Group Ring and Group Algebra

From Wikipedia we have for a ring $\mathcal{R}$ and a group $\mathcal{G}$ a new ring written as $\mathcal{RG}$ that is defined on the functions $\mathcal{G} \longrightarrow \mathcal{R}$ with finite support and addition and multiplication defined as follows.

$\sum\limits_{g \in \mathcal{G}} \phi_g g \ + \ \sum\limits_{g \in \mathcal{G}} \psi_g g \ := \ \sum\limits_{g \in \mathcal{G}} \left( \phi_g + \psi_g \right) g$

and

$\sum\limits_{g \in \mathcal{G}} \phi_g g \ \cdot \ \sum\limits_{g \in \mathcal{G}} \psi_g g \ := \ \sum\limits_{g \in \mathcal{G}} \left( \sum\limits_{u,v \in \mathcal{G} | uv = g} \phi_u \cdot \psi_v \right) g \ = \ \sum\limits_{g \in \mathcal{G}} \left( \sum\limits_{h \in \mathcal{G}} \phi_{h} \cdot \psi_{h^{-1}g} \right) g$

Here, $\sum\limits_{g \in \mathcal{G}} \phi_g g$ stands for a (formal) sum of elements of $\mathcal{G}$ with coefficients taken from the ring $\mathcal{R}$ . This makes sense as long as the functions in question have finite support, i.e. $\left| \left\{ g \in \mathcal{G} \left|\, \phi_g \neq 0 \right. \right\} \right| < \infty$ .

The group ring is written $\mathcal{R}\left[ \mathcal{G} \right]$ . It is clear that the group ring is also a module over the ring $\mathcal{R}$ . For an element $g \in \mathcal{G}$ let us define the following mapping (action) on the group ring

$\Phi_g \,:\, \mathcal{R}\left[ \mathcal{G} \right] \ni \sum\limits_{h \in \mathcal{G}} \phi_h h \longmapsto \sum\limits_{h \in \mathcal{G}} \phi_{h} (gh) \,=\, \sum\limits_{h \in \mathcal{G}} \phi_{g^{-1}h} h \in \mathcal{R}\left[ \mathcal{G} \right]$

This is nothing else but a representation of the group $\mathcal{G}$ on the linear mappings of the group ring as a module.

If $\mathcal{R}$ is a field the group ring is called a group algebra over the field. In this case the group action from above is nothing else but a representation of the group on the linear mappings of the group algebra as a vector space.

As an example let us investigate the (multiplicative) group of the $n$ -th unit roots, $\mathcal{G} \,:=\, \left\{ e^{\frac{2\pi i k}{n}} \left|\,k = 0, 1, \ldots, n-1 \right.\right\}$ . Then we have for an arbitrary ring $\mathcal{R}$ an element of $\mathcal{R}\left[\mathcal{G}\right]$ :

$r_0\,\mathbf{e}_0 + r_1\,\mathbf{e}_1 + r_2\,\mathbf{e}_2 + \ldots + r_{n-1} \,\mathbf{e}_{n-1}$

with

$\mathcal{G} \ni \mathbf{e}_k \,:=\, e^{\frac{2\pi i k}{n}} \quad\text{and}\quad r_k \in \mathcal{R} \ \left(k = 0, 1, \ldots, n-1 \right) \ .$

For a product of 2 elements we get

(1) $\begin{eqnarray*} \left( \sum\limits_{k=0}^{n-1} r_k\,\mathbf{e}_k \right) \cdot \left( \sum\limits_{l=0}^{n-1} q_l\,\mathbf{e}_l \right) \nonumber & = & \sum\limits_{k=0}^{n-1} \sum\limits_{l=0}^{n-1} r_k \, q_l \,\mathbf{e}_{k + l} \\ \nonumber & = & \sum\limits_{k=0}^{n-1} \left( \sum\limits_{j = 0}^{n-1} r_j\,q_{k-j} \right) \mathbf{e}_k \end{eqnarray*}$

taking $r_k = r_{k+n} = r_{k + 2n} = \ldots = r_{k-n} = r_{k - 2n} = \ldots$ and analogously for $q_k$ . The coefficients of the product are the result of a convolution of the two $n$ -element series $(r_k)$ and $(q_k)$ .

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