∞-Lie theory

# Contents

## Magnus algebras and Magnus group

Let $X$ be a set of symbols. The free associative k-algebra $k⟨X⟩$ on the set $X$ where $k$ is a commutative unital ring, will be denoted $A\left(X\right)$. It is clearly graded (by the length of the word) as $A\left(X\right)={\oplus }_{n}{A}^{n}\left(X\right)$. The product of $k$-modules $\stackrel{^}{A}\left(X\right)={\prod }_{n}{A}^{n}\left(X\right)$ has a natural multiplication

$\left(\mathrm{ab}{\right)}_{n}=\sum _{i=0}^{n}{a}_{i}{b}_{n-i}$(ab)_n = \sum_{i = 0}^n a_i b_{n-i}

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

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

Hausdorff series $H\left(U,V\right)$ is an element $\mathrm{log}\left(\mathrm{exp}\left(U\right)\mathrm{exp}\left(V\right)\right)$ in $\stackrel{^}{L}\left(\left\{U,V\right\}\right)$.

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

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 $\mathrm{exp}\left(U\right)\mathrm{exp}\left(V\right)=\mathrm{exp}\left(H\left(U,V\right)\right)$.

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 $H\left(X,Y\right)={\sum }_{N=0}^{\infty }{H}_{N}\left(X,Y\right)$ where Dynkin’s Lie polynomials ${H}_{N}={H}_{N}\left(X,Y\right)$ are defined recursively by ${H}_{1}=X+Y$ and

$\left(N+1\right){H}_{N+1}=\frac{1}{2}\left[X-Y,{H}_{N}\right]+\sum _{r=0}^{⌊N/2-1⌋}\frac{{B}_{2r}}{\left(2r\right)!}\sum _{s}\left[{H}_{{s}_{1}},\left[{H}_{{s}_{2}},\left[\dots ,\left[{H}_{{s}_{2r}},X+Y\right]\dots \right]\right]\right]$(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 $s=\left({s}_{1},\dots ,{s}_{2r}\right)$ of strictly positive integers whose sum ${s}_{1}+\dots +{s}_{2r}=N$.

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

First few terms of Hausdorff series are

$H\left(X,Y\right)=X+Y+\frac{1}{2}\left[X,Y\right]+\frac{1}{12}\left(\left[X,\left[X,Y\right]\right]+\left[Y,\left[Y,X\right]\right]\right)+\frac{1}{24}\left[Y,\left[X,\left[Y,X\right]\right]+\dots$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

## Literature

• 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).
• Anton Alekseev, Charles Torossian, The Kashiwara-Vergne conjecture and Drinfeld’s associators, pdf

category: algebra

Revised on November 29, 2013 02:11:14 by Zoran Škoda (161.53.130.104)