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alternative algebra
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(Definition)
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A non-associative algebra is alternative if
-
, and
-
,
for any , where is the associator on .
Remarks
- Let
be alternative and suppose
. From the fact that
, 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\ ]=-[\ b,a,c\ ]$ $ [\ a,b,c\ ]=-[\ b,a,c\ ]$](http://images.planetmath.org:8080/cache/objects/6349/l2h/img12.png)
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![$ [\ a,b,c\ ]=-[\ a,c,b\ ]$ $ [\ a,b,c\ ]=-[\ a,c,b\ ]$](http://images.planetmath.org:8080/cache/objects/6349/l2h/img13.png)
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![$ [\ a,b,c\ ]=-[\ c,b,a\ ]$ $ [\ a,b,c\ ]=-[\ c,b,a\ ]$](http://images.planetmath.org:8080/cache/objects/6349/l2h/img14.png)
Put more succinctly,
where
, the symmetric group on three letters, and
is the sign of .
- An alternative algebra is a flexible algebra, provided that the algebra is not Boolean (characteristic
). To see this, replace in the first anti-commutative identities above with
and the result follows.
- Artin's Theorem: If a non-associative algebra
is not Boolean, then is alternative iff every subalgebra of generated by two elements is associative. The proof is clear
from the above discussion.
- A commutative alternative algebra
is a Jordan algebra. This is true since
shows that the Jordan identity is satisfied.
- Alternativity can be defined for a general ring
: it is a ring such that for any ,
and
.
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"alternative algebra" is owned by CWoo.
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(view preamble)
Cross-references: ring, Jordan identity, Jordan algebra, commutative, clear, associative, generated by, iff, Boolean, identities, algebra, flexible algebra, symmetric group on three letters, fixed, coordinates, associator, non-associative algebra
There are 6 references to this entry.
This is version 6 of alternative algebra, born on 2004-10-10, modified 2006-02-24.
Object id is 6349, canonical name is AlternativeAlgebra.
Accessed 5493 times total.
Classification:
| AMS MSC: | 17D05 (Nonassociative rings and algebras :: Other nonassociative rings and algebras :: Alternative rings) |
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Pending Errata and Addenda
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