdorff_series}{Hausdorff group and Hausdorff series}\dotfill \pageref*{hausdorff_group_and_hausdorff_series} \linebreak
\noindent\hyperlink{literature}{Literature}\dotfill \pageref*{literature} \linebreak


\hypertarget{magnus_algebras_and_magnus_group}{}\subsection*{{Magnus algebras and Magnus group}}\label{magnus_algebras_and_magnus_group}

Let $X$ be a set of symbols. The [[free construction|free]] [[associative algebra|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 algebra|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

\begin{displaymath}
(ab)_n = \sum_{i = 0}^n a_i b_{n-i}
\end{displaymath}
where $a = (a_n)_n$ and $b = (b_n)_n$. Furthermore, $\hat{A}(X)$ has the topology of the product of [[discrete topological space]]s. This makes $\hat{A}(X)$ a [[Hausdorff space|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 \textbf{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 \textbf{Magnus group} is the (multiplicative) subgroup of the Magnus algebra consisting of all elements in the Magnus algebra with free term $1$.

\hypertarget{hausdorff_group_and_hausdorff_series}{}\subsubsection*{{Hausdorff group and Hausdorff series}}\label{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 $exp(\hat{L}(X))$.

\textbf{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 \textbf{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 number]]s.

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

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

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

First few terms of Hausdorff series are

\begin{displaymath}
H(X,Y) = X + Y + \frac{1}{2}[X,Y] + \frac{1}{12}([X,[X,Y]]+[Y,[Y,X]]) + \frac{1}{24}[Y,[X,[Y,X]]] + \ldots
\end{displaymath}
\hypertarget{literature}{}\subsection*{{Literature}}\label{literature}

Here we list mostly references about the classical part of the subject. The references connecting Hausdorff series to [[Drinfeld associator]]s, [[Grothendieck-Teichmueller group]] and [[Kashiwara-Vergne conjecture]] see in the corresponding entries.

\begin{itemize}%
\item [[N. Bourbaki]], \emph{Lie groups and algebras}, chapter II
\item [[M M Postnikov]], Lectures on geometry, Semester V, Lie groups and algebras
\item E. B. Dynkin, \emph{Calculation of the coefficents in the Campbell-Hausdorff formula}, Doklady Akad. Nauk SSSR (N.S.) 57, 323-326, (1947).

\end{itemize}
cf. [[Malcev completion]]

\begin{itemize}%
\item [[Terence Tao]], 254A, Notes 1, \emph{Lie groups, Lie algebras, and the Baker-Campbell-Hausdorff formula}, blog entry
\item V. Kurlin, \emph{Exponential Baker-Campbell-Hausdorff formula}, \href{http://arxiv.org/abs/math/0606330}{http://arxiv.org/abs/math/0606330}
\item Terry Tao's blog: \href{https://terrytao.wordpress.com/2011/06/21/the-c11-baker-campbell-hausdorff-formula}{the-c11-baker-campbell-hausdorff-formula}
\item \href{https://en.wikipedia.org/wiki/Baker%E2%80%93Campbell%E2%80%93Hausdorff_formula}{wikipedia}
\item Kuo-Tsai Chen, \emph{Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula}, Annals of Mathematics \textbf{65}:1 (1957) 163--178 \href{https://doi.org/10.2307/1969671}{doi} \href{http://www.jstor.org/stable/1969671}{jstor}
\item [[Wilhelm Magnus]], \emph{A connection between the Baker-Hausdorff formula and a problem of Burnside}, Ann. of Math. \textbf{52} (1950) 111-126

\end{itemize}
category: algebra [[!redirects Baker-Campbell-Hausdorff series]] [[!redirects BCH formula]] [[!redirects Baker-Campbell-Hausdorff formula]] [[!redirects BCH theorem]] [[!redirects Campbell-Hausdorff formula]] [[!redirects Hausdorff formula]]



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