On the wikipedia page on convolution, there is a section on convolution in bialgebras. It's completely mysterious to me. If it has something to do with the regular concept of convolution, can some one explain the connection? Or is it just a coincidence of names? My motivation is to get a better understanding of convolution.
Thanks!
The following is Example 8 at page 166 of Dascalescu, Nastasescu, Raianu, "Hopf algebras. An introduction".
Consider the $\mathbb{R}$-algebra of polynomials in one indeterminate $\mathbb{R}[X]$. Define a comultiplication and a counit by $$\Delta(X)=X\otimes 1+1\otimes X, \qquad \varepsilon(X)=0,$$ uniquely extended by the universal property of the polynomial algebra. With this structure, $\mathbb{R}[X]$ is even a Hopf algebra and it coincides with the universal enveloping Hopf algebra of the one-dimensional abelian Lie algebra and the Hopf algebra of representative functions on the affine algebraic group $(\mathbb{R},+,0)$.
Anyway, $\mathbb{R}[X]^*$ is an algebra with the convolution product. Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function with compact support. Then consider $f^*\in\mathbb{R}[X]^*$ as given by $$f^*(P):=\int f(x)\tilde{P}(x)dx$$ where $\tilde{P}$ denotes the real function associated to the polynomial $P$. Up to the isomorphism $\mathbb{R}[X]\otimes \mathbb{R}[X]\cong \mathbb{R}[X,Y]$ we have that $\Delta(P)=\sum P_1\otimes P_2=P(X+Y)$. Now, if $g$ is anothe real function with compact support then \begin{align} (f^**g^*)(P) & = \sum \left(\int f(x)\tilde{P}_1(x)dx\right)\left(\int g(y)\tilde{P}_2(y)dy\right) \\ & = \sum \left(\int \left(\int f(x)g(y)\tilde{P}_1(x)\tilde{P}_2(y)dx\right)dy\right) \\ & = \left(\int \left(\int f(x)g(y)\tilde{P}(x+y)dx\right)dy\right) \\ & = \left(\int \left(\int f(x)g(t-x)dx\right)\tilde{P}(t)dt\right) \\ & = h^*(P) \end{align} where $h(t) = \int f(x)g(t-x)dx$ is the convolution product of $f$ and $g$ in the classical sense.