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

Example: Space of all polynomial with two variables with rational coefficient, with co-multiplication [math]\displaystyle{ \Delta: \mathbb{Q}[x,y] \rightarrow \mathbb{Q}[x,y] \otimes \mathbb{Q}[x,y] }[/math] be [math]\displaystyle{ \Delta(f(x,y)) = f(x_1+x_2, y_1+y_2) }[/math] where [math]\displaystyle{ x_1 }[/math] corresponds to an [math]\displaystyle{ x }[/math] (or [math]\displaystyle{ y }[/math]) before the tensor sign and [math]\displaystyle{ x_2 }[/math] being after the tensor sign, similarly for [math]\displaystyle{ y }[/math]. 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)