Coalgebra automorphisms congruent to the identity are exactly coderivation exponentials

The complete degree-zero endomorphism algebra $E^0$ and its closed degree-zero coderivation subspace are supplied by Complete coderivations form a filtered differential graded Lie algebra. Thus Exponential and logarithm in a noncommutative complete filtered algebra defines inverse exponential and logarithm series on positive-order degree-zero endomorphisms. All subsequent identities can be checked in $C_n=C{\otimes }_{k}B\mathrm{/}{I}^n$ and its tensor target and then passed to limits.

Conversely, let $G$ be allowed and $N=G-\mathrm{id}$. On $C_n$, $N^n=0$. Define the endomorphism-valued polynomial $P_n(t)={\sum }_{j=0}^{n-1}\left(\genfrac{}{}{0px}{}{t}{j}\right)N^j$. For every nonnegative integer $m$, the finite binomial theorem gives $P_n(m)=G^m$. Since $G^m$ is a counital coaugmented coalgebra map, $\mathrm{\Delta }{P}_n(m)=(P_n(m)\otimes P_n(m))\mathrm{\Delta }$, $P_n(m)(1_C)=1_C$, and $ϵP_n(m)=ϵ$.

For $X\in F^1{\mathfrak{l}}^0$, induction on $j$ using its coderivation equation gives $\mathrm{\Delta }{X}^j=(X\otimes 1+1\otimes X{)}^j\mathrm{\Delta }$. The operators $X\otimes 1$ and $1\otimes X$ commute because $X$ has degree zero. Each is of positive coefficient order, so the exponential and binomial calculations are finite in each quotient. Consequently $\mathrm{\Delta }\exp X=(\exp X\otimes \exp X)\mathrm{\Delta }$. Also $\exp X(1_C)=1_C$ and $ϵ\exp X=ϵ$, because every positive power of $X$ kills $1_C$ and is killed by $ϵ$. Its inverse $\exp (-X)$ has the same properties. Hence exponential gives an allowed automorphism.

Each difference in these three identities is a polynomial in $t$ with coefficients in a $k$-vector space of maps. A polynomial of degree at most $a$ with such coefficients vanishing at $a+1$ distinct nonnegative integers has zero coefficients: apply the inverse of their scalar Vandermonde matrix over the characteristic-zero field $k$. This does not require finite-dimensional map spaces or any domain property of $B\mathrm{/}{I}^n$. Therefore the identities of step 1.2 hold as polynomial identities for the indeterminate $t$.

Differentiate those polynomial identities at $t=0$. Since $P_n(0)=\mathrm{id}$, the result is $\mathrm{\Delta }{P}_n^{\mathrm{\prime }}(0)=(P_n^{\mathrm{\prime }}(0)\otimes 1+1\otimes P_n^{\mathrm{\prime }}(0))\mathrm{\Delta }$, $P_n^{\mathrm{\prime }}(0)(1_C)=0$, and $ϵP_n^{\mathrm{\prime }}(0)=0$. For $j\ge 1$, differentiation of $t(t-1)\cdots (t-j+1)\mathrm{/}j!$ at zero gives $(-1{)}^{j-1}\mathrm{/}j$. Thus $P_n^{\mathrm{\prime }}(0)={\sum }_{j=1}^{n-1}(-1{)}^{j-1}{N}^j\mathrm{/}j$, exactly the reduction of $\log G$.

These reductions are compatible for all $n$, and the coderivation, counit, and coaugmentation identities are closed in the completed targets. Therefore $\log G\in F^1{\mathfrak{l}}^0$. Step 1.1 gives inverse identities for the underlying endomorphisms, and step 2.1 shows that every positive-order degree-zero coderivation exponentiates into the allowed group. The restrictions are therefore mutually inverse bijections onto the entire stated group.