𝕂-monoid

Group ring

The group ring K𝐺 of a group 𝐺 is a Hopf algebra constructed by applying the monoidal free module functor to the structure of the corresponding Hopf monoid 𝐺 in 𝖲𝖾𝗍, so

• its basis are the group elements;
• its multiplication on the basis is the group multiplication;
• its comultiplication is diagonal on basis elements, i.e. its basis are precisely the grouplikes of K𝐺;
• its antipous is inversion on basis elements.

As such, it is a specialization of the monoid ring.

Construction as maps

Let 𝐺 be a group, and K be a ring. The group ring K𝐺 may be identified with the set of maps of finite-support 𝐺 →K, with the convolution and conjugate operations defined below,¹ where we identify 𝑔 ∈𝐺 with 𝛿𝑔 :ℎ ↦[𝑔 =ℎ]. The convolution operation is defined by

If 𝑅 is an Involutive ring, the conjugate is defined by

Hilbert space

If K =ℂ, then the group ring can be made into a Hilbert space with some inner product, usually taken from those listed below.

Properties

• ∗-representation of the complex group ring
• Regular group representation
• Ideal of the complex group ring
• Idempotent of the complex group ring
• Isomorphism between the complex group ring and direct sum of matrix algebras on carriers of irreducible representations
• Centre of the group ring

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