Comultiplication in graded Hopf Algebras Let $H$ be a graded Hopf algebra over some commutative ring $k$.
I'm looking for a proof of the following result, which seems to be stated in various locations.
For $x$ in $H$ of degree $n$
$$\Delta(x) = x \otimes 1 + 1 \otimes x + \sum x_{(1)} \otimes x_{(2)}$$
where $x_{(i)}$ has degree between $1$ and $n-1$ inclusive.
I suspect it follows from the counital properties. In other words:
$$\mu \circ (\epsilon \otimes id_H) \circ \Delta = id_H = \mu \circ (id_H \otimes \epsilon) \circ \Delta$$
At first glance, it appears our result would follow, as it is an obvious way to ensure the above relation holds, however couldn't we also have the following case:
$$ \Delta(x) = \alpha \otimes a + \beta \otimes b + \dots $$
where $\alpha, \beta \in H$ s.t. $\epsilon(\alpha) = \epsilon(\beta)=1$, and  $x = a + b$. This certainly satisfies our counital relation.
Any help greatly appreciated.

There are two errors in the statement of the question:


*

*You need to assume $n > 0$. Otherwise, $x = 1$ would be a counterexample (usually).

*You need to assume that $H$ is a connected graded Hopf algebra, or another condition that rules out the following counterexample: Let $N$ be a positive integer. Let $q$ be a primitive $N$-th root of unity in $k$. Let $H$ be the $k$-algebra $k\left<g, x \mid g^N = 1,\  x^N = 0, \  gx = qxg\right>$ with comultiplication given by $\Delta\left(g\right) = g \otimes g$ and $\Delta\left(x\right) = g \otimes x + x \otimes 1$ and counity given by $\epsilon\left(g\right) = 1$ and $\epsilon\left(x\right) = 0$. This $H$ is a free $k$-module with basis $\left(g^i x^j\right)_{\left(i,j\right)\in\left\{0,1,\ldots,N-1\right\}^2}$ (this can be proven using the theory of Ore extensions) and is indeed a well-defined bialgebra and even a Hopf algebra; it is known as the Taft Hopf algebra. It can be made into a graded Hopf algebra by letting $g$ have degree $0$ and $x$ have degree $1$. The element $x$ is not primitive (for $N>1$), although your claim would make it seem that it was!
With these errors corrected, the claim indeed holds. For a proof, see the solution of Exercise 1.3.19 (g) in Darij Grinberg and Victor Reiner, Hopf algebras in Combinatorics, arXiv:1409.8356v5. More precisely, Exercise 1.26 (f) states that if $I = \bigoplus_{n > 0} H_n$ (where $H_n$ is the $n$-th degree component of $H$), then every $x \in I$ satisfies $\Delta\left(x\right) \in 1 \otimes x + x \otimes 1 + I \otimes I$. Hence, for every homogeneous element $x$ of $H$ of degree $n > 0$, we have $\Delta\left(x\right) \in 1 \otimes x + x \otimes 1 + I \otimes I$ and thus
$\Delta\left(x\right) - 1 \otimes x - x \otimes 1 \in I \otimes I$.
Combined with
$\Delta\left(x\right) - 1 \otimes x - x \otimes 1 \in \bigoplus_{j=0}^n H_j \otimes H_{n-j}$
(which follows from the gradedness of $H$), this becomes
$\Delta\left(x\right) - 1 \otimes x - x \otimes 1 \in \left(I\otimes I\right)\cap\left(\bigoplus_{j=0}^n H_j \otimes H_{n-j}\right) = \bigoplus_{j=1}^{n-1} H_j \otimes H_{n-j}$.
