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'alternative algebra'
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| Title of object: |
alternative algebra |
| Canonical Name: |
AlternativeAlgebra |
| Type: |
Definition |
| Created on: |
2004-10-10 14:24:08 |
| Modified on: |
2004-12-12 13:17:14 |
| Classification: |
msc:17D05 |
| Defines: |
Artin's theorem on alternative algebras |
Preamble:
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Content:
A not-necessarily-associative algebra $A$ is \emph{alternative} if
\begin{enumerate}
\item $[\ a,a,b\ ]=0$, and
\item $[\ b,a,a\ ]=0$,
\end{enumerate}
for any $a,b\in A$, where $[\ , , ]$ is the associator on $A$.
\par
\textbf{Remarks}
\begin{itemize}
\item Let $A$ be alternative and suppose $\operatorname{char}(A)\neq2$. From the fact that $[\ a+b,a+b,c\ ]=0$, we can deduce that the associator $[\ , , ]$ is \emph{anti-commutative}, when one of the three coordinates is held fixed. That is, for any $a,b,c\in A$,
\begin{enumerate}
\item $[\ a,b,c\ ]=-[\ b,a,c\ ]$
\item $[\ a,b,c\ ]=-[\ a,c,b\ ]$
\item $[\ a,b,c\ ]=-[\ c,b,a\ ]$
\end{enumerate}
Put more succinctly, $$[\ a_1,a_2,a_3\ ]=\operatorname{sgn}(\pi)[\ a_{\pi(1)},a_{\pi(2)},a_{\pi(3)}\ ],$$ where $\pi\in S_3$, the symmetric group on three letters, and $\operatorname{sgn}(\pi)$ is the \PMlinkname{sign}{SignatureOfAPermutation} of $\pi$.
\item An alternative algebra is a flexible algebra, provided that the algebra is not \PMlinkname{Boolean}{BooleanLattice} (\PMlinkname{characteristic}{Characteristic} $\neq2$). To see this, replace $c$ in the first anti-commutative identities above with $a$ and the result follows.
\item \textbf{Artin's Theorem}: If a not-necessarily-associative algebra $A$ is not Boolean, then $A$ is alternative iff every subalgebra of $A$ generated by two elements is associative. The proof is clear from the above discussion.
\item A commutative alternative algebra $A$ is a Jordan algebra. This is true since $a^2(ba)=a^2(ab)=(ab)a^2=((ab)a)a=(a(ab))a=(a^2b)a$ shows that the Jordan identity is satisfied.
\end{itemize} |
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