Spahn group algebra (changes)

Skip the Navigation Links | Home Page | All Pages | Latest Revisions | Authors |

Showing changes from revision #1 to #2: Added | Removed | Changed

Definition

For a kk-algebra AA and an (abelian) group GG , the space of group morphisms with finite support

Idea

A[G]:=hom finsupp(G,A)A[G]:=hom_{fin\,supp}(G,A)

Take (i.e. the space containing just those morphisms which send only finitely many elements of a group,GG , as not labelling to a basis for a vector space over a fieldk0 k 0 , ) then equipped that with the multiplication defined by convolution of the functions group is will called extend to give that vector space the structure of analgebra?group algebra of AA over GG . over With pointwise addition k K[G] k K[G] . This is usually an denoted associative k A[G] k[G] A . -algebra.

If Equivalently instead one can think of a field we used the ring multiplication of inthe integers, following wayℤ\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 ℤ[G]\mathbb{Z}[G].

As Take a further generalisation, the field elements can of be a replaced group, by any commutative ring.GG, as labelling a module basis and define the multiplication of two elements by x g⋅x h:=x ghx_g \cdot x_h:=x_{gh}.

The multiplication:

If AA is a ring one calls this construction also a group ring.

If The we notion denote of by group algebra is a special case of that of ae ge_gcategory algebra? , . the generator corresponding tog∈Gg\in G, then an arbitrary element of k[G]k[G] can be written as ∑ g∈Gn 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.

Any group algebra is in particular a Hopf alebra? and a graded algebra?.

The convolution product

If we denote by e ge_g, the generator corresponding to g∈Gg\in G, then an arbitrary element of k[G]k[G] can be written as ∑ g∈Gn 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,

(∑ g∈Gn geg)(∑ g∈Gm geg)=∑ g∈G(∑ g 1∈Gn 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 Properties

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.

Remark

There is an adjunction?

(R[−]⊣(−) ×):Alg R→(−) ×Grp(R[-]\dashv (-)^\times):Alg_R \stackrel{(-)^\times}{\to}Grp

where R[−]R[-] forms group rings and (−) ×(-)^\times assigns to an RR-algebra its group of units?.

Extra structure

Remark

Let VV be an abelian group. A morphism of rings K[G]→End(V)K[G]\to End(V) of the group ring to the endomorphism ring of VV is a K[G]K[G]-module. And any morphism of groups p:G→End(V)p:G\to End(V) can by extended to a morphism of rings P:K[G]→End(V)P:K[G]\to End (V) by p(g)↦P(e g)p(g)\mapsto P(e_g). This observation is used extensively in the theory of group representations?.

The group algebra is always a Hopf algebra?.

Remark

(Maschke’s theorem) Let GG be a finite group, let KK be a field.

Then K[G]K[G] is a semi-simple algebra? iff the order of GG is not divisible by the characteristic? of K.

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 ℤG\mathbb{Z} G is the most important example, extensively used in the representation theory? of finite groups.

Last revised on June 1, 2012 at 21:08:39. See the history of this page for a list of all contributions to it.

EditDiscussPrevious revisionHide changesHistory (1 revision) Print Source