Oh, I’m forgetting the commutativity of terms – should be looking for ’Jordan superalgebra’, then there are plenty of hits.

]]>John Baez has been thinking about the relation between Lie algebras, Jordan algebras and Noether’s theorem in

- John Baez,
*Getting to the Bottom of Noether’s Theorem*, (arXiv:2006.14741)

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

- Sultan Catto, Yasemin Gürcan, Amish Khalfan and Levent Kurt,
*Quantum Symmetries: From Clifford and Hurwitz Algebras to M-Theory and Leech Lattices*, pdf

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

- Čestmir Burdik and Sultan Catto,
*Hurwitz Algebras and the Octonion Algebra*, (paper)

There are some older papers, such as

- L.A.Ferreira, J.F.Gomes, P.Teotonio Sobrinho, A.H.Zimerman,
*Symplectic bosons, Fermi fields and super Jordan algebras*, (doi:10.1016/0370-2693(90)91933-3)

which I can’t access. Seems like a natural idea, no?

]]>An edit worth reporting rev 43.

]]>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!).

]]>sure, thanks.

]]>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 $1/2$.

]]>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 have added a bunch of links

]]>I wanted to record a result by Max Koecher on cones and Jordan algebras, and I wound up drastically expanding the page

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