alternative algebra (original) (raw)

Remarks

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  Let A be alternative and suppose char⁡(A)≠2. From the fact that [a+b,a+b,c]=0, we can deduce that the associator [,,] is anti-commutative, when one of the three coordinates is held fixed. That is, for any a,b,c∈A,
  1. (a)
     [a,b,c]=-[b,a,c]
  2. (b)
     [a,b,c]=-[a,c,b]
  3. (c)
     [a,b,c]=-[c,b,a]
     Put more succinctly,

     [a1,a2,a3]=sgn⁡(π)⁢[aπ⁢(1),aπ⁢(2),aπ⁢(3)],
     where π∈S3, the symmetric group on three letters, and sgn⁡(π) is the sign (http://planetmath.org/SignatureOfAPermutation) of π.

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• •
  Artin’s Theorem: If a non-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.
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  A commutative alternative algebra A is a Jordan algebra. This is true since a2⁢(b⁢a)=a2⁢(a⁢b)=(a⁢b)⁢a2=((a⁢b)⁢a)⁢a=(a⁢(a⁢b))⁢a=(a2⁢b)⁢a shows that the Jordan identity is satisfied.
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  Alternativity can be defined for a general ring R: it is a non-associative ring such that for any a,b∈R, (a⁢a)⁢b=a⁢(a⁢b) and (a⁢b)⁢b=a⁢(b⁢b). Equivalently, an alternative ring is an alternative algebra over ℤ.