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A Jordan algebra is a vector space with a commutative bilinear operation $\circ$ obeying an identity that's often written as

$$ (x \circ y) \circ (x \circ x) = x \circ (y \circ (x \circ x)) . $$

I find this identity rather obscure. If we write $x^2 = x \circ x$ and use $L_a$ to stand for left multiplication by $a$, we can rewrite it in a more appealing form:

$$ L_{x^2} L_x = L_x L_{x^2} .$$

However, I'd be even happier if this were a special case of a more general identity

$$ L_{x^m} L_{x^n} = L_{x^n} L_{x^m} \qquad (\ast) $$

holding for all $n, m \in \mathbb{N}$.

This more general identity parses in any Jordan algebra, because any Jordan algebra is power-associative: expressions like $x \circ \cdots \circ x$ are independent of how you parenthesize them, so $x^n$ is well-defined. But is this more general identity $(\ast)$ true in every Jordan algebra?

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    $\begingroup$ Seems not what you want but multilinear form of the Jordan identity does not look so obscure. It is$$[L_{ab},L_c]+[L_{bc},L_a]+[L_{ca},L_b]=0$$(where $[-,-]$ is usual commutator of operators). Another equivalent form:$$\operatorname{as}(ab,d,c)+\operatorname{as}(bc,d,a)+\operatorname{as}(ca,d,b)=0$$where $\operatorname{as}(x,y,z)=(xy)z-x(yz)$ is the associator. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jun 8, 2020 at 19:02
  • $\begingroup$ I'm sorry to be foolish, but aren't some of these right multiplications? For example, $L_{x^2}L_x y$ should be $x^2 \circ (x \circ y)$, but we want $(x \circ y) \circ x^2$. $\endgroup$
    – LSpice
    Commented Jun 8, 2020 at 20:56
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    $\begingroup$ A Jordan algebra is commutative so left multiplication is the same as right multiplication. $\endgroup$
    – John Baez
    Commented Jun 8, 2020 at 22:10

1 Answer 1

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This identity (*) is, indeed, true, and is, in fact, a step in one of the standard ways to prove that Jordan algebras are power-associative: see McCrimmon's 2004 book A Taste of Jordan Algebras, exercise 5.2.2A (question (2)) on page 201.

Edit: another reference, which has the better taste of not being an exercise and of being earlier in a book: Jacobson, Structure and Representations of Jordan Algebras (1968), page 35, just above formula (56). (I also just realized that the fact is mentioned in the Wikipedia article on Jordan algebras, with that reference.)

For completeness of MathOverflow, I might as well copy the essence of the argument: first prove the identity $$ L_{(b\circ d)\circ c} = L_{b\circ d}L_c + L_{c\circ d}L_b + L_{b\circ c}L_d - L_b L_c L_d - L_d L_c L_b $$ by linearizing the Jordan identity (here we use the fact that the characteristic is not two). Apply this to $b=a^k$ and $c=d=a$, giving $$ L_{a^{k+2}} = 2L_{a^{k+1}}L_a + L_{a^2}L_{a^k} - L_{a^k} L_a^2 - L_a^2 L_{a^k} $$ From this it follows by induction that all $L_{a^k}$ belong to the algebra generated by $L_a$ and $L_{a^2}$ and, since these commute, they all commute.

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    $\begingroup$ Wow, thanks! I don't know how I overlooked that. I now claim this is the true definition of Jordan algebra. $\endgroup$
    – John Baez
    Commented Jun 7, 2020 at 20:44
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    $\begingroup$ @JohnBaez assuming there should be a "true definition"... one advantage of the usual definition is its simplicity, and the fact that it is given by a finite number of identities. Certainly a student whose exercise is to check that some given Jordan is indeed Jordan will be grateful that the definition is the usual one. Also the usual one makes it easy to implement an algorithm checking whether a (non-associative) algebra given by structure constants (in a computable field) is Jordan. $\endgroup$
    – YCor
    Commented Jun 7, 2020 at 20:59
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    $\begingroup$ @YCor why not have a conceptually clear definition, like John would like, and then a theorem that gives a finite (short!) list of identities to check that suffices to imply the full definition? $\endgroup$
    – David Roberts ♦
    Commented Jun 8, 2020 at 10:41
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    $\begingroup$ @DavidRoberts why not, I have no problem with it, but don't see why it should be called the only "true" definition. From another intuition one might find the original definition simpler. Also, what do you think of associativity? $(ab)c=a(bc)$ may sound nice because we're used to it, but what's important is that all bracketings of products $x_1\dots x_n$ lead to the same element. In spite of this I'm happier with the usual short definition. $\endgroup$
    – YCor
    Commented Jun 8, 2020 at 10:47
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    $\begingroup$ [This addresses the edited post, not the current stream of comments] Actually I once checked that the naive definition of Jordan algebra in char 2 (commutative + Jordan identity) is so bad that it does not pass to extension of scalars (for extensions of fields of char 2, probably $F_2\subset F_4$). The correct definition of Jordan ring should be that a ring satisfying all identities with coefficients in $\mathbf{Z}$ satisfied by, say, all complex Jordan algebras. When 2 is not invertible it implies we should add the polarizations among the axioms. $\endgroup$
    – YCor
    Commented Jun 8, 2020 at 19:21

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