Kashiwara–Vergne conjecture

Let ${\bf k}={\mathbb R}$ or ${\mathbb C}$, let $\G$ be a finite-dimensional ${\bf k}$-Lie algebra, and let $\hat{\mathfrak f}_{2}$ denote the completed free Lie algebra on $x,y$. For $A(x,y),B(x,y)\in\hat{\mathfrak f}_{2}$, define $(\partial_xA)(x,y)\in\operatorname{End}(\G)$ by

$$(\partial_xA)(x,y)(a)=\left.\frac{d}{dt}\right|_{t=0}A(x+ta,y),$$

and define $(\partial_yB)(x,y)\in\operatorname{End}(\G)$ by

$$(\partial_yB)(x,y)(a)=\left.\frac{d}{dt}\right|_{t=0}B(x,y+ta).$$

Writing $z=\log e^xe^y$, the Kashiwara–Vergne conjecture. For every finite-dimensional ${\bf k}$-Lie algebra $\G$, there exists a pair of Lie series $A(x,y),B(x,y)\in\hat{\mathfrak f}_{2}$ such that

$$x+y-\log e^ye^x=(1-e^{-\operatorname{ad}x})(A(x,y))+(e^{\operatorname{ad}y}-1)(B(x,y)),$$

$A$ and $B$ give convergent power series in a neighborhood of $(0,0)\in\G^2$, and

$$\operatorname{tr}_{\G}\bigl((\operatorname{ad}x)\partial_xA+(\operatorname{ad}y)\partial_yB\bigr)=\frac12\operatorname{tr}_{\G}\left(\frac{\operatorname{ad}x}{e^{\operatorname{ad}x}-1}+\frac{\operatorname{ad}y}{e^{\operatorname{ad}y}-1}-\frac{\operatorname{ad}z}{e^{\operatorname{ad}z}-1}-1\right),$$

where the last identity is an identity of analytic functions on $\G^2$ near the origin. This conjecture is the original analytic form of the Kashiwara–Vergne equations, relating the Campbell–Baker–Hausdorff series to trace identities for adjoint operators. The source presents it as a conjecture; its resolution status is not specified in the supplied text.