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I have attempted to google for this, but the searching is marred by the close relationship of Jordan Algebras and Quantum Mechanics. I have been passively thinking of this question for ages, tonight I finally post it. :)

My question is fairly simple. Similar to the universal enveloping algebra for Lie algebras, we should be able to form a universal enveloping algebra for Jordan algebras. So question one:

Can we indeed construct this "Universal enveloping algebra"?(Thanks Matt)

We can form a Universal enveloping algebra for Jordan algebras. So;

Are there some sort of Serre relations for Jordan algebras?

and then as we do for Lie algebras,

Can we find "quantum Jordan 'Serre' relations" for quantum Jordan algebras?

or just;

Can we deform the Jordan Universal Enveloping algebra in a similar way that we deform Lie Enveloping algebras to their quantized versions?

I would be happy with an explanation or a reference.

Thanks in advance!

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Yes, there is a notion of universal enveloping algebra for Jordan algbera; it is left adjoint to the functor that associates a Jordan algebra to an associative algebra.

However, there is no analogue of the Poincare--Birkhoff--Witt theorem for Jordan algebras, and so the map from the Jordan algebra to its universal enveloping algebra need not be injective. (I forget where I learned this, but it should be discussed in most texts that discuss Jordan algebras; indeed, there is an adjective --- which I also forget --- used to separate those Jordan algebras that do embed into their enveloping algebra from those that don't.)

As for deformation quantization, I don't know.

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