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I wanted to record a result by Max Koecher on cones and Jordan algebras, and I wound up drastically expanding the page
I have added a bunch of links
I have edited a bit at Jordan algebra:
moved the in-line references to the References-section and replaced them with corresponding pointers;
renamed what used be titled “Idea” into “Definition” and instead added an attempt at an actual “Idea”-section.
split up what used to be the section “Formally real Jordan algebras” into “Formally real Jordan algebras and their origin in quantum physics” and “Classification of formally real Jordan algebras”
to the new “Formally real Jordan algebras and their origin in quantum physics” I have added remarks on how the symmetrized Jordan product relates to the more famous commutator and how both can be seen to be two pieces of the deformation quantization of a Poisson manifold.
I made the commutator into the usual one (no $1/2$, requiring a $1/2$ elsewhere). Although one could have differing conventions, I think that this best fits with the bit on deformation quantization: the Poisson bracket deforms to the commutator without $1/2$, while the pointwise multiplication deforms to the anticommutator (the Jordan multiplication) with $1/2$.
sure, thanks.
OTOH, the Lie product in a $JLB$-algebra uses half the commutator, so I put in a remark about that (including the important link to JLB-algebra!).
An edit worth reporting rev 43.
John Baez has been thinking about the relation between Lie algebras, Jordan algebras and Noether’s theorem in
Since super-Lie algebras are needed in physics, that got me wondering if there are super-Jordan algebras. I see there are one or two mentions, such as
We could also have started from a super Jordan algebra [14,15] instead of a Jordan algebra
in
and
It is also possible to consider super Jordan algebras for generalized Jordan algebras involving both bosonic and fermionic observables. In this case the automorphism group is a supergroup.
in
There are some older papers, such as
which I can’t access. Seems like a natural idea, no?
Oh, I’m forgetting the commutativity of terms – should be looking for ’Jordan superalgebra’, then there are plenty of hits.
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