quadratic Jordan algebra

A quadratic Jordan algebra consists of a vector space A over a field K with a distinguished element 1 and a quadratic map of A into the K-endomorphisms of A, a ↦ Q(a), satisfying the conditions:

• {{math|Q(1) {{=}} id}};
• {{math|Q(Q(a)b) {{=}} Q(a)Q(b)Q(a)}} ("fundamental identity");
• {{math|Q(a)R(b,a) {{=}} R(a,b)Q(a)}} ("commutation identity"), where {{math|R(a,b)c {{=}} (Q(a + c) − Q(a) − Q(c))b.}}

Further, these properties are required to hold under any extension of scalars.{{harvnb|Racine|1973|p=1}}

An element a is invertible if {{math|Q(a)}} is invertible and there exists {{math|b}} such that {{math|Q(b)}} is the inverse of {{math|Q(a)}} and {{math|Q(a)b {{=}} a}}: such b is unique and we say that b is the inverse of a. A Jordan division algebra is one in which every non-zero element is invertible.

Let B be a subspace of A. Define B to be a quadratic ideal{{harvnb|Jacobson|1968|p=153}} or an inner ideal if the image of Q(b) is contained in B for all b in B; define B to be an outer ideal if B is mapped into itself by every Q(a) for all a in A. An ideal of A is a subspace which is both an inner and an outer ideal. A quadratic Jordan algebra is simple if it contains no non-trivial ideals.{{harvnb|Racine|1973|p=2}}

For given b, the image of Q(b) is an inner ideal: we call this the principal inner ideal on b.{{harvnb|Jacobson|1968|p=154}}

The centroid Γ of A is the subset of End_K(A) consisting of endomorphisms T which "commute" with Q in the sense that for all a

• T Q(a) = Q(a) T;
• Q(Ta) = Q(a) T².

The centroid of a simple algebra is a field: A is central if its centroid is just K.{{harvnb|Racine|1973|p=3}}

If A is a unital associative algebra over K with multiplication × then a quadratic map Q can be defined from A to End_K(A) by Q(a) : b ↦ a × b × a. This defines a quadratic Jordan algebra structure on A. A quadratic Jordan algebra is special if it is isomorphic to a subalgebra of such an algebra, otherwise exceptional.

Let A be a vector space over K with a quadratic form q and associated symmetric bilinear form q(x,y) = q(x+y) - q(x) - q(y). Let e be a "basepoint" of A, that is, an element with q(e) = 1. Define a linear functional T(y) = q(y,e) and a "reflection" y^∗ = T(y)e - y. For each x we define Q(x) by

:Q(x) : y ↦ q(x,y^∗)x − q(x) y^∗ .

Then Q defines a quadratic Jordan algebra on A.{{harvnb|Jacobson|1968|p=35}}{{harvnb|Racine|1973|pp=5–6}}

Let A be a unital Jordan algebra over a field K of characteristic not equal to 2. For a in A, let L denote the left multiplication map in the associative enveloping algebra

:$L(a)\; :\; x\; \backslash mapsto\; a\; x\; \backslash$

and define a K-endomorphism of A, called the quadratic representation, by

:$\backslash displaystyle\{Q(a)=2L(a)^2\; -L(a^2).\}$

Then Q defines a quadratic Jordan algebra.

The quadratic identities can be proved in a finite-dimensional Jordan algebra over R or C following Max Koecher, who used an invertible element. They are also easy to prove in a Jordan algebra defined by a unital associative algebra (a "special" Jordan algebra) since in that case Q(a)b = aba.

See:

• {{harvnb|Koecher|1999|pp=72–76}}
• {{harvnb|Faraut|Koranyi|1994|pp=32–34}} They are valid in any Jordan algebra over a field of characteristic not equal to 2. This was conjectured by Jacobson and proved in {{harvtxt|Macdonald|1960}}: Macdonald showed that if a polynomial identity in three variables, linear in the third, is valid in any special Jordan algebra, then it holds in all Jordan algebras.See:
• {{harvnb|Jacobson|1968|pp=40–47,52}}

In {{harvtxt|Jacobson|1969|pp=19–21}} an elementary proof, due to McCrimmon and Meyberg, is given for Jordan algebras over a field of characteristic not equal to 2.

Koecher's arguments apply for finite-dimensional Jordan algebras over the real or complex numbers.See:

• {{harvnb|Koecher|1999}}
• {{harvnb|Faraut|Koranyi|1994|pp=32–35}}

An element a in A is called invertible if it is invertible in R[a] or C[a]. If b denotes the inverse, then power associativity of a shows that L(a) and L(b) commute.

In fact a is invertible if and only if Q(a) is invertible. In that case

{{quote box|align=left|::$\backslash displaystyle\{Q(a)^\{-1\}a=a^\{-1\},\backslash ,\backslash ,\backslash ,\; Q(a^\{-1\})=Q(a)^\{-1\}.\}$}}

{{Clear}}

Indeed, if Q(a) is invertible it carries R[a] onto itself. On the other hand Q(a)1 = a², so

:$\backslash displaystyle\{(Q(a)^\{-1\}\; a)a=a\; Q(a)^\{-1\}\; a=L(a)Q(a)^\{-1\}a=Q(a)^\{-1\}a^2\; =1.\}$

The Jordan identity

:$\backslash displaystyle\{[L(a),L(a^2)](b)=0\}$

can be polarized by replacing a by a + tc and taking the coefficient of t. Rewriting this as an operator applied to c yields

:$\backslash displaystyle\{2L(ab)L(a)\; +\; L(a^2)L(b)=2L(a)L(b)L(a)\; +\; L(a^2b).\}$

Taking b = a⁻¹ in this polarized Jordan identity yields

:$\backslash displaystyle\{Q(a)L(a^\{-1\})=L(a).\}$

Replacing a by its inverse, the relation follows if L(a) and L(a⁻¹) are invertible. If not it holds for a + ε1 with ε arbitrarily small and hence also in the limit.

{{quote box|align=left|*If a and b are invertible then so is Q(a)b and it satisfies the inverse identity:

::$\backslash displaystyle\{(Q(a)b)^\{-1\}=Q(a^\{-1\})b^\{-1\}.\}$

• The quadratic representation satisfies the following fundamental identity:

::$\backslash displaystyle\{Q(Q(a)b)=Q(a)Q(b)Q(a).\}$

}}

{{Clear}}

For c in A and F(a) a function on A with values in End A, let

D_cF(a) be the derivative at t = 0 of F(a + tc). Then

:$\backslash displaystyle\{c=D\_c(Q(a)a^\{-1\})=2Q(a,c)a^\{-1\}\; +Q(a)D\_c(a^\{-1\}),\}$

where Q(a,b) if the polarization of Q

:$\backslash displaystyle\{Q(a,c)=\{1\backslash over\; 2\}\; (Q(a+c)\; -Q(a)-Q(c))=L(a)L(c)\; +\; L(c)L(a)\; -\; L(ac).\}$

Since L(a) commutes with L(a⁻¹)

:$\backslash displaystyle\{Q(a,c)a^\{-1\}=\; (L(a)L(c)\; +\; L(c)L(a)\; -\; L(ac))a^\{-1\}=c.\}$

Hence

:$\backslash displaystyle\{c=2c\; +Q(a)D\_c(a^\{-1\}),\}$

so that

{{quote box|align=left|::$\backslash displaystyle\{D\_c(a^\{-1\})=\; -\; Q(a)^\{-1\}c.\}$}}

{{Clear}}

Applying D_c to L(a⁻¹)Q(a) = L(a) and acting on b = c⁻¹ yields

:$\backslash displaystyle\{(Q(a)b)(Q(a^\{-1\})b^\{-1\})=1.\}$

On the other hand L(Q(a)b) is invertible on an open dense set where Q(a)b must also be invertible with

:$\backslash displaystyle\{(Q(a)b)^\{-1\}=Q(a^\{-1\})b^\{-1\}.\}$

Taking the derivative D_c in the variable b in the expression above gives

:$\backslash displaystyle\{-Q(Q(a)b)^\{-1\}Q(a)c\; =-Q(a)^\{-1\}Q(b)^\{-1\}c.\}$

This yields the fundamental identity for a dense set of invertible elements, so it follows in general by continuity. The fundamental identity implies that c = Q(a)b is invertible if a and b are invertible and gives a formula for the inverse of Q(c). Applying it to c gives the inverse identity in full generality.

As shown above, if a is invertible,

:$\backslash displaystyle\{Q(a,b)a^\{-1\}\; =b.\}$

Taking D_c with a as the variable gives

:$\backslash displaystyle\{0=Q(c,b)a^\{-1\}\; +\; Q(a,b)D\_c(a^\{-1\})=Q(c,b)a^\{-1\}\; -Q(a,b)Q(a)^\{-1\}c.\}$

Replacing a by a⁻¹ gives, applying Q(a) and using the fundamental identity gives

:$\backslash displaystyle\{Q(a)Q(b,c)a=\; Q(a)Q(a^\{-1\},b)Q(a)c\; =\; \backslash frac\{1\}\{2\}Q(a)[Q(a^\{-1\}+b)\; -Q(a^\{-1\})\; -Q(b)]Q(a)c\; =\; Q(Q(a)b,a)c.\}$

Hence

:$\backslash displaystyle\{Q(a)Q(b,c)a=Q(Q(a)b,a)c.\}$

Interchanging b and c gives

:$\backslash displaystyle\{Q(a)Q(b,c)a=Q(a,Q(a)c)b.\}$

On the other hand {{math|1=R(x,y)}} is defined by {{math|1=R(x,y)z = 2 Q(x,z)y}}, so this implies

:$\backslash displaystyle\{Q(a)R(b,a)c=R(a,b)Q(a)c,\}$

so that for a invertible and hence by continuity for all a

{{quote box|align=left|

::$\backslash displaystyle\{Q(a)R(b,a)=R(a,b)Q(a).\}$

}}

{{Clear}}

The Jordan identity {{math|1=a(a²b) = a²(ab)}} can be polarized by replacing a by a + tc and taking the coefficient of t. This gives{{harvnb|Meyberg|1972|pp=66–67}}

:$\backslash displaystyle\{c(a^2b)\; +\; 2a\; (ac)b)\; =\; a^2(cb)\; +\; 2\; (ac)(ab).\}$

In operator notation this implies

{{quote box|align=left|::$\backslash displaystyle\{[L(a^2),L(c)]+2[L(ac),L(a)]=0.\}$}}

{{Clear}}

Polarizing in a again gives

:$\backslash displaystyle\{c((ad)b)\; +\; d((ac)b)\; +\; a((dc)b)=\; (cd)(ab)+(da)(cb)+(ac)(db).\}$

Written as operators acting on d, this gives

:$\backslash displaystyle\{L(c)L(b)L(a)\; +L((ac)b)\; +L(a)L(b)L(c)\; =\; L(ab)L(c)+L(cb)L(a)\; +L(ac)L(b).\}$

{{Clear}}

Replacing c by b and b by a gives

{{quote box|align=left|::$\backslash displaystyle\{L(b)L(a)^2\; +L(a(ab))+L(a)^2L(b)=L(a^2)L(b)+2L(ab)L(a).\}$}}

{{Clear}}

Also, since the right hand side is symmetric in b and 'c, interchanging b and c on the left and subtracting , it follows that the commutators [L(b),L(c)] are derivations of the Jordan algebra.

Let

:$\backslash displaystyle\{Q(a)=2L(a)^2\; -\; L(a^2).\}$

Then Q(a) commutes with L(a) by the Jordan identity.

From the definitions if {{math|1=Q(a,b) = ½ (Q(a = b) − Q(a) − Q(b))}} is the associated symmetric bilinear mapping, then {{math|1=Q(a,a) = Q(a)}} and

:$\backslash displaystyle\{Q(a,b)=L(a)L(b)+L(b)L(a)-L(ab).\}$

Moreover

{{quote box|align=left|:$\backslash displaystyle\{2Q(ab,a)=L(b)Q(a)\; +Q(a)L(b).\}$}}

{{Clear}}

Indeed

:{{math|1=2Q(ab,a) − L(b)Q(a) − Q(a)L(b) = 2L(ab)L(a) + 2L(a)L(ab) − 2L(a(ab)) − 2L(a)²L(b) − 2L(b)L(a)² + L(a²)L(b) + L(b)L(a²).}}

By the second and first polarized Jordan identities this implies

:{{math|1=2Q(ab,a) − L(b)Q(a) − Q(a)L(b) = 2[L(a),L(ab)] + [L(b),L(a²)] = 0.}}

The polarized version of {{math|1=[Q(a),L(a)] = 0}} is

{{quote box|align=left|::$\backslash displaystyle\{2[Q(a,b),L(a)]=2[L(b),Q(a)].\}$}}

{{Clear}}

Now with {{math|1=R(a,b) = 2[L(a),L(b)] + 2L(ab)}}, it follows that

:$\backslash displaystyle\{Q(a)R(b,a)=2[Q(a)L(b),L(a)]\; +2Q(a)L(ab)=2[Q(ab,a),L(a)]\; +2[L(a),L(b)]Q(a)+2Q(a)L(ab).\}$

So by the last identity with ab in place of b this implies the commutation identity:

:$\backslash displaystyle\{Q(a)R(b,a)=2[L(a),L(b)]Q(a)+2L(ab)Q(a)=R(a,b)Q(a).\}$

The identity Q(a)R(b,a) = R(a,b)Q(a) can be strengthened to

{{quote box|align=left|::$\backslash displaystyle\{Q(a)R(b,a)=R(a,b)Q(a)=2Q(Q(a)b,a).\}$}}

{{Clear}}

Indeed applied to c, the first two terms give

:$\backslash displaystyle\{2Q(a)Q(b,c)a=2Q(Q(a)c,a)b.\}$

Switching b and c then gives

:$\backslash displaystyle\{Q(a)R(b,a)c=2Q(Q(a)b,a)c.\}$

The identity {{math|1=Q(Q(a)b) = Q(a)Q(b)Q(a)}} is proved using the Lie bracket relations{{harvnb|Meyberg|1972}}

{{quote box|align=left|

::$\backslash displaystyle\{[R(a,b),R(c,d)]=R(R(a,b)c,d)\; -\; R(c,R(b,a)d).\}$}}

{{Clear}}

Indeed the polarization in c of the identity {{math|1=Q(c)L(x) + L(x)Q(c) = 2Q(cx,c)}} gives

:$\backslash displaystyle\{Q(c,y)L(x)\; +L(x)Q(c,y)\; =\; Q(yx,c)\; +\; Q(cx,y).\}$

Applying both sides to d, this shows that

:$\backslash displaystyle\{[L(x),R(c,d)]=\; R(xc,d)\; -\; R(c,xd).\}$

In particular these equations hold for x = ab. On the other hand if T = [L(a),L(b)] then D(z) = Tz is a derivation of the Jordan algebra, so that

:$\backslash displaystyle\{[T,R(c,d)]\; =\; R(Dc,d)\; +\; R(c,Dd).\}$

The Lie bracket relations follow because R(a,b) = T + L(ab).

Since the Lie bracket on the left hand side is antisymmetric,

{{quote box|align=left|::$\backslash displaystyle\{R(R(a,b)c,d)\; -\; R(c,R(b,a)d)=-\; R(R(c,d)a,b)\; +R(a,R(d,c)b).\}$}}

{{Clear}}

As a consequence

{{quote box|align=left|:$\backslash displaystyle\{R(y,Q(x)y)=R(y,x)^2\; -\; 2Q(y)Q(x).\}$}}

{{Clear}}

Indeed set a = y, b = x, c = z, d = x and make both sides act on y.

On the other hand

{{quote box|align=left|::$\backslash displaystyle\{2Q(Q(a)b)=2R(a,b)Q(Q(a)b,a)-Q(a)R(b,Q(a)b).\}$}}

{{Clear}}

Indeed this follows by setting x = Q(a)b in

:$\backslash displaystyle\{[R(a,b),R(x,y)]a=-\; R(R(x,y)a,b)a\; +R(a,R(y,x)b)a.\}$

Hence, combining these equations with the strengthened commutation identity,

:$\backslash displaystyle\{2Q(Q(a)b)=2R(a,b)Q(Q(a)b,a)-R(b,a)Q(a)R(a,b)\; +2Q(a)Q(b)Q(a)=2Q(a)Q(b)Q(a).\}$

Let A be a quadratic Jordan algebra over R or C. Following {{harvtxt|Jacobson|1969}}, a linear Jordan algebra structure can be associated with A such that, if L(a) is Jordan multiplication, then the quadratic structure is given by Q(a) = 2L(a)² − L(a²).

Firstly the axiom Q(a)R(b,a) = R

(a,b)Q(a) can be strengthened to

:$\backslash displaystyle\{Q(a)R(b,a)=R(a,b)Q(a)=2Q(Q(a)b,a).\}$

Indeed applied to c, the first two terms give

:$\backslash displaystyle\{2Q(a)Q(b,c)a=2Q(Q(a)c,a)b.\}$

Switching b and c then gives

:$\backslash displaystyle\{Q(a)R(b,a)c=2Q(Q(a)b,a)c.\}$

Now let

:$\backslash displaystyle\{L(a)=\backslash frac\{1\}\{2\}R(a,1).\}$

Replacing b by a and a by 1 in the identity above gives

:$\backslash displaystyle\{R(a,1)=R(1,a)=2Q(a,1).\}$

In particular

:$\backslash displaystyle\{L(a)=Q(a,1),\backslash ,\backslash ,\backslash ,L(1)=Q(1,1)=I.\}$

If furthermore a is invertible then

:$\backslash displaystyle\{R(a,b)=2Q(Q(a)b,a)Q(a)^\{-1\}=\; 2Q(a)Q(b,a^\{-1\}).\}$

Similarly if 'b is invertible

:$\backslash displaystyle\{R(a,b)=\; 2Q(a,b^\{-1\})Q(b).\}$

The Jordan product is given by

:$\backslash displaystyle\{a\backslash circ\; b\; =\; L(a)b=\backslash frac\{1\}\{2\}R(a,1)b=Q(a,b)1,\}$

so that

:$\backslash displaystyle\{a\backslash circ\; b\; =\; b\; \backslash circ\; a.\}$

The formula above shows that 1 is an identity. Defining a² by a∘a = Q(a)1, the only remaining condition to be verified is the Jordan identity

:$\backslash displaystyle\{[L(a),L(a^2)]=0.\}$

In the fundamental identity

:$\backslash displaystyle\{Q(Q(a)b)=\; Q(a)Q(b)Q(a),\}$

Replace a by a + t, set b = 1 and compare the coefficients of t² on both sides:

:$\backslash displaystyle\{Q(a)=2Q(a,1)^2\; -\; Q(a^2,1)=\; 2L(a)^2\; -\; L(a^2).\}$

Setting b = 1 in the second axiom gives

:$\backslash displaystyle\{Q(a)L(a)=L(a)Q(a),\}$

and therefore L(a) must commute with L(a²).

In a unital linear Jordan algebra the shift identity asserts that

{{quote box|align=left|:$\backslash displaystyle\{R(Q(a)b,b)=R(a,Q(b)a).\}$}}

{{Clear}}

Following {{harvtxt|Meyberg|1972}}, it can be established as a direct consequence of polarized forms of the fundamental identity and the commutation or homotopy identity. It is also a consequence of Macdonald's theorem since it is an operator identity involving only two variables.See:

• {{harvnb|Meyberg|1972|pp=85–86}}
• {{harvnb|McCrimmon|2004|pp=202–203}}

For a in a unital linear Jordan algebra A the quadratic representation is given by

:$\backslash displaystyle\{Q(a)=2L(a)^2\; -\; L(a^2),\}$

so the corresponding symmetric bilinear mapping is

:$\backslash displaystyle\{Q(a,b)=L(a)L(b)+L(b)L(a)\; -L(ab).\}$

The other operators are given by the formula

:$\backslash displaystyle\{\backslash frac\{1\}\{2\}R(a,b)=L(a)L(b)\; -L(b)L(a)\; +L(ab),\}$

so that

:$\backslash displaystyle\{Q(a,b)=\; 2L(a)L(b)-\backslash frac\{1\}\{2\}R(a,b).\}$

The commutation or homotopy identity

:$\backslash displaystyle\{R(a,b)Q(a)=Q(a)R(b,a)=2Q(Q(a)b,a),\}$

can be polarized in a. Replacing a by a + t1 and taking the coefficient of t gives

{{quote box|align=left|:$\backslash displaystyle\{L(b)Q(a)+R(a,b)L(a)=Q(a)L(b)\; +L(a)R(b,a)=L(Q(a)b)\; +2Q(ab,a).\}$}}

{{Clear}}

The fundamental identity

:$\backslash displaystyle\{Q(Q(a)b)=Q(a)Q(b)Q(a)\}$

can be polarized in a. Replacing a by a +t1 and taking the coefficients of t gives (interchanging a and b)

{{quote box|align=left|:$\backslash displaystyle\{2Q(ab,a)=\; Q(a)L(b)\; +\; L(b)Q(a).\}$}}

{{Clear}}

Combining the two previous displayed identities yields

{{quote box|align=left|:$\backslash displaystyle\{\; L(Q(a)b)+L(b)Q(a)=L(a)R(b,a),\backslash ,\backslash ,\backslash ,\backslash ,L(Q(b)a)\; +Q(b)L(a)=\; R(b,a)L(b).\}$}}

{{Clear}}

Replacing a by a +t1 in the fundamental identity and taking the coefficient of t² gives

:$\backslash displaystyle\{2Q(Q(a)b,b)-L(a)Q(b)L(a)=\; Q(a)Q(b)+Q(b)Q(a)\; -Q(ab).\}$

Since the right hand side is symmetric this implies

{{quote box|align=left|:$\backslash displaystyle\{2Q(Q(a)b,b)-2Q(Q(b)a,a)=\; L(a)Q(b)L(a)\; -\; L(b)Q(a)L(b).\}$}}

{{Clear}}

These identities can be used to prove the shift identity:

:$\backslash displaystyle\{R(Q(a)b,b)=R(a,Q(b)a).\}$

It is equivalent to the identity

:$\backslash displaystyle\{2L(Q(a)b)L(b)-\; Q(Q(a)b,b)\; =\; 2L(a)L(Q(b)a)\; -Q(a,Q(b)a).\}$

By the previous displayed identity this is equivalent to

:$\backslash displaystyle\{[L(Q(a)b)+L(b)Q(a)]L(b)=\; L(a)[L(Q(b)a)\; +\; Q(b)L(a)].\}$

On the other hand, the bracketed terms can be simplified by the third displayed identity. It implies that both sides are equal to {{math|1=½ L(a)R(b,a)L(b)}}.

For finite-dimensional unital Jordan algebras, the shift identity can be seen more directly using mutations.{{harvnb|Koecher|1999}} Let a and b be invertible, and let

{{math|1=L_b(a)=R(a,b)}} be the Jordan multiplication in A^b. Then

{{math|1=Q(b)L_b(a) = L_a(b)Q(b)}}. Moreover

{{math|1=Q(b)Q_b(a) = Q(b)Q(a)Q(b) =Q_a(b)Q(b)}}.

on the other hand {{math|1=Q_b(a)=2L_b(a)² − L_b(a^2,b)}} and similarly with a and b interchanged. Hence

:$\backslash displaystyle\{L\_a(b^\{2,a\})Q(b)\; =Q(b)L\_b(a^\{2,b\}).\}$

Thus

:$\backslash displaystyle\{R(Q(b)a,a)Q(b)=Q(b)R(Q(a)b,b)=R(b,Q(a)b)Q(b),\}$

so the shift identity follows by cancelling Q(b). A density argument allows the invertibility assumption to be dropped.

{{see also|Jordan pair}}

A linear unital Jordan algebra gives rise to a quadratic mapping Q and associated mapping R satisfying the fundamental identity, the commutation of homotopy identity and the shift identity. A Jordan pair {{math|1=(V₊,V₋)}} consists of two vector space {{math|1=V_±}} and two quadratic mappings {{math|1=Q_±}} from {{math|1=V_±}} to {{math|1=V_∓}}. These determine bilinear mappings {{math|1=R_±}} from {{math|1=V_± × V_∓}} to {{math|1=V_±}} by the formula {{math|1=R(a,b)c = 2Q(a,c)b}} where {{math|1=2Q(a,c) = Q(a + c) − Q(a) − Q(c)}}. Omitting ± subscripts, these must satisfy{{harvnb|Loos|2006}}

the fundamental identity

:$\backslash displaystyle\{Q(Q(a)b)=Q(a)Q(b)Q(a),\}$

the commutation or homotopy identity

:$\backslash displaystyle\{R(a,b)Q(a)=Q(a)R(b,a)=2Q(Q(a)b,a),\}$

and the shift identity

:$\backslash displaystyle\{R(Q(a)b,b)=R(a,Q(b)a).\}$

A unital Jordan algebra A defines a Jordan pair by taking V_± = A with its quadratic structure maps Q and R.

• Mutation (Jordan algebra)

{{reflist|2}}

• {{citation|last1=Faraut|first1= J.|last2= Koranyi|first2= A. | author2-link = Ádám Korányi|title= Analysis on symmetric cones|series= Oxford Mathematical Monographs|publisher= Oxford University Press|year= 1994|isbn= 0198534779}}
• {{citation|last=Jacobson|first= N.|title= Structure and representations of Jordan algebras|series= American Mathematical Society Colloquium Publications|volume=39|publisher= American Mathematical Society|year= 1968 |url=https://books.google.com/books?id=aAGWAwAAQBAJ |isbn=978-0-8218-4640-7 }}
• {{citation|url=http://www.math.tifr.res.in/~publ/ln/tifr45.pdf|mr=0325715|last=Jacobson|first= N.|title=Lectures on quadratic Jordan algebras|series=Tata Institute of Fundamental Research Lectures on Mathematics|volume= 45|publisher= Tata Institute of Fundamental Research|place= Bombay|year= 1969}}
• {{citation|title=The Minnesota Notes on Jordan Algebras and Their Applications |first=M.|last= Koecher |series=Lecture Notes in Mathematics|volume=1710|publisher=Springer|isbn=3-540-66360-6|year=1999| zbl=1072.17513}}
• {{citation|first=Ottmar|last= Loos |title=Jordan pairs|series= Lecture Notes in Mathematics|volume=460|url=https://books.google.com/books?id=6Zl8CwAAQBAJ&pg=PR1 |date=2006 |publisher=Springer |isbn=978-3-540-37499-2 |orig-year=1975 }}
• {{citation|last=Loos|first=Ottmar|title=Bounded symmetric domains and Jordan pairs|series=Mathematical lectures|publisher=University of California, Irvine|year=1977|url=http://molle.fernuni-hagen.de/~loos/jordan/archive/irvine/irvine.pdf|url-status=dead|archiveurl=https://web.archive.org/web/20160303234008/http://molle.fernuni-hagen.de/~loos/jordan/archive/irvine/irvine.pdf|archivedate=2016-03-03}}
• {{citation | last=Macdonald | first= I. G.| authorlink=Ian G. Macdonald | title=Jordan algebras with three generators | journal=Proc. London Math. Soc. | volume= 10 | year=1960 | pages=395–408 |doi = 10.1112/plms/s3-10.1.395 | url=http://plms.oxfordjournals.org/content/s3-10/1/395| archive-url=https://archive.today/20130615201237/http://plms.oxfordjournals.org/content/s3-10/1/395| url-status=dead| archive-date=2013-06-15}}
• {{citation|mr=0202783|last=McCrimmon|first= Kevin|title=A general theory of Jordan rings|journal=Proc. Natl. Acad. Sci. U.S.A. |volume=56|year= 1966|issue=4|pages= 1072–9|jstor=57792|doi=10.1073/pnas.56.4.1072 | zbl=0139.25502 |pmc=220000 |pmid=16591377|doi-access=free}}
• {{citation|last=McCrimmon|first= Kevin|chapter=Quadratic methods in nonassociative algebras|title= Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1|pages= 325–330|year=1975|chapter-url=http://www.mathunion.org/ICM/ICM1974.1/Main/icm1974.1.0325.0330.ocr.pdf}}
• {{Citation | last1=McCrimmon | first1=Kevin | title=A taste of Jordan algebras | url=https://books.google.com/books?isbn=0387954473 | publisher=Springer-Verlag | series=Universitext | isbn=978-0-387-95447-9 | doi=10.1007/b97489 | id=[http://www.math.virginia.edu/Faculty/McCrimmon/ Errata] | year=2004 | mr=2014924 | zbl=1044.17001}}
• {{citation|last=McCrimmon|first= Kevin|title= Jordan algebras and their applications|journal= Bull. Amer. Math. Soc. |volume=84| year=1978|issue= 4|pages= 612–627|url=https://www.ams.org/journals/bull/1978-84-04/S0002-9904-1978-14503-0/home.html|doi=10.1090/s0002-9904-1978-14503-0|doi-access=free}}
• {{citation|last=Meyberg|first= K.|title=Lectures on algebras and triple systems|publisher=University of Virginia|year= 1972|url=http://www.math.uci.edu/~brusso/Meyberg(Reduced2).pdf}}
• {{citation | last=Racine | first=Michel L. | title=The arithmetics of quadratic Jordan algebras | series=Memoirs of the American Mathematical Society |volume=136 | isbn=978-0-8218-1836-7 | year=1973 | publisher=American Mathematical Society | zbl=0348.17009 }}

• {{citation | title=Octonion Planes Defined by Quadratic Jordan Algebras | volume=104 | series=Memoirs of the American Mathematical Society | first=John R. | last=Faulkner | publisher=American Mathematical Society | year=1970 | isbn=0-8218-5888-2 | zbl=0206.23301 }}

Category:Non-associative algebras