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Revision difference : power-associative algebra |
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Version 11 |
| Let $A$ be a non-associative algebra. A subalgebra $B$ of $A$ is said to be \emph{cyclic} if it is generated by one element. |
Let $A$ be a non-associative algebra. A subalgebra $B$ of $A$ is said to be \emph{cyclic} if it is generated by one element. |
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| A non-associative algebra is \emph{power-associative} if, $[B,B,B]=0$ for any cyclic subalgebra $B$ of $A$, where $[-,-,-]$ is the associator. |
A non-associative algebra is \emph{power-associative} if, $[B,B,B]=0$ for any cyclic subalgebra $B$ of $A$, where $[-,-,-]$ is the associator. |
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| If we inductively define the powers of an element $a\in A$ by |
If we inductively define the powers of an element $a\in A$ by |
| \begin{enumerate} |
\begin{enumerate} |
| \item (when $A$ is unital with $1\neq0$) $a^0:=1$, |
\item (when $A$ is unital with $1\neq0$) $a^0:=1$, |
| \item $a^1:=a$, and |
\item $a^1:=a$, and |
| \item $a^n:=a(a^{n-1})$ for $n>1$, |
\item $a^n:=a(a^{n-1})$ for $n>1$, |
| \end{enumerate} |
\end{enumerate} |
| then power-associativity of $A$ means that $[a^i,a^j,a^k]=0$ for any non-negative integers $i,j$ and $k$, since the associator is trilinear (linear in each of the three coordinates). This implies that $a^ma^n=a^{m+n}$. In addition, $(a^m)^n=a^{mn}$. |
then power-associativity of $A$ means that $[a^i,a^j,a^k]=0$ for any non-negative integers $i,j$ and $k$, since the associator is trilinear (linear in each of the three coordinates). This implies that $a^ma^n=a^{m+n}$. In addition, $(a^m)^n=a^{mn}$. |
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| A theorem, due to A. Albert, states that any finite power-associative division algebra over the integers of characteristic not equal to 2, 3, or 5 is a field. This is a generalization of the Wedderburn's Theorem on finite division rings. |
A theorem, due to A. Albert, states that any finite power-associative division algebra over the integers of characteristic not equal to 2, 3, or 5 is a field. This is a generalization of the Wedderburn's Theorem on finite division rings. |
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| \begin{thebibliography}{8} |
\begin{thebibliography}{8} |
| \bibitem{Shafer} R. D. Schafer, {\em An Introduction on Nonassociative Algebras}, Dover, New York (1995). |
\bibitem{Shafer} R. D. Schafer, {\em An Introduction on Nonassociative Algebras}, Dover, New York (1995). |
| \end{thebibliography} |
\end{thebibliography} |
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