MATH 415 Lecture 23

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Endomorphism Examples

$\phi ((m,n))=(m+n,0)$ and $\psi ((m,n))=(0,n)$ are elements of $\mathrm {End} (\left\langle \mathbb {Z} \times \mathbb {Z} ,+\right\rangle )$

Weyl Algebra

Let $\left\langle F[x],+\right\rangle$ be the additive group of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F[x]} . Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi : p(x) \mapsto x \, p(x)} and $\psi :p(x)\mapsto {\frac {\mathrm {d} p}{\mathrm {d} x}}$

Thus we have $\psi (\phi (p(x)))-\phi (\psi (p(x)))=1$

Hence $\left\langle \phi ,\psi \right\rangle \subset \mathrm {End} (F[x])$.

Group Rings and Group Algebras

We start with a group $G=\left\{g_{i}~\mid ~i\in I\right\}$. Let $R$ be a commutative ring with nonzero unity.

$RG=\left\{\sum _{i}a_{i}\,g_{i}~\mid ~a_{i}\in R,\ g_{i}\in G,\ {\mbox{all but finite number of}}\ a_{i}\ {\mbox{are}}\ 0\right\}$

We define addition of elements of $RG$ as

$\left(\sum _{i}a_{i}\,g_{i}\right)+\left(\sum _{i}b_{i}\,g_{i}\right)=\sum _{i}\left(a_{i}+b_{i}\right)\,g_{i}$

We define multiplication of elements of $RG$ as

$\left(\sum _{i}a_{i}\,g_{i}\right)\,\left(\sum _{i}b_{i}\,g_{i}\right)=\sum _{i}\left(\sum _{g_{j}\,g_{k}=g_{i}}a_{j}\,b_{k}\right)\,g_{i}$

(similar to multiplication of polynomials)

Theorem 24.4

$\left\langle RG,+,\cdot \right\rangle$ is a ring. In particular, it is called a group ring of $G$ over $R$.

If $R=F$ is a field, then $\left\langle FG,+,\cdot \right\rangle$ is a group algebra of $G$ over $F$.