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Alternative Algebra


Let A denote an R-algebra, so that A is a vector space over R and

 A×A->A
(1)
 (x,y)|->x·y.
(2)

Then A is said to be alternative if, for all x,y in A,

 (x·y)·y=x·(y·y)
(3)
 (x·x)·y=x·(x·y).
(4)

Here, vector multiplication x·y is assumed to be bilinear.

The associator (x,y,z) is an alternating function, and the subalgebra generated by two elements is associative.


See also

Associator

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References

Finch, S. "Zero Divisor Structure in Real Algebras." http://algo.inria.fr/csolve/zerodiv/.Schafer, R. D. An Introduction to Non-Associative Algebras. New York: Dover, p. 5, 1995.

Referenced on Wolfram|Alpha

Alternative Algebra

Cite this as:

Weisstein, Eric W. "Alternative Algebra." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AlternativeAlgebra.html

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