Group ring
In mathematical representation theory,group ringan abstract algebra construction, that allows representations ofgroup G, overfield K,be treated as modules.The ring constructed (notation K[G], or sometimes just KG) can be described asvector space over Kbasiselements gG,ring multiplicationgroup operationG extended by bilinearity towhole space. That is, g1g2 = g3 as an equationG still holds trueK[G], andwhole structureK[G] as an algebra over K follows when we applydistributive lawK-linearity.
Itthen true thatmodule MK[G]just whatnormally meant aslinear representationG overfield K. Thereno particular reasonlimit Kbefield here; butclassical results that were obtained first when K iscomplex number fieldGfinite group justify close attentionthis case. It was shown that K\[G] issemisimple ring, under those conditions,profound implications forrepresentationsfinite groups.
When G isfinite abelian group,group ringcommutative,its structuer easyexpresstermsrootsunity. When K isfieldcharacteristic p, andprime number p dividesorder offinite group G, thengroup ringnot semisimple:hasnon-zero radical,this givescorresponding subjectmodular representation theory its own, deeper character.
An example ofgroup ringan infinite group isringLaurent polynomials: thisexactlygroup ringan infinite cyclic group.
There isneat characterisation from category theory ofgroup ring construction as adjoint tofunctor takingK-algebraits groupunits.
Continuous case
Forpurposesfunctional analysis,in particularharmonic analysis, one wishescarry overgroup ring constructiontopological groups G. In this casegroup ring should consistenough functions on G,productconvolution: inalgebraic case one already sees that if group ring elementswritten as functions F(g)group elements,valuesK, thenring product istypeconvolution.
In practice therevarious candidatesthe complex group algebraG. One can takealgebracontinuous complex functions on G thatzero outsidecompact set: thenconvolution product will present no convergence difficulties. In case Gcompact thisjust C(G),ringall continuous functions on G withconvolution product. In this way one will getC-star algebra.
In generalneedapply Fubini's theorem leads touseL1(G), L2(G),their intersection. That is,deal withanalytical difficulties one needsrangealgebras, insteadjust one.
