∞-Lie theory (higher geometry)
synthetic differential ∞-groupoid?
Let $X$ be a set of symbols. 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
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 $1$.
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
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
invariant hyperfunctions_, Inventiones math. 47, 249–272 (1978) pdf
Last revised on April 3, 2018 at 10:23:07. See the history of this page for a list of all contributions to it.