nLab convolution algebra

Idea

There are many related constructions of algebras, topological algebras and so on which bear the name of a convolution algebra.

The basic mechanism is usually the

• Convolution of maps from a coalgebra to an algebra .

The probably most widespread example of this is the

• Convolution algebra of a group

This is a special case of the

• Convolution algebra of a groupoid/category

Of maps from a coalgebra to an algebra

Let $k$ be a commutative unital ring, $(C,\Delta)$ a (counital) $k$-coalgebra and $(A,m)$ an associative (unital) $k$-algebra. Then the set of linear maps

$$\mathrm{Hom}_k(C,A)$$

has a structure of an associative (unital) algebra, called convolution algebra, in which the product of two linear maps $f,g$ is given by

$$f\star g = m\circ(f\otimes g)\circ\Delta.$$

Of maps from a comonoid to a monoid in a closed monoidal category

Proposition

Let $(\mathcal{C}, \otimes, I, \multimap)$ be a closed monoidal category, $(A,\Delta,\eta)$ a comonoid in $\mathcal{C}$ and $(B,\nabla,\epsilon)$ a monoid. Then $A \multimap B$ is a monoid.

Proposition

Let $(\mathcal{C}, \otimes, I, \multimap)$ be a closed monoidal category, $Mon(\mathcal{C})$ the category of monoids of $\mathcal{C}$ and $Comon(\mathcal{C})$ the category of comonoids? of $\mathcal{C}$. We then have a functor:

$$Comon(\mathcal{C})^{op} \times Mon(\mathcal{C}) \rightarrow Mon(\mathcal{C})$$

which associate $A \multimap B$ to $(A,B)$

Of a group

Given a finite group $G$ and a ring $R$, the space of functions $C(G,R)$ inherits the convolution product defined by

$$f_1 \star f_2 \colon g \mapsto \sum_{g_1 \cdot g_2 = g} f_1(g_1) \cdot f_2(g_2) \,.$$

This is the non-commutative product operation that appears in the Hopf algebra structure on $C(G,R)$.

Of a groupoid/category

More generally, there is convolution of functions on morphisms of a groupoid. See at groupoid convolution algebra for details.