Talk:Jordan algebra

Non-Associative Algebra

It might be helpful to explain how a "non-associative" algebra can satisfy commutativity (as is stated in the given definition). —Preceding unsigned comment added by 132.170.54.156 (talk) 05:18, 6 February 2008 (UTC)

Jordan superalgebra

Jordan superalgebra is not defined in the article and the link Jordan superalgebra redirects back to this article. A definition would help. Deltahedron (talk) 11:16, 21 April 2013 (UTC)

Quadratic Jordan algebras

This is one of the main things missing from this article. There is no discussion of the quadratic representation. It is treated in great detail in textbooks, for example McCrimmon. It usefully describes various automorphism groups and structure groups. That certainly could be placed in the generalizations section for example. Mathsci (talk) 12:41, 5 May 2013 (UTC)

yes, please can somebody, who knows to write Wiki, fill in the quadratic representation and how it leads to a symmetric multiplication (in the sense of Ottmar Loos) on the invertible elements. — Preceding unsigned comment added by 130.133.155.66 (talk) 08:47, 5 September 2013 (UTC)

I subsequently wrote the article on quadratic Jordan algebra and it is now in the "see also" section of this article. It is also discussed in detail in Symmetric cone#Quadratic representation. Mathsci (talk) 12:55, 5 September 2013 (UTC)

Thank you - nice articles - Wiki now even beats the text-books. — Preceding unsigned comment added by 130.133.155.68 (talk) 14:39, 26 October 2013 (UTC)

Nevertheless, the article should include the data of a Jordan algebra, that means Killing-form, left-multiplication L, quadratic representation P and that the fundamental formula of P corresponds exactly to the adjoint representation of Lie algebras. — Preceding unsigned comment added by 130.133.155.68 (talk) 16:52, 14 May 2016 (UTC)

Forms of the exceptional jordan algebra over R

I believe there are 3 forms, corresponding to the 3 forms of the group F4. The obvious forms use the split/non-split octonions. The third (less obvious form) comes from a different subalgebra of the unique exceptional Jordan algebra over C. See On Reduced Exceptional Simple Jordan Algebras, A. A. Albert and N. Jacobson, Annals of Mathematics, Second Series, Vol. 66, No. 3 (Nov., 1957), pp. 400-417

The current reference given to the claim about 2 forms over R (article of Albert) classifies the reduced algebras after finite extension (here this just means up to C). A 130.15.101.115 (talk) 20:26, 4 November 2013 (UTC)

Yes, quite right. Thanks for catching this. Mark M (talk) 23:38, 14 December 2013 (UTC)

Application to quantum mechanics

The article states Jordan algebras were first introduced by Pascual Jordan (1933) to formalize the notion of an algebra of observables in quantum mechanics but does not say anything further about where that went. McCrimmon (2004) indicates in a rather informal way that the programme faltered after the 1934 classification of simple algebras, showing that there were insufficiently many exceptional finite dimensional algebras, and that the theorem of Zelmanov put paid to the hopes of finding infinite dimensional candidates. So is there a more formal treatment of the history somewhere? Deltahedron (talk

This remark is not clear. There is only one exceptional real simple Jordan algebra called M(3,8). But this has nothing to do with physics, except for the fact, that this Jordan algebra failed for any use in physics. This does not say anything on general Jordan algebras. Note that in a natural way, Minkowski space has a Jordan algebra structure, more general every pseudoorthogonal vector space has one. And every Hilbert space also, even if it is infinite-dimensional. And even if the non-degenerate sesqui-linear form is indefinite, i.e. we have a pseudo-Hilbert space. These examples should be included in the article.