Spahn group algebra (Rev #1)

Definition

For a kk-algebra AA and an (abelian) group GG the group

Idea

Take the elements of a group, GG, as labelling a basis for a vector space over a field kk, then that multiplication of the group will extend to give that vector space the structure of an algebra? over kk. This is usually denoted k[G]k[G].

If instead of a field we used the ring of integers, \mathbb{Z}, it is usual to call the result the group ring. For this we take the free abelian group on the set of elements of the group and extend the multiplication to give a ring structure on the result. This is usually denoted [G]\mathbb{Z}[G].

As a further generalisation, the field can be replaced by any commutative ring.

The multiplication:

If we denote by e ge_g, the generator corresponding to gGg\in G, then an arbitrary element of k[G]k[G] can be written as gGn geg\sum_{g\in G}n_ge_g where the n gn_g are elements of kk, and only finitely many of them are non-zero.

The multiplication is then by what is sometimes called a ‘convolution’ product, that is,

( gGn geg)( gGm geg)= gG( g 1Gn g 1m g 1 1ge g).\Big(\sum_{g\in G}n_ge_g\Big)\Big(\sum_{g\in G}m_ge_g\Big) = \sum_{g\in G}\Big(\sum_{g_1\in G}n_{g_1}m_{g_1^{-1}g}e_g\Big).

Thoughts

Thinking of a group?, GG, as a special sort of category?, the group algebra (denoted k[G]k[G] or kGk G) of a group is just the category algebra? of that category.

Extra structure

The group algebra is always a Hopf algebra?.

The group algebra is always a graded algebra?.

Sometimes instead of working over a ground field? kk, one allows kk to be a commutative unital ring?. Then we talk about group ring (though it is in fact a commutative unital kk-algebra?). The integer group ring G\mathbb{Z} G is the most important example, extensively used in the representation theory? of finite groups.

Revision on June 1, 2012 at 19:08:26 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.