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Hausdorff series

Context

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Magnus algebras and Magnus group

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

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

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

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

The Magnus group is the (multiplicative) subgroup of the Magnus algebra consisting of all elements in the Magnus algebra with free term 11.

Hausdorff group and Hausdorff series

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

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

The formula exp(X)exp(Y)=(exp(Y)exp(X)) 1exp(X)exp(Y) = (exp(Y)exp(X))^{-1} implies the basic symmetry of the Hausdorff series: H(Y,X)=H(X,Y)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))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)= N=0 H N(X,Y)H(X,Y) = \sum_{N=0}^\infty H_N(X,Y) where Dynkin’s Lie polynomials H N=H N(X,Y)H_N = H_N(X,Y) are defined recursively by H 1=X+YH_1 = X+Y and

(N+1)H N+1=12[XY,H N]+ r=0 N/21B 2r(2r)! s[H s 1,[H s 2,[,[H s 2r,X+Y]]]] (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 ss is the sum over all 2r2r-tuples s=(s 1,,s 2r)s = (s_1,\ldots,s_{2r}) of strictly positive integers whose sum s 1++s 2r=Ns_1 +\ldots+s_{2r} = N.

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

First few terms of Hausdorff series are

H(X,Y)=X+Y+12[X,Y]+112([X,[X,Y]]+[Y,[Y,X]])+124[Y,[X,[Y,X]]+ 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).

cf. Malcev completion

  • 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)