Baker–Campbell–Hausdorff series

Let $\mathfrak{g}$ be a finite-dimensional real Lie algebra and let $X,Y\in \mathfrak{g}$. For a nonempty word $w=Z_1\cdots Z_N$ in the two letters $X,Y$, define its right-nested commutator by

$[Z_1{]}_R=Z_1,[Z_1\cdots Z_N{]}_R=[Z_1,[Z_2\cdots Z_N{]}_R](N\ge 2)\mathrm{.}$

Equivalently, when $N\ge 2$, this is ${\mathrm{ad}}_{Z_1}\circ \cdots \circ {\mathrm{ad}}_{Z_{N-1}}(Z_N)$ in the notation of Adjoint representation of a Lie algebra.

For $N\ge 1$, the degree-$N$ Dynkin polynomial is

$H_N(X,Y)={\sum }_{k=1}^N\frac{(-1{)}^{k-1}}{kN}{\sum }_{\begin{array}{c}{m}_i,n_i\ge 0,m_i+n_i>0(1\le i\le k)\\ {\sum }_{i=1}^k(m_i+n_i)=N\end{array}}\frac{[X^{m_1}{Y}^{n_1}\cdots X^{m_k}{Y}^{n_k}{]}_R}{m_1!n_1!\cdots m_k!n_k!}\mathrm{.}$

Here $X^m{Y}^n$ means a block of $m$ copies of $X$ followed by $n$ copies of $Y$; it is word notation, not multiplication in $\mathfrak{g}$. For fixed $N$ both sums are finite, so $H_N(X,Y)$ is well defined using only the vector space operations and Lie bracket supplied by Finite-dimensional Lie algebra.

The formal Baker–Campbell–Hausdorff series is the degree-indexed formal sum

$\mathrm{BCH}(X,Y):={\sum }_{N=1}^{\mathrm{\infty }}{H}_N(X,Y)\mathrm{.}$

It begins

$X+Y+\frac{1}{2}[X,Y]+\frac{1}{12}[X,[X,Y]]+\frac{1}{12}[Y,[Y,X]]+\cdots \mathrm{.}$

Indeed, the $N=1$ part has the two one-letter blocks and gives $X+Y$. In degree two, the $k=1$ block $(m_1,n_1)=(1,1)$ contributes $\frac{1}{2}[X,Y]$; the two mixed $k=2$ words contribute $-\frac{1}{4}([X,Y]+[Y,X])=0$, and all repeated-letter brackets vanish. Direct collection of the finite degree-three sum gives the two displayed $1\mathrm{/}12$ terms.

Until convergence is proved, $\mathrm{BCH}(X,Y)$ means this formal sequence of homogeneous Lie polynomials, not an element obtained by summing infinitely many vectors. Wherever the series converges, the same notation denotes its sum. The next convergence lemma justifies this analytic meaning on a neighborhood of $(0,0)$.

For the zero Lie algebra every $H_N$ is zero. For a one-dimensional real Lie algebra the bracket vanishes, so the formal series is $X+Y$. No metric, nondegeneracy, interval, endpoint, choice principle, or biconditional is part of this definition.