Notes for AKT-090924-2/0:15:13

Example: Space of all polynomial with two variables with rational coefficient, with co-multiplication $\Delta :\mathbb {Q} [x,y]\rightarrow \mathbb {Q} [x,y]\otimes \mathbb {Q} [x,y]$ be $\Delta (f(x,y))=f(x_{1}+x_{2},y_{1}+y_{2})$ where $x_{1}$ corresponds to an $x$ (or $y$) before the tensor sign and $x_{2}$ being after the tensor sign, similarly for $y$. Co-unit being the map sending each polynomial to its constant term. Check that this is a co-algebra (together with the usual polynomial multiplication, this space is a bi-algebra)