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E0054 proposition
Primitives and group-likes
The primitives and group-likes of the symmetric bialgebra \((\SS(V), \Delta)\) E0052 are classified by exp/log.
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An element \(p \in \SH(V)\) is primitive (\(\Delta(p) = p \otimes 1 + 1 \otimes p\)) if and only if \(p \in V = \SS^1(V)\).
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An element \(g \in \SH(V)\) is group-like (\(\Delta(g) = g \otimes g\), \(\varepsilon(g) = 1\)) if and only if \(g = \exp(v) = \sum_{k \geq 0} v^k / k!\) for a unique \(v \in V\).
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\(\exp\) and \(\log\) are inverse bijections between primitives and group-likes.
Proof. Ad primitives) Expand \(a = \sum_k a_k\) and \(\Delta(a) = \sum_k \Delta(a_k)\). For \(k \geq 2\), \(\Delta(a_k)\) has a nonzero middle component in \(\bigoplus_{i,j \geq 1} \SS^i \otimes \SS^j\), but the primitive condition forces this to vanish. Hence \(a = a_1 \in V\).
Ad group-likes) If \(g\) is group-like, \(h = g - 1\) lies in \(\SS_+(V)\) and \(\log(1 + h) = \sum_{k \geq 1} (-1)^{k+1} h^k / k\) converges in \(\SH(V)\). The coproduct of \(\log(g)\) is \(\log(g \otimes g) = \log(g) \otimes 1 + 1 \otimes \log(g)\) by the functional equation, so \(\log(g)\) is primitive, hence \(\log(g) \in V\).
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