Symmetric cone

The trace form has the advantage of being manifestly invariant under automorphisms of the Jordan algebra, which is thus a closed subgroup of O(E) and thus a compact Lie group.

In practical examples, however, it is often easier to produce an inner product for which the L(a) are self-adjoint than verify directly positive-definiteness of the trace form.

In fact polarizing of the Jordan relation—replacing a by a + tb and taking the coefficient of t—yields This identity implies that L(am) is a polynomial in L(a) and L(a2) for all m. In fact, assuming the result for lower exponents than m, Setting b = am – 1 in the polarized Jordan identity gives: a recurrence relation showing inductively that L(am + 1) is a polynomial in L(a) and L(a2).

If e is a non-zero idempotent then the eigenvalues of L(e) can only be 0, 1/2 and 1, since taking a = b = e in the polarized Jordan identity yields In particular the operator norm of L(e) is 1 and its trace is strictly positive.

Let Ei and Eij be the subspaces of the Peirce decomposition of E. For x in O, set πij = Pij π(xeij), regarded as an operator on Vi.

For a in E, define an endomorphism of E, called the quadratic representation, by[9] Note that for self-adjoint matrices L(X)Y = ⁠1/2⁠(XY + YX), so that Q(X)Y = XYX.

On the other hand, Q(a)1 = a2, so Taking b = a−1 in the polarized Jordan identity, yields Replacing a by its inverse, the relation follows if L(a) and L(a−1) are invertible.

Finally it can be verified immediately from the definitions that, if u = 1 − 2e for some idempotent e, then Q(u) is the period 2 automorphism constructed above for the centralizer algebra and module of e. If a is an invertible operator and b is in the positive cone C, then so is Q(a)b.

Let C be a symmetric cone in the Euclidean space E. As above, Aut C denotes the closed subgroup of GL(E) taking C (or equivalently its closure) onto itself.

Making both sides act on c yields On the other hand, and likewise Combining these expressions gives which implies the Jordan identity.

Let C be the positive cone in a simple Euclidean Jordan algebra E. Aut C is the closed subgroup of GL(E) taking C (or its closure) onto itself.

be the Lie algebras of G and K. G is closed under taking adjoints and K is the fixed point subgroup of the period 2 automorphism σ(g) = (g*)−1.

If N is the closed subgroup of S such that nx = x modulo ⊕(p,q) > (i,j) Epq, then S = AN = NA, a semidirect product with A normalizing N. Moreover, G has the following Iwasawa decomposition: For i ≠ j let Then the Lie algebra of N is Taking ordered orthonormal bases of the Eij gives a basis of E, using the lexicographic order on pairs (i,j).

The Jordan product on E extends bilinearly to EC, so that (a + ib)(c + id) = (ac − bd) + i(ad + bc).

The automorphism groups of E and EC consist of invertible real and complex linear operators g such that L(ga) = gL(a)g−1 and g1 = 1.

In particular e and 1 − e are orthogonal central idempotents with L(e) = P and L(1 − e) = I − P. So simplicity follows from the fact that the center of EC is the complexification of the center of E. According to the "elementary approach" to bounded symmetric space of Koecher,[20] Hermitian symmetric spaces of noncompact type can be realized in the complexification of a Euclidean Jordan algebra E as either the open unit ball for the spectral norm, a bounded domain, or as the open tube domain T = E + iC, where C is the positive open cone in E. In the simplest case where E = R, the complexification of E is just C, the bounded domain corresponds to the open unit disk and the tube domain to the upper half plane.

They both lie in the Riemann sphere C ∪ {∞}, the standard one-point compactification of C. Moreover, the symmetry groups are all particular cases of Möbius transformations corresponding to matrices in SL(2,C).

This complex Lie group and its maximal compact subgroup SU(2) act transitively on the Riemann sphere.

[21] The compactification and complex Lie group are described in the next section and correspond to the dual Hermitian symmetric space of compact type.

Each Jordan frame gives rise to a product of copies of R and C. The symmetry groups of the corresponding open domains and the compactification—polydisks and polyspheres—can be deduced from the case of the unit disk, the upper halfplane and Riemann sphere.

The analysis can also be reduced to this case because all points in the complex algebra (or its compactification) lie in an image of the polydisk (or polysphere) under the unitary structure group.

Conversely if ||U|| < 1 then I − U is invertible and Since the Cayley transform and its inverse commute with the transpose, they also establish a bijection for symmetric matrices.

In A = EC the above resolvent identities take the following form:[27] and equivalently where the Bergman operator B(x,y) is defined by B(x,y) = I − 2R(x,y) + Q(x)Q(y) with R(x,y) = [L(x),L(y)] + L(xy).

The orbit of these points under the unitary structure group is the whole of D. The Cartan decomposition follows because KD is the stabilizer of 0 in GD.

The center of KD is isomorphic to the circle group: a rotation through θ corresponds to multiplication by eiθ on D so lies in SU(1,1)/{±1}.

On the other hand, if K is a compact subgroup of U(n,n), there is a K-invariant inner product on C2n obtained by averaging any inner product with respect to Haar measure on K. The Hermitian form corresponds to an orthogonal decomposition into two subspaces of dimension n both invariant under K with the form positive definite on one and negative definite on the other.

By Sylvester's law of inertia, given two subspaces of dimension n on which the Hermitian form is positive definite, one is carried onto the other by an element of U(n,n).

Let HT be the group of biholomorphisms of the tube T. The Cayley transform shows that is isomorphic to the group HD of biholomorphisms of the bounded domain D. Since ANT acts simply transitively on the tube T while KT fixes ic, they have trivial intersection.

[34] The account in Koecher (1969) develops the theory of bounded symmetric domains starting from the standpoint of 3-graded Lie algebras.

These Jordan triple systems correspond to irreducible Hermitian symmetric spaces given by Siegel domains of the second kind.