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Groupprops β

Group ring

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Definition

Given a group G and a ring R, the group ring or group algebra of G over R, denoted R(G) is defined as the following ring:

  • Additively, it is a free R-module with basis indexed by elements of G
  • The multiplication is defined as follows: the product of the basis element for g and the basis element for h is the basis element for gh. Multiplication on arbitrary elements is obtained by extending this rule R-linearly.

Note that the group ring R(G) is an algebra over R, with R naturally sitting as the subring Re (e being the identity element).

Functoriality

Fix a base ring R. We can then talk of the map sending any arbitrary group G, to its group ring R(G). This map is a functor from the category of groups to the category of R-algebras. In other words, given any homomorphism of groups G \to H we geta corresponding homomorphism of algebras R(G) \to R(H).

Further, this functor preserves injectivity and surjectivity: if H \le G the undiced map R(H) \to R(G) in injective. Similarly, if H is a quotient of G the induced map R(G) \to R(H) is surjective.

Representations of the group as modules over the group ring

Let M be a R-module with an action of G on M as R-module automorphisms. Then, M naturally acquires the structure of a R(G)- module.

In particular:

  • Any action of G as automorphisms of an abelian group is equivalent to viewing the Abelian group as a module over \Z(G) (the group ring over the ring of integers).
  • Any linear representation of G over a field k turns the vector space into a k(G) module.

Additional structure

We can equip the group ring with some additional structure, namely a coalgebra structure and an antipode map, thus turning it into a Hopf algebra.

Further information: Group algebra as a Hopf algebra

Related notions

  • Skew group ring is an analogous notion where we use an action of the group on the ring.

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