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In mathematics, '''Glennie's identity''' is an identity used by Charles M. Glennie to establish some s-identities that are valid in [[Jordan algebra#Special Jordan algebras|special Jordan algebras]] but not in all [[Jordan algebra]]s. A Jordan s-identity ("s" for special) is a Jordan polynomial<ref>In this context, Jordan polynomial is a polynomial operator on a Jordan algebra. The Jordan algebra is named after [[Pascual Jordan]] and not the [[Camille Jordan]] famous for the [[Jordan normal form]]. Jordan polynomial has a different meaning in the context of the Jordan normal form.</ref> which vanishes in all special Jordan algebras but not in all Jordan algebras. What is now known as Glennie's identity first appeared in his 1963 Yale PhD thesis with [[Nathan Jacobson]] as thesis advisor.
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==Formal definition==
 
Let • denote the product in a special Jordan algebra <math>A</math>. For all ''X'', ''Y'', ''Z'' in ''A'', define the Jordan triple product
# {''X'',''Y'',''Z''} = (''X''•''Y'')•''Z'' &minus; (''Y''•''Z'')•''X'' + (''Z''•''X'')•''Y'' then Glennie's identity ''G''<sub>8</sub> holds in the form:
#2{ {''Z'',{''X'',''Y'',''X''},''Z''}, ''Y'', ''Z''•''X''} &minus; {''Z'', {''X'', {''Y'', ''X''•''Z'', ''Y''}, ''X''}, ''Z''} = 2{ ''X''•''Z'', ''Y'', {''X'', {''Z'',''Y'',''Z''}, ''X''} } &minus; {''X'', {''Z'', {''Y'',''X''•''Z'',''Y''}, ''Z''}, ''X''}.<ref>{{cite journal|title=Some identities valid in special Jordan algebras but not in all Jordan algebras|journal=Pacific J. Math|author=Glennie, C.M.|year=1966|volume=16|pages=47–59|url=http://projecteuclid.org/euclid.pjm/1102995084}}</ref>
 
==References==
{{reflist}}
 
[[Category:Non-associative algebras]]

Latest revision as of 02:06, 8 May 2014

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