The quotient of a connected graded bialgebra

Question

Let $k$ be a field. $H$ is called a connected graded bialgebra, if there are k-submodules $H^{n}$, $n \geq 0$, of $H$ such that:

1. $H^0=k$;
2. $H=\oplus _{n=0} ^{\infty} H^n$;
3. $H^p H^q \subseteq H^{p+q}, p,q \geq 0$;
4. $\Delta(H^n) \subseteq \oplus_{p+q=n} H^p \otimes H^q, n \geq 0$.

Elements of $H^n$ are said to have degree $n$. Now suppose $I$ is a biideal of $H$, is $H/I$ still connected graded? If $I$ is generated by elements of the same degree, could we get $H/I$ connected graded?

Answer 1

Yes, if $I$ is a graded biideal, i.e. if $I$ is a biideal generated by homogeneous elements (which do not need to be all of the same degree, necessarily), then $H/I$ is a graded bialgebra, which is connected if $I\neq H$ and $0$ otherwise.