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Let $G$ by a unipotent linear algebraic group defined over a field of characteristic $0$, with Lie algebra $\mathfrak{g}$. The exponential map $\mathfrak{g}\to G$ is bijective, and we can recover the group structure on $G$ from $\mathfrak{g}$ with the Baker–Campbell–Hausdorff formula.

Write $A$ for the coordinate ring of $G$, and let $\mathfrak{m}\subset A$ be the ideal of the identity element. There is a Hopf algebra structure on $A$. We can identify $\mathfrak{g}$ with the linear dual of $\mathfrak{m}/\mathfrak{m}^2$, so the algebra of regular functions on $\mathfrak{g}$ is the symmetric algebra $\mathrm{Sym}\,\,\mathfrak{m}/\mathfrak{m}^2$. The exponential map induces a ring map $\mathrm{exp}^*:A\to\mathrm{Sym}\,\,\mathfrak{m}/\mathfrak{m}^2$.

Is there an explicit formula for $\mathrm{exp}^*$ in terms of the comultiplication on $A$?

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