|
|
|
|
alternative algebra
|
(Definition)
|
|
|
A non-associative algebra  is alternative if
-
![$ [\ a,a,b\ ]=0$](http://images.planetmath.org:8080/cache/objects/6349/l2h/img2.png "$ [\ a,a,b\ ]=0$") , and
-
![$ [\ b,a,a\ ]=0$](http://images.planetmath.org:8080/cache/objects/6349/l2h/img3.png "$ [\ b,a,a\ ]=0$") ,
for any  , where ![$ [\ , , ]$](http://images.planetmath.org:8080/cache/objects/6349/l2h/img5.png "$ [\ , , ]$") is the associator on  .
Remarks
- Let
 be alternative and suppose
 . From the fact that
![$ [\ a+b,a+b,c\ ]=0$](http://images.planetmath.org:8080/cache/objects/6349/l2h/img9.png "$ [\ a+b,a+b,c\ ]=0$") , we can deduce that the associator ![$ [\ , , ]$](http://images.planetmath.org:8080/cache/objects/6349/l2h/img10.png "$ [\ , , ]$") is anti-commutative, when one of the three coordinates is held fixed. That is, for any
 ,
-
![$ [\ a,b,c\ ]=-[\ b,a,c\ ]$](http://images.planetmath.org:8080/cache/objects/6349/l2h/img12.png "$ [\ a,b,c\ ]=-[\ b,a,c\ ]$")
-
![$ [\ a,b,c\ ]=-[\ a,c,b\ ]$](http://images.planetmath.org:8080/cache/objects/6349/l2h/img13.png "$ [\ a,b,c\ ]=-[\ a,c,b\ ]$")
-
![$ [\ a,b,c\ ]=-[\ c,b,a\ ]$](http://images.planetmath.org:8080/cache/objects/6349/l2h/img14.png "$ [\ a,b,c\ ]=-[\ c,b,a\ ]$")
Put more succinctly,
where
 , the symmetric group on three letters, and
 is the sign of  .
- An alternative algebra is a flexible algebra, provided that the algebra is not Boolean (characteristic
 ). To see this, replace  in the first anti-commutative identities above with 
and the result follows.
- Artin's Theorem: If a non-associative algebra
 is not Boolean, then  is alternative iff every subalgebra of  generated by two elements is associative. The proof is clear
from the above discussion.
- A commutative alternative algebra
 is a Jordan algebra. This is true since
 shows that the Jordan identity is satisfied.
- Alternativity can be defined for a general ring
 : it is a ring such that for any  ,
 and
 .
|
"alternative algebra" is owned by CWoo.
|
|
(view preamble)
Cross-references: ring, Jordan identity, Jordan algebra, commutative, clear, associative, generated by, iff, Boolean, identities, algebra, flexible algebra, symmetric group on three letters, fixed, coordinates, associator, non-associative algebra
There are 6 references to this entry.
This is version 6 of alternative algebra, born on 2004-10-10, modified 2006-02-24.
Object id is 6349, canonical name is AlternativeAlgebra.
Accessed 5493 times total.
Classification:
| AMS MSC: | 17D05 (Nonassociative rings and algebras :: Other nonassociative rings and algebras :: Alternative rings) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
![$\displaystyle [\ a_1,a_2,a_3\ ]=\operatorname{sgn}(\pi)[\ a_{\pi(1)},a_{\pi(2)},a_{\pi(3)}\ ],$](http://images.planetmath.org:8080/cache/objects/6349/l2h/img15.png "$\displaystyle [\ a_1,a_2,a_3\ ]=\operatorname{sgn}(\pi)[\ a_{\pi(1)},a_{\pi(2)},a_{\pi(3)}\ ],$"){kind=small}