Mutation (Jordan algebra)

In mathematics, a mutation, also called a homotope, of a unital Jordan algebra is a new Jordan algebra defined by a given element of the Jordan algebra. The mutation has a unit if and only if the given element is invertible, in which case the mutation is called a proper mutation or an isotope. Mutations were first introduced by Max Koecher in his Jordan algebraic approach to Hermitian symmetric spaces and bounded symmetric domains of tube type. Their functorial properties allow an explicit construction of the corresponding Hermitian symmetric space of compact type as a compactification of a finite-dimensional complex semisimple Jordan algebra. The automorphism group of the compactification becomes a complex subgroup, the complexification of its maximal compact subgroup. Both groups act transitively on the compactification. The theory has been extended to cover all Hermitian symmetric spaces using the theory of Jordan pairs or Jordan triple systems. Koecher obtained the results in the more general case directly from the Jordan algebra case using the fact that only Jordan pairs associated with period two automorphisms of Jordan algebras are required.

Definitions

Template:See also Let A be a unital Jordan algebra over a field k of characteristic ≠ 2. For a in A define the Jordan multiplication operator on A by

$L(a)b=ab$

and the quadratic representation Q(a) by

$Q(a)=2L(a{)}^2-L(a^2).$

It satisfies

$Q(1)=I.$

the fundamental identity

$Q(Q(a)b)=Q(a)Q(b)Q(a)$

the commutation or homotopy identity

$Q(a)R(b,a)=R(a,b)Q(a)=2Q(Q(a)b,a),$

where

$R(a,b)c=2Q(a,c)b,Q(x,y)=\frac{1}{2}(Q(x+y)-Q(x)-Q(y)).$

In particular if a or b is invertible then

$R(a,b)=2Q(a)Q(a^{-1},b)=2Q(a,b^{-1})Q(b).$

It follows that A with the operations Q and R and the identity element defines a quadratic Jordan algebra, where a quadratic Jordan algebra consists of a vector space A with a distinguished element 1 and a quadratic map of A into endomorphisms of A, a ↦ Q(a), satisfying the conditions:

• Template:Math
• Template:Math ("fundamental identity")
• Template:Math ("commutation or homotopy identity"), where Template:Math

The Jordan triple product is defined by

$\{a,b,c\}=(ab)c+(cb)a-(ac)b,$

so that

$Q(a)b=\{a,b,a\},Q(a,c)b=\{a,b,c\},R(a,b)c=\{a,b,c\}.$

There are also the formulas

$Q(a,b)=L(a)L(b)+L(b)L(a)-L(ab),R(a,b)=[L(a),L(b)]+L(ab).$

For y in A the mutation A^y is defined to the vector space A with multiplication

$a\circ b=\{a,y,b\}.$

If Q(y) is invertible, the mutual is called a proper mutation or isotope.

Quadratic Jordan algebras

Template:Main Let A be a quadratic Jordan algebra over a field k of characteristic ≠ 2. Following Template:Harvtxt, 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

$Q(a)R(b,a)=R(a,b)Q(a)=2Q(Q(a)b,a).$

Indeed, applied to c, the first two terms give

$2Q(a)Q(b,c)a=2Q(Q(a)c,a)b.$

Switching b and c then gives

$Q(a)R(b,a)c=2Q(Q(a)b,a)c.$

Now let

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

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

$R(a,1)=R(1,a)=2Q(a,1).$

In particular

$L(a)=Q(a,1),L(1)=Q(1,1)=I.$

The Jordan product is given by

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

so that

$a\circ b=b\circ a.$

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

$[L(a),L(a^2)]=0.$

In the fundamental identity

$Q(Q(a)b)=Q(a)Q(b)Q(a),$

Replace a by a + t1, set b = 1 and compare the coefficients of t² on both sides:

$Q(a)=2Q(a,1{)}^2-Q(a^2,1)=2L(a{)}^2-L(a^2).$

Setting b = 1 in the second axiom gives

$Q(a)L(a)=L(a)Q(a),$

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

Inverses

Let A be a unital Jordan algebra over a field k of characteristic ≠ 2. An element a in a unital Jordan algebra A is said to be invertible if there is an element b such that ab = 1 and a²b = a.

Properties.

Template:Quote box Template:Clear If Template:Math and Template:Math, then Template:Math. The Jordan identity Template:Math can be polarized by replacing Template:Math by Template:Math and taking the coefficient of Template:Math. This gives

$[L(x^2),L(y)]+2[L(xy),L(x)]=0.$

Taking Template:Math or Template:Math and Template:Math or Template:Math shows that Template:Math commutes with Template:Math and Template:Math commutes with Template:Math. Hence Template:Math. Applying Template:Math gives Template:Math. Hence Template:Math. Conversely if Template:Math and Template:Math, then the second relation gives Template:Math. So both Template:Math and Template:Math are invertible. The first gives Template:Math so that Template:Math and Template:Math are each other's inverses. Since Template:Math commutes with Template:Math it commutes with its inverse Template:Math. Similarly Template:Math commutes with Template:Math. So Template:Math and Template:Math.

Template:Quote box Template:Clear Indeed, if Template:Math is invertible then the above implies Template:Math is invertible with inverse Template:Math. Any inverse b satisfies Template:Math, so Template:Math. Conversely if Template:Math is invertible let Template:Math. Then Template:Math. The fundamental identity then implies that Template:Math and Template:Math are each other's inverses so that Template:Math.

Template:Quote box Template:Clear This follows from the formula Template:Math.

Template:Quote box Template:Clear Suppose that Template:Math. Then by the fundamental identity Template:Math is invertible, so Template:Math is invertible.

Template:Quote box Template:Clear This is an immediate consequence of the fundamental identity and the fact that Template:Math is invertible if and only Template:Math and Template:Math are invertible. Template:Quote box Template:Clear In the commutation identity Template:Math, set Template:Math with Template:Math. Then Template:Math and Template:Math. Since Template:Math commutes with Template:Math, Template:Math.

Template:Quote box Template:Clear If Template:Math and Template:Math commute, then Template:Math implies Template:Math. Conversely suppose that Template:Math is invertible with inverse Template:Math. Then Template:Math. Morevoer Template:Math commutes with Template:Math and hence its inverse Template:Math. So it commutes with Template:Math.

Template:Quote box Template:Clear The algebra Template:Math is commutative and associative, so if Template:Math is an inverse there Template:Math and Template:Math. Conversely Template:Math leaves Template:Math invariant. So if it is bijective on Template:Math it is bijective there. Thus Template:Math lies in Template:Math.

Elementary properties of proper mutations

Template:Quote box Template:Clear In fact multiplication in the algebra A^y is given by

$a\circ b=\{a,y,b\},$

so by definition is commutative. It follows that

$a\circ b=L_y(a)b,$

with

$L_y(a)=[L(a),L(y)]+L(ay).$

If e satisfies Template:Math, then taking a = 1 gives

$ye=1.$

Taking a = e gives

$e(ya)=y(ea)$

so that L(y) and L(e) commute. Hence y is invertible and e = y⁻¹.

Now for y invertible set

$Q_y(a)=Q(a)Q(y),R_y(a,b)=R(a,Q(y)b).$

Then

$Q_y(e)=Q_y(y^{-1})=Q(y^{-1})Q(y)=I.$

Moreover,

$Q_y(a)Q_y(b)Q_y(a)=Q(a)Q(y)Q(b)Q(y)Q(a)Q(y)=Q(a)Q(Q(y)b)Q(a)Q(y)=Q(Q(a)Q(y)b)Q(y)=Q_y(Q_y(a)b).$

Finally

$Q(y)R(c,Q(y)d)Q(y{)}^{-1}=R(Q(y)c,d),$

since

$Q(y)R(c,Q(y)d)x=2Q(y)Q(c,x)Q(y)d=2Q(Q(y)c,Q(y)x)d=R(Q(y)c,d)Q(y)x.$

Hence

$Q_y(a)R_y(b,a)=Q(a)Q(y)R(b,Q(y)a)=Q(y^{-1})Q(Q(y)a)R(b,Q(y)a)=Q(y{)}^{-1}R(Q(y)a,b)Q(Q(y)a)=R_y(a,b)Q_y(a).$

Thus Template:Math is a unital quadratic Jordan algebra. It therefore corresponds to a linear Jordan algebra with the associated Jordan multiplication operator M(a) given by

$M(a)b=\frac{1}{2}{R}_y(a,e)b=\frac{1}{2}R(a,Q(y)e)b=\frac{1}{2}R(a,y)b=\{a,y,b\}=L_y(a)b.$

This shows that the operators Template:Math satisfy the Jordan identity so that the proper mutation or isotope Template:Math is a unital Jordan algebra. The correspondence with quadratic Jordan algebras shows that its quadratic representation is given by Template:Math.

Nonunital mutations

The definition of mutation also applies to non-invertible elements y. If A is finite-dimensional over R or C, invertible elements a in A are dense, since invertibility is equivalent to the condition that det Q(a) ≠ 0. So by continuity the Jordan identity for proper mutations implies the Jordan identity for arbitrary mutations. In general the Jordan identity can be deduced from Macdonald's theorem for Jordan algebras because it involves only two elements of the Jordan algebra. Alternatively, the Jordan identity can be deduced by realizing the mutation inside a unital quadratic algebra.

For a in A define a quadratic structure on A₁ = A ⊕ k by

$Q_1(a\oplus \alpha 1)(b\oplus \beta 1)={\alpha }^2\beta 1\oplus [{\alpha }^{2}a+{\alpha }^{2}b+2\alpha \beta a+\alpha \{a,y,b\}+\beta Q(a)y+Q(a)Q(y)b].$

It can then be verified that Template:Math is a unital quadratic Jordan algebra. The unital Jordan algebra to which it corresponds has A^y as an ideal, so that in particular A^y satisfies the Jordan identity. The identities for a unital quadratic Jordan algebra follow from the following compatibility properties of the quadratic map Template:Math and the squaring map Template:Math:

• Template:Math
• Template:Math
• Template:Math
• Template:Math
• Template:Math
• Template:Math.

Hua's identity

Let A be a unital Jordan algebra. If a, b and a – b are invertible, then Hua's identity holds:

Template:Quote box Template:Clear In particular if x and 1 – x are invertible, then so too is 1 – x⁻¹ with

$(1-x{)}^{-1}+(1-x^{-1}{)}^{-1}=1.$

To prove the identity for x, set Template:Math. Then Template:Math. Thus Template:Math commutes with Template:Math and Template:Math. Since Template:Math, it also commutes with Template:Math and Template:Math. Since Template:Math, Template:Math also commutes with Template:Math and Template:Math.

It follows that Template:Math. Moreover, Template:Math since Template:Math. So Template:Math commutes with Template:Math and hence Template:Math. Thus Template:Math has inverse Template:Math.

Now let Template:Math be the mutation of A defined by a. The identity element of Template:Math is Template:Math. Moreover, an invertible element c in A is also invertible in Template:Math with inverse Template:Math.

Let Template:Math in Template:Math. It is invertible in A, as is Template:Math. So by the special case of Hua's identity for x in Template:Math

$a^{-1}=Q(a{)}^{-1}(a^{-1}-Q(a{)}^{-1}b{)}^{-1}+Q(a{)}^{-1}(a^{-1}-b^{-1}{)}^{-1}=(a-b{)}^{-1}+(a-Q(a)b^{-1}{)}^{-1}.$

Bergman operator

If A is a unital Jordan algebra, the Bergman operator is defined for a, b in A by

$B(a,b)=I-R(a,b)+Q(a)Q(b).$

If a is invertible then

$B(a,b)=Q(a)Q(a^{-1}-b);$

while if b is invertible then

$B(a,b)=Q(a-b^{-1})Q(b).$

In fact if a is invertible

Template:Math

and similarly if b is invertible.

More generally the Bergman operator satisfies a version of the commutation or homotopy identity:

Template:Quote box Template:Clear and a version of the fundamental identity:

Template:Quote box Template:Clear There is also a third more technical identity:

Quasi-invertibility

Let A be a finite-dimensional unital Jordan algebra over a field k of characteristic ≠ 2. For a pair Template:Math with Template:Math and Template:Math invertible define

Template:Quote box Template:Clear In this case the Bergman operator Template:Math defines an invertible operator on A and

Template:Quote box Template:Clear In fact

$B(a,b{)}^{-1}(a-Q(a)b)=Q(a^b)Q(a^{-1})(a-Q(a)b)=Q(a^b)(a^b{)}^{-1}=a^b.$

Moreover, by definition Template:Math is invertible if and only if Template:Math is invertible. In that case

Template:Quote box Template:Clear Indeed,

$a^{b+c}=((a^{-1}-b)-c{)}^{-1}=((a^b{)}^{-1}-c{)}^{-1}=(a^b{)}^c.$

The assumption that Template:Math be invertible can be dropped since Template:Math can be defined only supposing that the Bergman operator Template:Math is invertible. The pair Template:Math is then said to be quasi-invertible. In that case Template:Math is defined by the formula

$a^b=B(a,b{)}^{-1}(a-Q(a)b).$

If Template:Math is invertible, then Template:Math for some Template:Math. The fundamental identity implies that Template:Math. So by finite-dimensionality Template:Math is invertible. Thus Template:Math is invertible if and only if Template:Math is invertible and in this case

Template:Quote box Template:Clear In fact

Template:Math

so the formula follows by applying Template:Math to both sides.

As before Template:Math is quasi-invertible if and only if Template:Math is quasi-invertible; and in that case

$a^{b+c}=(a^b{)}^c.$

If k = R or C, this would follow by continuity from the special case where Template:Math and Template:Math were invertible. In general the proof requires four identities for the Bergman operator:

Template:Quote box Template:Clear In fact applying Template:Math to the identity Template:Math yields

$B(a,b)Q(a^b)B(b,a)=B(a,b)Q(a)=Q(a)B(b,a).$

The first identity follows by cancelling Template:Math and Template:Math. The second identity follows by similar cancellation in

Template:Math.

The third identity follows by applying the second identity to an element d and then switching the roles of c and d. The fourth follows because

Template:Math.

In fact Template:Math is quasi-invertible if and only if Template:Math is quasi-invertible in the mutation Template:Math. Since this mutation might not necessarily unital this means that when an identity is adjoint Template:Math becomes invertible in Template:Math. This condition can be expressed as follows without mentioning the mutation or homotope:

Template:Quote box Template:Clear In fact if Template:Math is quasi-invertible, then Template:Math satisfies the first identity by definition. The second follows because Template:Math. Conversely the conditions state that in Template:Math the conditions imply that Template:Math is the inverse of Template:Math. On the other hand, Template:Math for Template:Math in Template:Math. Hence Template:Math is invertible.

Equivalence relation

Let A be a finite-dimensional unital Jordan algebra over a field k of characteristic ≠ 2. Two pairs Template:Math with Template:Math invertible are said to be equivalent if Template:Math is invertible and Template:Math.

This is an equivalence relation, since if Template:Math is invertible Template:Math so that a pair Template:Math is equivalent to itself. It is symmetric since from the definition Template:Math. It is transitive. For suppose that Template:Math is a third pair with Template:Math invertible and Template:Math. From the above

$a_1^{-1}-b_1+b_3=(a_1^{-1}-b_1+b_2)-b_2+b_3=a_2^{-1}-b_2+b_3$

is invertible and

$a_3=a_2^{b_2-b_3}=(a_1^{b_1-b_2}{)}^{b_2-b_3}=a_1^{b_1-b_3}.$

As for quasi-invertibility, this definition can be extended to the case where Template:Math and Template:Math are not assumed to be invertible.

Two pairs Template:Math are said to be equivalent if Template:Math is quasi-invertible and Template:Math. When k = R or C, the fact that this more general definition also gives an equivalence relation can deduced from the invertible case by continuity. For general k, it can also be verified directly:

• The relation is reflexive since Template:Math is quasi-invertible and Template:Math.
• The relation is symmetric, since Template:Math.
• The relation is transitive. For suppose that Template:Math is a third pair with Template:Math quasi-invertible and Template:Math. In this case

$B(a_1,b_1-b_3)=B(a_1,b_1-b_2)B(a_2,b_2-b_3),$

so that Template:Math is quasi-invertible with

$a_3=a_2^{b_2-b_3}=(a_1^{b_1-b_2}{)}^{b_2-b_3}=a_1^{b_1-b_3}.$

The equivalence class of Template:Math is denoted by Template:Math.

Structure groups

Template:See also Let Template:Math be a finite-dimensional complex semisimple unital Jordan algebra. If Template:Math is an operator on Template:Math, let Template:Math be its transpose with respect to the trace form. Thus Template:Math, Template:Math, Template:Math and Template:Math. The structure group of A consists of g in Template:Math such that

$Q(ga)=gQ(a)g^t.$

They form a group Template:Math. The automorphism group Aut A of A consists of invertible complex linear operators g such that L(ga) = gL(a)g⁻¹ and g1 = 1. Since an automorphism g preserves the trace form, g⁻¹ = g^t.

• The structure group is closed under taking transposes g ↦ g^t and adjoints g ↦ g*.
• The structure group contains the automorphism group. The automorphism group can be identified with the stabilizer of 1 in the structure group.
• If a is invertible, Q(a) lies in the structure group.
• If g is in the structure group and a is invertible, ga is also invertible with (ga)⁻¹ = (g^t)⁻¹a⁻¹.
• The structure group Γ(A) acts transitively on the set of invertible elements in A.
• Every g in Γ(A) has the form g = h Q(a) with h an automorphism and a invertible.

The complex Jordan algebra A is the complexification of a real Euclidean Jordan algebra E, for which the trace form defines an inner product. There is an associated involution Template:Math on Template:Math which gives rise to a complex inner product on Template:Math. The unitary structure group Γ_u(A) is the subgroup of Γ(A) consisting of unitary operators, so that Template:Math. The identity component of Template:Math is denoted by Template:Math. It is a connected closed subgroup of Template:Math.

• The stabilizer of 1 in Γ_u(A) is Aut E.
• Every g in Γ_u(A) has the form g = h Q(u) with h in Aut E and u invertible in A with u* = u⁻¹.
• Γ(A) is the complexification of Γ_u(A).
• The set S of invertible elements u in A such that u* = u⁻¹ can be characterized equivalently either as those u for which L(u) is a normal operator with uu* = 1 or as those u of the form exp ia for some a in E. In particular S is connected.
• The identity component of Γ_u(A) acts transitively on S
• Given a Jordan frame (e_i) and v in A, there is an operator u in the identity component of Γ_u(A) such that uv = Σ α_i e_i with α_i ≥ 0. If v is invertible, then α_i > 0.

The structure group Γ(A) acts naturally on X. For g in Γ(A), set

$g(a,b)=(ga,(g^t{)}^{-1}b).$

Then Template:Math is quasi-invertible if and only if Template:Math is quasi-invertible and

$g(x^y)=(gx{)}^{(g^t{)}^{-1}y}.$

In fact the covariance relations for g with Q and the inverse imply that

$gB(x,y)g^{-1}=B(gx,(g^t{)}^{-1}y)$

if x is invertible and so everywhere by density. In turn this implies the relation for the quasi-inverse. If a is invertible then Q(a) lies in Γ(A) and if (a,b) is quasi-invertible B(a,b) lies in Γ(A). So both types of operators act on X.

The defining relations for the structure group show that it is a closed subgroup of ${𝔤}_0$ of Template:Math. Since Template:Math, the corresponding complex Lie algebra contains the operators Template:Math. The commutators Template:Math span the complex Lie algebra of derivations of Template:Math. The operators Template:Math span ${𝔤}_0$ and satisfy Template:Math and Template:Math.

Geometric properties of quotient space

Let A be a finite-dimensional complex unital Jordan algebra which is semisimple, i.e. the trace form Tr L(ab) is non-degenerate. Let Template:Math be the quotient of Template:Math by the equivalence relation. Let Template:Math be the subset of X of classes Template:Math. The map Template:Math, Template:Math is injective. A subset Template:Math of Template:Math is defined to be open if and only if Template:Math is open for all Template:Math.

Template:Quote box Template:Clear The transition maps of the atlas with charts Template:Math are given by

${\phi }_{cb}={\phi }_c\circ {\phi }_b^{-1}:{\phi }_b(X_b\cap X_c)\to {\phi }_c(X_b\cap X_c).$

and are injective and holomorphic since

${\phi }_{cb}(a)=a^{b-c}$

with derivative

${\phi }_{cb}^{\prime }(a)=B(a,b-c{)}^{-1}.$

This defines the structure of a complex manifold on X because Template:Math on Template:Math.

Template:Quote box Template:Clear Indeed, all the polynomial functions Template:Math are non-trivial since Template:Math. Therefore, there is a Template:Math such that Template:Math for all i, which is precisely the criterion for Template:Math to lie in Template:Math.

Template:Harvtxt uses the Bergman operators to construct an explicit biholomorphism between X and a closed smooth algebraic subvariety of complex projective space. This implies in particular that Template:Math is compact. There is a more direct proof of compactness using symmetry groups.

Given a Jordan frame (e_i) in E, for every a in A there is a k in U = Γ_u(A) such that Template:Math with Template:Math (and Template:Math if a is invertible). In fact, if (a,b) is in X then it is equivalent to k(c,d) with c and d in the unital Jordan subalgebra Template:Math, which is the complexification of Template:Math. Let Template:Math be the complex manifold constructed for Template:Math. Because Template:Math is a direct sum of copies of C, Z is just a product of Riemann spheres, one for each Template:Math. In particular it is compact. There is a natural map of Z into X which is continuous. Let Y be the image of Z. It is compact and therefore coincides with the closure of Y₀ = A_e ⊂ A = X₀. The set U⋅Y is the continuous image of the compact set U × Y. It is therefore compact. On the other hand, U⋅Y₀ = X₀, so it contains a dense subset of X and must therefore coincide with X. So X is compact.

The above argument shows that every (a,b) in X is equivalent to k(c,d) with c and d in Template:Math and k in Template:Math. The mapping of Z into X is in fact an embedding. This is a consequence of Template:Math being quasi-invertible in Template:Math if and only if it is quasi-invertible in Template:Math. Indeed, if Template:Math is injective on A, its restriction to Template:Math is also injective. Conversely, the two equations for the quasi-inverse in Template:Math imply that it is also a quasi-inverse in Template:Math.

Möbius transformations

Let Template:Math be a finite-dimensional complex semisimple unital Jordan algebra. The group SL(2,C) acts by Möbius transformation on the Riemann sphere C ∪ {∞}, the one-point compactification of C. If g in SL(2,C) is given by the matrix

$g=\left(\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right),$

then

$g(z)=(\alpha z+\beta )(\gamma z+\delta {)}^{-1}.$

There is a generalization of this action of SL(2,C) to A and its compactification X. In order to define this action, note that SL(2,C) is generated by the three subgroups of lower and upper unitriangular matrices and the diagonal matrices. It is also generated by the lower (or upper) unitriangular matrices, the diagonal matrices and the matrix

$J=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right).$

The matrix J corresponds to the Möbius transformation Template:Math and can be written

$J=\left(\begin{array}{cc}1& 0\\ -1& 1\end{array}\right)\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)\left(\begin{array}{cc}1& 0\\ -1& 1\end{array}\right).$

The Möbius transformations fixing ∞ are just the upper triangular matrices. If g does not fix ∞, it sends ∞ to a finite point a. But then g can be composed with an upper unitriangular to send a to 0 and then with J to send 0 to infinity.

For an element Template:Math of Template:Math, the action of Template:Math in SL(2,C) is defined by the same formula

$g(a)=(\alpha a+\beta 1)(\gamma a+\delta 1{)}^{-1}.$

This defines an element of Template:Math provided that Template:Math is invertible in Template:Math. The action is thus defined everywhere on Template:Math if g is upper triangular. On the other hand, the action on X is simple to define for lower triangular matrices.

• For diagonal matrices g with diagonal entries Template:Math and Template:Math, Template:Math is a well-defined holomorphic action on Template:Math which passes to the quotient X. On Template:Math it agrees with the Möbius action.
• For lower unitriangular matrices, with off-diagonal parameter γ, define Template:Math. Again this is holomorphic on Template:Math and passes to the quotient X. When Template:Math and Template:Math,

$g(a:0)=(a:-\gamma )=(a^{-\gamma }:0)=(a(\gamma a+1{)}^{-1}:0)$

if Template:Math is invertible, so this is an extension of the Möbius action.

• For upper unitriangular matrices, with off-diagonal parameter β, the action on Template:Math is defined by Template:Math. Template:Harvtxt showed that this defined a complex one-parameter flow on Template:Math. The corresponding holomorphic complex vector field extended to Template:Math, so that the action on the compact complex manifold Template:Math could be defined by the associated complex flow. A simpler method is to note that the operator Template:Math can be implemented directly using its intertwining relations with the unitary structure group.

In fact on the invertible elements in A, the operator Template:Math satisfies Template:Math. To define a biholomorphism j on X such that Template:Math, it is enough to define these for Template:Math in some suitable orbit of Γ(A) or Γ_u(A). On the other hand, as indicated above, given a Jordan frame (e_i) in E, for every a in A there is a k in U = Γ_u(A) such that Template:Math with Template:Math.

The computation of Template:Math in the associative commutative algebra Template:Math is straightforward since it is a direct product. For Template:Math and Template:Math, the Bergman operator on Template:Math has determinant Template:Math. In particular Template:Math for some λ ≠ 0. So that Template:Math is equivalent to Template:Math. Let Template:Math. On Template:Math, for a dense set of Template:Math, the pair Template:Math is equivalent to Template:Math with b invertible. Then Template:Math is equivalent to Template:Math. Since Template:Math is holomorphic it follows that j has a unique continuous extension to X such that Template:Math for Template:Math in Template:Math, the extension is holomorphic and for Template:Math, Template:Math

Template:Quote box Template:Clear The holomorphic transformations corresponding to upper unitriangular matrices can be defined using the fact that they are the conjugates by J of lower unitriangular matrices, for which the action is already known. A direct algebraic construction is given in Template:Harvtxt.

This action of Template:Math is compatible with inclusions. More generally if Template:Math is a Jordan frame, there is an action of Template:Math on A_e which extends to A. If Template:Math and Template:Math, then Template:Math and Template:Math give the action of the product of the lower and upper unitriangular matrices. If Template:Math is invertible, the corresponding product of diagonal matrices act as Template:Math. In particular the diagonal matrices give an action of Template:Math and Template:Math.

Holomorphic symmetry group

Let Template:Math be a finite-dimensional complex semisimple unital Jordan algebra. There is a transitive holomorphic action of a complex matrix group Template:Math on the compact complex manifold Template:Math. Template:Harvtxt described Template:Math analogously to Template:Math in terms of generators and relations. Template:Math acts on the corresponding finite-dimensional Lie algebra of holomorphic vector fields restricted to Template:Math, so that Template:Math is realized as a closed matrix group. It is the complexification of a compact Lie group without center, so a semisimple algebraic group. The identity component Template:Math of the compact group acts transitively on Template:Math, so that Template:Math can be identified as a Hermitian symmetric space of compact type.

The group G is generated by three types of holomorphic transformation on Template:Math:

• Operators W corresponding to elements W in Template:Math given by Template:Math. These were already described above. On Template:Math, they are given by Template:Math.
• Operators S_c defined by Template:Math. These are the analogue of lower unitriangular matrices and form a subgroup isomorphic to the additive group of Template:Math, with the given parametrization. Again these act holomorphically on Template:Math and the action passes to the quotient X. On Template:Math the action is given by Template:Math if Template:Math is quasi-invertible.
• The transformation Template:Math corresponding to Template:Math in Template:Math. It was constructed above as part of the action of Template:Math} on Template:Math. On invertible elements in Template:Math it is given by Template:Math.

The operators Template:Math normalize the group of operators Template:Math. Similarly the operator Template:Math normalizes the structure group, Template:Math. The operators Template:Math also form a group of holomorphic transformations isomorphic to the additive group of Template:Math. They generalize the upper unitriangular subgroup of Template:Math. This group is normalized by the operators W of the structure group. The operator Template:Math acts on Template:Math as Template:Math. If Template:Math is a scalar the operators Template:Math and Template:Math coincide with the operators corresponding to lower and upper unitriangular matrices in Template:Math. Accordingly, there is a relation Template:Math and Template:Math is a subgroup of G. Template:Harvtxt defines the operators Template:Math in terms of the flow associated to a holomorphic vector field on Template:Math, while Template:Harvtxt give a direct algebraic description.

Template:Quote box Template:Clear Indeed, Template:Math.

Let Template:Math and Template:Math be the complex Abelian groups formed by the symmetries Template:Math and Template:Math respectively. Let Template:Math.

Template:Quote box Template:Clear The two expressions for Template:Math are equivalent as follows by conjugating by Template:Math.

For Template:Math invertible, Hua's identity can be rewritten

$Q(a)=T_a\circ j\circ T_{a^{-1}}\circ j\circ T_a\circ j.$

Moreover, Template:Math and Template:Math.

The convariance relations show that the elements of Template:Math fall into sets Template:Math, Template:Math, Template:Math, Template:Math. ... The first expression for Template:Math follows once it is established that no new elements appear in the fourth or subsequent sets. For this it suffices to show that

Template:Math.

For then if there are three or more occurrences of Template:Math, the number can be recursively reduced to two. Given Template:Math in Template:Math, pick Template:Math so that Template:Math and Template:Math are invertible. Then

$j{T}_{a}j{T}_{b}j=j{T}_c{T}_{\lambda }j{T}_{{\lambda }^{-1}}{T}_d\circ j={\lambda }^{2}j{T}_{c}j{T}_{-\lambda }j{T}_{d}j={\lambda }^2{T}_{-c^{-1}}jQ(c^{-1})T_{-c^{-1}-\lambda -d^{-1}}jQ(d^{-1})j{T}_{-d^{-1}},$

which lies in Template:Math.

Template:Quote box Template:Clear It suffices to check that if Template:Math, then Template:Math. If so Template:Math, so Template:Math.

Exchange relations

Template:Quote box Template:Clear For Template:Math invertible, Hua's identity can be rewritten

$Q(a)=T_a\circ j\circ T_{a^{-1}}\circ j\circ T_a\circ j.$

Since Template:Math, the operators Template:Math belong to the group generated by Template:Math.

For quasi-invertible pairs Template:Math, there are the "exchange relations"

This identity shows that Template:Math is in the group generated by Template:Math. Taking inverses, it is equivalent to the identity Template:Math.

To prove the exchange relations, it suffices to check that it valid when applied to points the dense set of points Template:Math in Template:Math for which Template:Math is quasi-invertible. It then reduces to the identity: Template:Quote box Template:Clear In fact, if Template:Math is quasi-invertible, then Template:Math is quasi-invertible if and only if Template:Math is quasi-invertible. This follows because Template:Math is quasi-invertible if and only if Template:Math is. Moreover, the above formula holds in this case.

For the proof, two more identities are required:

$B(c+b,a)=B(c,a^b)B(b,a)$

$R(a,b)=R(a^b,b-Q(b)a)=R(a-Q(a)b,b^a)$

The first follows from a previous identity by applying the transpose. For the second, because of the transpose, it suffices to prove the first equality. Setting Template:Math in the identity Template:Math yields

Template:Math

so the identity follows by cancelling Template:Math.

To prove the formula, the relations Template:Math and Template:Math show that it is enough to prove that

Template:Math

Indeed, Template:Math. On the other hand, Template:Math and Template:Math. So Template:Math.

Now set Template:Math. Then the exchange relations imply that Template:Math lies in Template:Math if and only if Template:Math is quasi-invertible; and that Template:Math lies in Template:Math if and only if Template:Math is in Template:Math.

In fact if Template:Math lies in Template:Math, then Template:Math is equivalent to Template:Math, so it a quasi-invertible pair; the converse follows from the exchange relations. Clearly Template:Math. The converse follows from Template:Math and the criterion for Template:Math to lie in Template:Math.

Lie algebra of holomorphic vector fields

Template:See also The compact complex manifold Template:Math is modelled on the space Template:Math. The derivatives of the transition maps describe the tangent bundle through holomorphic transition functions Template:Math. These are given by Template:Math, so the structure group of the corresponding principal fiber bundle reduces to Template:Math, the structure group of Template:Math. The corresponding holomorphic vector bundle with fibre Template:Math is the tangent bundle of the complex manifold Template:Math. Its holomorphic sections are just holomorphic vector fields on X. They can be determined directly using the fact that they must be invariant under the natural adjoint action of the known holomorphic symmetries of X. They form a finite-dimensional complex semisimple Lie algebra. The restriction of these vector fields to X₀ can be described explicitly. A direct consequence of this description is that the Lie algebra is three-graded and that the group of holomorphic symmetries of X, described by generators and relations in Template:Harvtxt and Template:Harvtxt, is a complex linear semisimple algebraic group that coincides with the group of biholomorphisms of X.

The Lie algebras of the three subgroups of holomorphic automorphisms of Template:Math give rise to linear spaces of holomorphic vector fields on Template:Math and hence Template:Math.

• The structure group Template:Math has Lie algebra ${𝔤}_0$ spanned by the operators Template:Math. These define a complex Lie algebra of linear vector fields Template:Math on Template:Math.
• The translation operators act on Template:Math as Template:Math. The corresponding one-parameter subgroups are given by Template:Math and correspond to the constant vector fields Template:Math. These give an Abelian Lie algebra ${𝔤}_{-1}$ of vector fields on Template:Math.
• The operators Template:Math defined on Template:Math by Template:Math. The corresponding one-parameter groups Template:Math define quadratic vector fields Template:Math on Template:Math. These give an Abelian Lie algebra ${𝔤}_1$ of vector fields on Template:Math.

Let

$𝔤={𝔤}_{-1}\oplus {𝔤}_0\oplus {𝔤}_1.$

Then, defining ${𝔤}_i=(0)$ for Template:Math, $𝔤$ forms a complex Lie algebra with

$[{𝔤}_p,{𝔤}_q]\subseteq {𝔤}_{p+q}.$

This gives the structure of a 3-graded Lie algebra. For elements Template:Math in $𝔤$, the Lie bracket is given by

$[(a_1,T_1,b_1),(a_2,T_2,b_2)]=(T_1{a}_2-T_2{a}_1,[T_1,T_2]+R(a_1,b_2)-R(a_2,b_1),T_2^t{b}_1-T_1^t{b}_2)$

The group Template:Math of Möbius transformations of X normalizes the Lie algebra $𝔤$. The transformation Template:Math corresponding to the Weyl group element Template:Math induces the involutive automorphism Template:Math given by

$\sigma (a,T,b)=(b,-T^t,a).$

More generally the action of a Möbius transformation

$g=\left(\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right)$

can be described explicitly. In terms of generators diagonal matrices act as

$\left(\begin{array}{cc}\alpha & 0\\ 0& {\alpha }^{-1}\end{array}\right)(a,T,b)=({\alpha }^{2}a,T,{\alpha }^{-2}b),$

upper unitriangular matrices act as

$\left(\begin{array}{cc}1& \beta \\ 0& 1\end{array}\right)(a,T,b)=(a+\beta T(1)-{\beta }^{2}b,T-\beta L(a),b),$

and lower unitriangular matrices act as

$\left(\begin{array}{cc}1& 0\\ \gamma & 1\end{array}\right)(a,T,b)=(a,T-\gamma L(b),b-\gamma T^t(1)-{\gamma }^{2}a).$

This can be written uniformly in matrix notation as

$\left(\begin{array}{cc}g(T)& g(a)\\ g(b)& g(T{)}^t\end{array}\right)=g\left(\begin{array}{cc}T& a\\ b& T^t\end{array}\right)g^{-1}.$

In particular the grading corresponds to the action of the diagonal subgroup of Template:Math, even with |α| = 1, so a copy of T.

The Killing form is given by

$𝐁((a_1,T_1,b_1),(a_2,T_2,b_2))=(a_1,b_2)+(b_1,a_2)+\beta (T_1,T_2),$

where Template:Math is the symmetric bilinear form defined by

$\beta (R(a,b),R(c,d))=(R(a,b)c,d)=(R(c,d)a,b),$

with the bilinear form Template:Math corresponding to the trace form: Template:Math.

More generally the generators of the group Template:Math act by automorphisms on $𝔤$ as

• $W(a,T,b)=(Wa,WT{W}^{-1},(W^t{)}^{-1}b),$
• $J(a,T,b)=(-b,-T^t,-a),$
• $T_x(a,T,b)=(a+Tx-Q(x)b,T-R(x,b),b),$
• $S_y(a,T,b)=(a,T-R(a,y),b-T^{t}y-Q(y)a).$

Template:Quote box Template:Clear The nondegeneracy of the Killing form is immediate from the explicit formula. By Cartan's criterion, $𝔤$ is semisimple. In the next section the group Template:Math is realized as the complexification of a connected compact Lie group Template:Math with trivial center, so semisimple. This gives a direct means to verify semisimplicity. The group H also acts transitively on X.

Template:Quote box Template:Clear To prove that $𝔤$ exhausts the holomorphic vector fields on Template:Math, note the group T acts on holomorphic vector fields. The restriction of such a vector field to Template:Math gives a holomorphic map of A into A. The power series expansion around 0 is a convergent sum of homogeneous parts of degree Template:Math. The action of Template:Math scales the part of degree Template:Math by Template:Math. By taking Fourier coefficients with respect to T, the part of degree m is also a holomorphic vector field. Since conjugation by Template:Math gives the inverse on Template:Math, it follows that the only possible degrees are 0, 1 and 2. Degree 0 is accounted for by the constant fields. Since conjugation by Template:Math interchanges degree 0 and degree 2, it follows that ${𝔤}_{\pm 1}$ account for all these holomorphic vector fields. Any further holomorphic vector field would have to appear in degree 1 and so would have the form Template:Math for some Template:Math in Template:Math. Conjugation by J would give another such map N. Moreover, Template:Math. But then

$e^{tM}(0,0,b)=J{e}^{tN}J(0,0,b)=J{e}^{tN}(b,0,0)=(0,0,e^{tN}b).$

Set Template:Math and Template:Math. Then

$Q(U_{t}a)b=U_{t}Q(a)V_{-t}b.$

It follows that Template:Math lies in Template:Math for all Template:Math and hence that Template:Math lies in ${𝔤}_0$. So $𝔤$ is exactly the space of holomorphic vector fields on X.

Compact real form

Template:Quote box Template:Clear Suppose Template:Math acts trivially on $𝔤$. Then Template:Math must leave the subalgebra Template:Math invariant. Hence so must Template:Math. This forces Template:Math, so that Template:Math. But then Template:Math must leave the subalgebra Template:Math invariant, so that Template:Math and Template:Math. If Template:Math acts trivially, Template:Math.

The group Template:Math can thus be identified with its image in GL $𝔤$.

Let Template:Math be the complexification of a Euclidean Jordan algebra Template:Math. For Template:Math, set Template:Math. The trace form on Template:Math defines a complex inner product on Template:Math and hence an adjoint operation. The unitary structure group Template:Math consists of those Template:Math in Template:Math that are in Template:Math, i.e. satisfy Template:Math. It is a closed subgroup of U(A). Its Lie algebra consists of the skew-adjoint elements in ${𝔤}_0$. Define a conjugate linear involution Template:Math on $𝔤$ by

$\theta (a,T,b)=(b^{*},-T^{*},a^{*}).$

This is a period 2 conjugate-linear automorphism of the Lie algebra. It induces an automorphism of Template:Math, which on the generators is given by

$\theta (S_a)=T_{a^{*}},\theta (j)=j,\theta (T_b)=S_{b^{*}},\theta (W)=(W^{*}{)}^{-1}.$

Let Template:Math be the fixed point subgroup of Template:Math in Template:Math. Let $𝔥$ be the fixed point subalgebra of Template:Math in $𝔤$. Define a sesquilinear form on $𝔤$ by Template:Math. This defines a complex inner product on $𝔤$ which restricts to a real inner product on $𝔥$. Both are preserved by Template:Math. Let Template:Math be the identity component of Template:Math. It lies in Template:Math. Let Template:Math be the diagonal torus associated with a Jordan frame in E. The action of Template:Math is compatible with Template:Math which sends a unimodular matrix $\left(\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right)$ to $\left(\begin{array}{cc}\bar{\delta }& -\bar{\gamma }\\ -\bar{\beta }& \bar{\alpha }\end{array}\right)$. In particular this gives a homomorphism of Template:Math into Template:Math.

Now every matrix Template:Math in Template:Math can be written as a product

$M=\left(\begin{array}{cc}{\zeta }_1& 0\\ 0& {\zeta }_1^{-1}\end{array}\right)\left(\begin{array}{cc}\cos \phi & \sin \phi \\ -\sin \phi & \cos \phi \end{array}\right)\left(\begin{array}{cc}{\zeta }_2& 0\\ 0& {\zeta }_2^{-1}\end{array}\right).$

The factor in the middle gives another maximal torus in Template:Math obtained by conjugating by Template:Math. If Template:Math with |α_i| = 1, then Template:Math gives the action of the diagonal torus Template:Math and corresponds to an element of Template:Math. The element Template:Math lies in Template:Math and its image is a Möbius transformation Template:Math lying in Template:Math. Thus Template:Math is another torus in Template:Math and Template:Math coincides with the image of Template:Math.

Template:Quote box Template:Clear Since Template:Math for the compact complex manifold corresponding to Template:Math, if follows that Template:Math, where Template:Math is the image of Template:Math. On the other hand, Template:Math, so that Template:Math. On the other hand, the stabilizer of Template:Math in Template:Math is Template:Math, since the fixed point subgroup of Template:Math under Template:Math is Template:Math. Hence Template:Math. In particular H is compact and connected since both K and S are. Because it is a closed subgroup of U $𝔤$, it is a Lie group. It contains K and hence its Lie algebra contains the operators Template:Math with Template:Math. It contains the image of Template:Math and hence the elements Template:Math with Template:Math in Template:Math. Since Template:Math and Template:Math, it follows that the Lie algebra ${𝔥}_1$ of Template:Math contains Template:Math for all Template:Math in Template:Math. Thus it contains $𝔥$.

They are equal because all skew-adjoint derivations of $𝔥$ are inner. In fact, since Template:Math normalizes $𝔥$ and the action by conjugation is faithful, the map of ${𝔥}_1$ into the Lie algebra $𝔡$ of derivations of $𝔥$ is faithful. In particular $𝔥$ has trivial center. To show that $𝔥$ equals ${𝔥}_1$, it suffices to show that $𝔡$ coincides with $𝔥$. Derivations on $𝔥$ are skew-adjoint for the inner product given by minus the Killing form. Take the invariant inner product on $𝔡$ given by Template:Math. Since $𝔥$ is invariant under $𝔡$ so is its orthogonal complement. They are both ideals in $𝔡$, so the Lie bracket between them must vanish. But then any derivation in the orthogonal complement would have 0 Lie bracket with $𝔥$, so must be zero. Hence $𝔥$ is the Lie algebra of Template:Math. (This also follows from a dimension count since Template:Math.)

Template:Quote box Template:Clear The formulas above for the action of Template:Math and Template:Math show that the image of Template:Math is closed in GL $𝔤$. Since Template:Math acts transitively on Template:Math and the stabilizer of Template:Math in Template:Math is Template:Math, it follows that Template:Math. The compactness of Template:Math and closedness of Template:Math implies that Template:Math is closed in GL $𝔤$.

Template:Quote box Template:Clear Template:Math is a closed subgroup of GL $𝔤$ so a real Lie group. Since it contains Template:Math with Template:Math or Template:Math, its Lie algebra contains $𝔤$. Since $𝔤$ is the complexification of $𝔥$, like $𝔥$ all its derivations are inner and it has trivial center. Since the Lie algebra of Template:Math normalizes $𝔤$ and o is the only element centralizing $𝔤$, as in the compact case the Lie algebra of Template:Math must be $𝔤$. (This can also be seen by a dimension count since Template:Math.) Since it is a complex subspace, Template:Math is a complex Lie group. It is connected because it is the continuous image of the connected set Template:Math. Since $𝔤$ is the complexification of $𝔥$, Template:Math is the complexification of Template:Math.

Noncompact real form

For Template:Math in Template:Math the spectral norm ||a|| is defined to be Template:Math if Template:Math with Template:Math and Template:Math in Template:Math. It is independent of choices and defines a norm on Template:Math. Let Template:Math be the set of Template:Math with ||a|| < 1 and let Template:Math be the identity component of the closed subgroup of G carrying Template:Math onto itself. It is generated by Template:Math, the Möbius transformations in Template:Math and the image of Template:Math corresponding to a Jordan frame. Let τ be the conjugate-linear period 2 automorphism of $𝔤$ defined by

$\tau (a,T,b)=(-a^{*},-T^{*},-b^{*}).$

Let ${𝔥}^{*}$ be the fixed point algebra of τ. It is the Lie algebra of Template:Math. It induces a period 2 automorphism of Template:Math with fixed point subgroup Template:Math. The group Template:Math acts transitively on Template:Math. The stabilizer of 0 is Template:Math.

The noncompact real semisimple Lie group Template:Math acts on X with an open orbit Template:Math. As with the action of Template:Math on the Riemann sphere, it has only finitely many orbits. This orbit structure can be explicitly described when the Jordan algebra Template:Math is simple. Let Template:Math be the subset of Template:Math consisting of elements Template:Math with exactly Template:Math of the α_i less than one and exactly Template:Math of them greater than one. Thus Template:Math. These sets are the intersections of the orbits Template:Math of Template:Math with Template:Math. The orbits with Template:Math are open. There is a unique compact orbit Template:Math. It is the Shilov boundary S of D consisting of elements Template:Math with Template:Math in Template:Math, the underlying Euclidean Jordan algebra. Template:Math is in the closure of Template:Math if and only if Template:Math and Template:Math. In particular Template:Math is in the closure of every orbit.

Jordan algebras with involution

The preceding theory describes irreducible Hermitian symmetric spaces of tube type in terms of unital Jordan algebras. In Template:Harvtxt general Hermitian symmetric spaces are described by a systematic extension of the above theory to Jordan pairs. In the development of Template:Harvtxt, however, irreducible Hermitian symmetric spaces not of tube type are described in terms of period two automorphisms of simple Euclidean Jordan algebras. In fact any period 2 automorphism defines a Jordan pair: the general results of Template:Harvtxt on Jordan pairs can be specialized to that setting.

Let τ be a period two automorphism of a simple Euclidean Jordan algebra E with complexification A. There are corresponding decompositions E = E₊ ⊕ E₋ and A = A₊ ⊕ A₋ into ±1 eigenspaces of τ. Let Template:Math. τ is assumed to satisfy the additional condition that the trace form on Template:Math defines an inner product. For Template:Math in Template:Math, define Template:Math to be the restriction of Template:Math to V. For a pair Template:Math in Template:Math, define Template:Math and Template:Math to be the restriction of Template:Math and Template:Math to Template:Math. Then Template:Math is simple if and only if the only subspaces invariant under all the operators Template:Math and Template:Math are Template:Math and Template:Math.

The conditions for quasi-invertibility in Template:Math show that Template:Math is invertible if and only if Template:Math is invertible. The quasi-inverse Template:Math is the same whether computed in Template:Math or Template:Math. A space of equivalence classes Template:Math can be defined on pairs Template:Math. It is a closed subspace of Template:Math, so compact. It also has the structure of a complex manifold, modelled on Template:Math. The structure group Template:Math can be defined in terms of Template:Math and it has as a subgroup the unitary structure group Template:Math with identity component Template:Math. The group Template:Math is the identity component of the fixed point subgroup of τ in Template:Math. Let Template:Math be the group of biholomorphisms of Template:Math generated by Template:Math in Template:Math, the identity component of Template:Math, and the Abelian groups Template:Math consisting of the Template:Math and Template:Math consisting of the Template:Math with Template:Math and Template:Math in Template:Math. It acts transitively on Template:Math with stabilizer Template:Math and Template:Math. The Lie algebra ${𝔤}_{\tau }$ of holomorphic vector fields on Template:Math is a 3-graded Lie algebra,

${𝔤}_{\tau }={𝔤}_{\tau ,+1}\oplus {𝔤}_{\tau ,0}\oplus {𝔤}_{\tau ,-1}.$

Restricted to Template:Math the components are generated as before by the constant functions into Template:Math, by the operators Template:Math and by the operators Template:Math. The Lie brackets are given by exactly the same formula as before.

The spectral decomposition in Template:Math and Template:Math is accomplished using tripotents, i.e. elements Template:Math such that Template:Math. In this case Template:Math is an idempotent in Template:Math. There is a Pierce decomposition Template:Math into eigenspaces of Template:Math. The operators Template:Math and Template:Math commute, so Template:Math leaves the eigenspaces above invariant. In fact Template:Math acts as 0 on Template:Math, as 1/4 on Template:Math and 1 on Template:Math. This induces a Pierce decomposition Template:Math. The subspace Template:Math becomes a Euclidean Jordan algebra with unit Template:Math under the mutation Jordan product Template:Math}.

Two tripotents Template:Math and Template:Math are said to be orthogonal if all the operators Template:Math when a and b are powers of Template:Math and Template:Math and if the corresponding idempotents Template:Math and Template:Math are orthogonal. In this case Template:Math and Template:Math generate a commutative associative algebra and Template:Math, since Template:Math. Let Template:Math be in Template:Math. Let Template:Math be the finite-dimensional real subspace spanned by odd powers of Template:Math. The commuting self-adjoint operators Template:Math with Template:Math odd powers of Template:Math act on Template:Math, so can be simultaneously diagonalized by an orthonormal basis Template:Math. Since Template:Math is a positive multiple of Template:Math, rescaling if necessary, Template:Math can be chosen to be a tripotent. They form an orthogonal family by construction. Since Template:Math is in Template:Math, it can be written Template:Math with Template:Math real. These are called the eigenvalues of Template:Math (with respect to τ). Any other tripotent Template:Math in Template:Math has the form Template:Math with Template:Math, so the Template:Math are up to sign the minimal tripotents in Template:Math.

A maximal family of orthogonal tripotents in Template:Math is called a Jordan frame. The tripotents are necessarily minimal. All Jordan frames have the same number of elements, called the rank of Template:Math. Any two frames are related by an element in the subgroup of the structure group of Template:Math preserving the trace form. For a given Jordan frame Template:Math, any element Template:Math in Template:Math can be written in the form Template:Math with Template:Math and Template:Math an operator in Template:Math. The spectral norm of Template:Math is defined by ||a|| = sup α_i and is independent of choices. Its square equals the operator norm of Template:Math. Thus Template:Math becomes a complex normed space with open unit ball Template:Math.

Note that for Template:Math in Template:Math, the operator Template:Math is self-adjoint so that the norm ||Template:Math|| = ||Template:Math||ⁿ. Since Template:Math, it follows that ||Template:Math|| = ||Template:Math||ⁿ. In particular the spectral norm of Template:Math in Template:Math is the square root of the spectral norm of Template:Math. It follows that the spectral norm of Template:Math is the same whether calculated in Template:Math or Template:Math. Since Template:Math preserves both norms, the spectral norm on Template:Math is obtained by restricting the spectral norm on Template:Math.

For a Jordan frame Template:Math, let Template:Math. There is an action of Template:Math on Template:Math which extends to V. If Template:Math and Template:Math, then Template:Math and Template:Math give the action of the product of the lower and upper unitriangular matrices. If Template:Math with Template:Math, then the corresponding product of diagonal matrices act as Template:Math, where Template:Math. In particular the diagonal matrices give an action of Template:Math and Template:Math.

As in the case without an automorphism τ, there is an automorphism θ of Template:Math. The same arguments show that the fixed point subgroup Template:Math is generated by Template:Math and the image of Template:Math. It is a compact connected Lie group. It acts transitively on Template:Math; the stabilizer of Template:Math is Template:Math. Thus Template:Math, a Hermitian symmetric space of compact type.

Let Template:Math be the identity component of the closed subgroup of Template:Math carrying Template:Math onto itself. It is generated by Template:Math and the image of Template:Math corresponding to a Jordan frame. Let ρ be the conjugate-linear period 2 automorphism of ${𝔤}_{\tau }$ defined by

$\rho (a,T,b)=(-a^{*},-T^{*},-b^{*}).$

Let ${𝔥}_{\tau }^{*}$ be the fixed point algebra of ρ. It is the Lie algebra of Template:Math. It induces a period 2 automorphism of Template:Math with fixed point subgroup Template:Math. The group Template:Math acts transitively on Template:Math. The stabilizer of 0 is Template:Math. Template:Math is the Hermitian symmetric space of noncompact type dual to Template:Math.

The Hermitian symmetric space of non-compact type have an unbounded realization, analogous the upper half-plane in C. Möbius transformations in Template:Math corresponding to the Cayley transform and its inverse give biholomorphisms of the Riemann sphere exchanging the unit disk and the upper halfplane. When the Hermitian symmetric space is of tube type the same Möbius transformations map the disk Template:Math in Template:Math onto the tube domain Template:Math were Template:Math is the open self-dual convex cone of squares in the Euclidean Jordan algebra Template:Math.

For Hermitian symmetric space not of tube type there is no action of Template:Math on X, so no analogous Cayley transform. A partial Cayley transform can be defined in that case for any given maximal tripotent Template:Math in Template:Math. It takes the disk Template:Math in Template:Math onto a Siegel domain of the second kind.

In this case Template:Math is a Euclidean Jordan algebra and there is symmetric Template:Math-valued bilinear form on Template:Math such that the corresponding quadratic form Template:Math takes values in its positive cone Template:Math. The Siegel domain consists of pairs Template:Math such that Template:Math lies in Template:Math. The quadratic form Template:Math on Template:Math and the squaring operation on Template:Math are given by Template:Math. The positive cone Template:Math corresponds to Template:Math with Template:Math invertible.

Examples

For simple Euclidean Jordan algebras Template:Math with complexication Template:Math, the Hermitian symmetric spaces of compact type Template:Math can be described explicitly as follows, using Cartan's classification.

Type I_n. A is the Jordan algebra of n × n complex matrices Template:Math with the operator Jordan product Template:Math. It is the complexification of Template:Math, the Euclidean Jordan algebra of self-adjoint n × n complex matrices. In this case Template:Math acting on Template:Math with $g=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ acting as Template:Math. Indeed, this can be verified directly for diagonal, upper and lower unitriangular matrices which correspond to the operators Template:Math, Template:Math and Template:Math. The subset Template:Math corresponds to the matrices Template:Math with Template:Math invertible. In fact consider the space of linear maps from Template:Math to Template:Math. It is described by a pair (Template:Math|Template:Math) with Template:Math in Template:Math. This is a module for Template:Math acting on the target space. There is also an action of Template:Math induced by the action on the source space. The space of injective maps Template:Math is invariant and Template:Math acts freely on it. The quotient is the Grassmannian Template:Math consisting of n-dimensional subspaces of Template:Math. Define a map of Template:Math into Template:Math by sending Template:Math to the injective map (Template:Math|Template:Math). This map induces an isomorphism of Template:Math onto Template:Math.

In fact let Template:Math be an n-dimensional subspace of Template:Math. If it is in general position, i.e. it and its orthogonal complement have trivial intersection with Template:Math and Template:Math, it is the graph of an invertible operator Template:Math. So the image corresponds to (Template:Math|Template:Math) with Template:Math and Template:Math.

At the other extreme, Template:Math and its orthogonal complement Template:Math can be written as orthogonal sums Template:Math, Template:Math, where Template:Math and Template:Math are the intersections with Template:Math and Template:Math and Template:Math with Template:Math. Then Template:Math and Template:Math. Moreover, Template:Math and Template:Math. The subspace Template:Math corresponds to the pair (Template:Math|Template:Math), where Template:Math is the orthogonal projection of Template:Math onto Template:Math. So Template:Math and Template:Math.

The general case is a direct sum of these two cases. Template:Math can be written as an orthogonal sum Template:Math where Template:Math and Template:Math are the intersections with Template:Math and Template:Math and Template:Math is their orthogonal complement in Template:Math. Similarly the orthogonal complement Template:Math of Template:Math can be written Template:Math. Thus Template:Math and Template:Math, where Template:Math are orthogonal complements. The direct sum Template:Math is of the second kind and its orthogonal complement of the first.

Maps Template:Math in the structure group correspond to Template:Math in Template:Math, with Template:Math. The corresponding map on Template:Math sends (Template:Math|Template:Math) to (Template:Math|Template:Math). Similarly the map corresponding to Template:Math sends (Template:Math|Template:Math) to (Template:Math|Template:Math), the map corresponding to Template:Math sends (Template:Math|Template:Math) to (Template:Math|Template:Math) and the map corresponding to Template:Math sends (Template:Math|Template:Math) to (Template:Math|Template:Math). It follows that the map corresponding to Template:Math sends (Template:Math|Template:Math) to (Template:Math|Template:Math). On the other hand, if Template:Math is invertible, (Template:Math|Template:Math) is equivalent to (Template:Math|Template:Math), whence the formula for the fractional linear transformation.

Type III_n. A is the Jordan algebra of n × n symmetric complex matrices Template:Math with the operator Jordan product Template:Math. It is the complexification of Template:Math, the Euclidean Jordan algebra of n × n symmetric real matrices. On Template:Math, define a nondegenerate alternating bilinear form by Template:Math. In matrix notation if $J=\left(\begin{array}{cc}0& I\\ -I& 0\end{array}\right)$,

$\omega (z_1,z_2)=zJ{z}^t.$

Let Template:Math denote the complex symplectic group, the subgroup of Template:Math preserving ω. It consists of Template:Math such that Template:Math and is closed under Template:Math. If $g=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ belongs to Template:Math then

$g^{-1}=\left(\begin{array}{cc}{d}^t& -c^t\\ -b^t& a^t\end{array}\right).$

It has center Template:Math}. In this case Template:Math} acting on Template:Math as Template:Math. Indeed, this can be verified directly for diagonal, upper and lower unitriangular matrices which correspond to the operators Template:Math, Template:Math and Template:Math. The subset Template:Math corresponds to the matrices Template:Math with Template:Math invertible. In fact consider the space of linear maps from Template:Math to Template:Math. It is described by a pair (Template:Math|Template:Math) with Template:Math in Template:Math. This is a module for Template:Math acting on the target space. There is also an action of Template:Math induced by the action on the source space. The space of injective maps Template:Math with isotropic image, i.e. ω vanishes on the image, is invariant. Moreover, Template:Math acts freely on it. The quotient is the symplectic Grassmannian Template:Math consisting of n-dimensional Lagrangian subspaces of Template:Math. Define a map of Template:Math into Template:Math by sending Template:Math to the injective map (Template:Math|Template:Math). This map induces an isomorphism of Template:Math onto Template:Math.

In fact let Template:Math be an n-dimensional Lagrangian subspace of Template:Math. Let Template:Math be a Lagrangian subspace complementing Template:Math. If they are in general position, i.e. they have trivial intersection with Template:Math and Template:Math, than Template:Math is the graph of an invertible operator Template:Math with Template:Math. So the image corresponds to (Template:Math|Template:Math) with Template:Math and Template:Math.

At the other extreme, Template:Math and Template:Math can be written as direct sums Template:Math, Template:Math, where Template:Math and Template:Math are the intersections with Template:Math and Template:Math and Template:Math with Template:Math. Then Template:Math and Template:Math. Moreover, Template:Math and Template:Math. The subspace Template:Math corresponds to the pair (Template:Math|Template:Math), where Template:Math is the projection of Template:Math onto Template:Math. Note that the pair (Template:Math, Template:Math) is in duality with respect to ω and the identification between them induces the canonical symmetric bilinear form on Template:Math. In particular V₁ is identified with U₂ and V₂ with U₁. Moreover, they are V₁ and U₁ are orthogonal with respect to the symmetric bilinear form on (Template:Math. Hence the idempotent Template:Math satisfies Template:Math. So Template:Math and Template:Math lie in Template:Math and Template:Math is the image of (Template:Math|Template:Math).

The general case is a direct sum of these two cases. Template:Math can be written as a direct sum Template:Math where Template:Math and Template:Math are the intersections with Template:Math and Template:Math and Template:Math is a complement in Template:Math. Similarly Template:Math can be written Template:Math. Thus Template:Math and Template:Math, where Template:Math are complements. The direct sum Template:Math is of the second kind. It has a complement of the first kind.

Maps Template:Math in the structure group correspond to Template:Math in Template:Math, with Template:Math. The corresponding map on Template:Math sends (Template:Math|Template:Math) to (Template:Math|Template:Math). Similarly the map corresponding to Template:Math sends (Template:Math|Template:Math) to (Template:Math|Template:Math), the map corresponding to Template:Math sends (Template:Math|Template:Math) to (Template:Math|Template:Math) and the map corresponding to Template:Math sends (Template:Math|Template:Math) to (Template:Math|Template:Math). It follows that the map corresponding to Template:Math sends (Template:Math|Template:Math) to (Template:Math|Template:Math). On the other hand, if Template:Math is invertible, (Template:Math|Template:Math) is equivalent to (Template:Math|Template:Math), whence the formula for the fractional linear transformation.

Type II_2n. A is the Jordan algebra of 2n × 2n skew-symmetric complex matrices Template:Math and Jordan product Template:Math where the unit is given by $J=\left(\begin{array}{cc}0& I\\ -I& 0\end{array}\right)$. It is the complexification of Template:Math, the Euclidean Jordan algebra of self-adjoint n × n matrices with entries in the quaternions. This is discussed in Template:Harvtxt and Template:Harvtxt.

Type IV_n. A is the Jordan algebra Template:Math with Jordan product Template:Math. It is the complexication of the rank 2 Euclidean Jordan algebra defined by the same formulas but with real coefficients. This is discussed in Template:Harvtxt.

Type VI. The complexified Albert algebra. This is discussed in Template:Harvtxt, Template:Harvtxt and Template:Harvtxt.

The Hermitian symmetric spaces of compact type Template:Math for simple Euclidean Jordan algebras Template:Math with period two automorphism can be described explicitly as follows, using Cartan's classification.

Type I_p,q. Let F be the space of q by p matrices over R with p ≠ q. This corresponds to the automorphism of E = H_{p + q}(R) given by conjugating by the diagonal matrix with p diagonal entries equal to 1 and q to −1. Without loss of generality Template:Math can be taken greater than Template:Math. The structure is given by the triple product Template:Math. The space X can be identified with the Grassmannian of Template:Math-dimensional subspace of Template:Math. This has a natural embedding in Template:Math by adding 0's in the last Template:Math coordinates. Since any Template:Math-dimensional subspace of Template:Math can be represented in the form [[[:Template:Math]]|Template:Math], the same is true for subspaces lying in Template:Math. The last Template:Math rows of Template:Math must vanish and the mapping does not change if the last Template:Math rows of Template:Math are set equal to zero. So a similar representation holds for mappings, but now with q by p matrices. Exactly as when Template:Math, it follows that there is an action of Template:Math by fractional linear transformations.

Type II_n F is the space of real skew-symmetric m by m matrices. After removing a factor of Template:Radic, this corresponds to the period 2 automorphism given by complex conjugation on E = H_n(C).

Type V. F is the direct sum of two copies of the Cayley numbers, regarded as 1 by 2 matrices. This corresponds to the canonical period 2 automorphism defined by any minimal idempotent in E = H₃(O).

See also

• Mutation (algebra)
• Symmetric cone

References

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