Group Algebra

definitionring-theory

Definition

For $R$ commutative ring, $(G,\cdot )$ finite group, and $G=\{e=g_1,g_2,\dots ,g_n\}$ with $x_e,x_{g_2},x_{g_3},\dots \in R[G]$. The group algebra $R[G]=\{r_e{x}_e+r_{g_2}{x}_{g_2}+\dots +r_{g_n}{x}_{g_n}\mid r_{g_i}\in R\}$ is a ring along with the following operations,

• $+$ is “component-wise” → $\left({\sum }_{g\in G}{r}_g{x}_g\right)+\left({\sum }_{g\in G}{r}_g^{\prime }{x}_g\right)={\sum }_{g\in G}(r_g+r_g^{\prime })x_g$
• $\cdot$ is induced from
  • distributivity
  • asking $r\in R$, $r=r\cdot x_e+{\sum }_{g\ne e}0x_g\in R$ to commute with all $R[G]$
  • $x_g{x}_h=x_{gh}$

$R[G]$ has additive identity $0_{R[G]}=0_R$.

If $R$ is unital, $R[G]$ is also unital.

$R[G]$ is commutative if and only if $G$ is commutative.

Examples