describe the general setup for our later applications to renormalization of MZVs. In the following an algebra means a $\bfk$-algebra where $\bfk$ is a unitary commutative ring that we usually take to be $\CC$. Denote the unit of $\bfk$ by $\bfone$.

The algebraic Birkhoff decomposition
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We review the algebraic framework of Connes and Kreimer for renormalization of perturbative quantum field theory.

A [**connected filtered Hopf algebra**]{} is a Hopf algebra $(H, \Delta)$ with $\bfk$-submodules $H^{(n)},\ n\geq 0$ of $H$ such that $$\begin{aligned}
&H^{(n)}\subseteq H^{(n+1)}, \quad
\cup_{n\geq 0} H^{(n)} = H, \quad
H^{(p)} H^{(q)}\subseteq H^{(p+q)}, &\\
&
\Delta(H^{(n)}) \subseteq \sum_{p+q=n} H^{(p)}\otimes H^{(q)}, \quad
H^{(0)}=\bfk {\rm \ (connectedness)}.&\end{aligned}$$ Let $\lambda\in \bfk$. A [**Rota–Baxter algebra**]{} of weight $\lambda$ is a pair $(R,P)$ where $R$ is a unitary $\bfk$-algebra and $P:R\to R$ is a linear operator such that $$P(x)P(y)=P(xP(y))+P(P(x)y)+\lambda P(xy),
\mlabel{eq:rbe}$$ for any $x,\,y\in R$. Often $\theta=-\lambda$ is used, especially in the physics literature. It follows from the definition that $P(R)$ and $(-\lambda-P)(R)$ are non-unitary subalgebras of $R$. So $\bfk+P(R)$ and $\bfk +(-\lambda-P)(R)$ are unitary subalgebras.

Let $H$ be a commutative connected filtered Hopf algebra. Let $(R,P)$ be a Rota-Baxter algebra of weight $-1$. Let $\phi: H \to R$ be an algebra homomorphism.

