nLab convolution algebra




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

The probably most widespread example of this is the

This is a special case of the

Of maps from a coalgebra to an algebra

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

Hom k(C,A) \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,gf,g is given by

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

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


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


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

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

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

Of a group

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

f 1f 2:g g 1g 2=gf 1(g 1)f 2(g 2). 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)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.

Last revised on November 25, 2022 at 18:04:35. See the history of this page for a list of all contributions to it.