Glennie's identity

In mathematics, Glennie's identity is an identity used by Charles M. Glennie to establish some s-identities that are valid in special Jordan algebras but not in all Jordan algebras. A Jordan s-identity ("s" for special) is a Jordan polynomial^[1] 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.

Formal definition

Let • denote the product in a special Jordan algebra $A$ . For all X, Y, Z in A, define the Jordan triple product

{X,Y,Z} = X•(Y•Z) − Y•(Z•X) + Z•(X•Y) then Glennie's identity G₈ holds in the form:
2{ {Z,{X,Y,X},Z}, Y, Z•X} − {Z, {X, {Y, X•Z, Y}, X}, Z} = 2{ X•Z, Y, {X, {Z,Y,Z}, X} } − {X, {Z, {Y,X•Z,Y}, Z}, X}.^[2]

References

^ 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.
^ Glennie, C.M. (1966). "Some identities valid in special Jordan algebras but not in all Jordan algebras". Pacific J. Math. 16: 47–59. doi:10.2140/pjm.1966.16.47.

[1] 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.

[2] Glennie, C.M. (1966). "Some identities valid in special Jordan algebras but not in all Jordan algebras". Pacific J. Math. 16: 47–59. doi:10.2140/pjm.1966.16.47.

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