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.
[1] For a in A define the Jordan multiplication operator on A by and the quadratic representation Q(a) by It satisfies the fundamental identity the commutation or homotopy identity where In particular if a or b is invertible then 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: The Jordan triple product is defined by so that There are also the formulas For y in A the mutation Ay is defined to the vector space A with multiplication If Q(y) is invertible, the mutual is called a proper mutation or isotope.
Alternatively, the Jordan identity can be deduced by realizing the mutation inside a unital quadratic algebra.
This condition can be expressed as follows without mentioning the mutation or homotope: In fact if (a,b) is quasi-invertible, then c = ab satisfies the first identity by definition.
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.
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.
Let A be a finite-dimensional complex unital Jordan algebra which is semisimple, i.e. the trace form Tr L(ab) is non-degenerate.
Loos (1977) uses the Bergman operators to construct an explicit biholomorphism between X and a closed smooth algebraic subvariety of complex projective space.
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 Ae = ⊕ Cei, which is the complexification of Ee = ⊕ Rei.
G acts on the corresponding finite-dimensional Lie algebra of holomorphic vector fields restricted to X0 = A, so that G is realized as a closed matrix group.
Accordingly, there is a relation j = S1 ∘ T1 ∘ S1 and PSL(2,C) is a subgroup of G. Loos (1977) defines the operators Tc in terms of the flow associated to a holomorphic vector field on X, while Dineen, Mackey & Mellon (1999) give a direct algebraic description.
[19] In fact if Sb Ta lies in Ω, then (a,b) is equivalent to (x,0), so it a quasi-invertible pair; the converse follows from the exchange relations.
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.
Any further holomorphic vector field would have to appear in degree 1 and so would have the form a ↦ Ma for some M in End A. Conjugation by J would give another such map N. Moreover, etM(a,0,0)= (etMa,0,0).
It lies in H. Let Ke = Tm be the diagonal torus associated with a Jordan frame in E. The action of SL(2,C)m is compatible with θ which sends a unimodular matrix
If a = Σ αiei with |αi| = 1, then Q(a) gives the action of the diagonal torus T = Tm and corresponds to an element of K ⊆ H. The element J lies in SU(2)m and its image is a Möbius transformation j lying in H. Thus S = j ∘ T ∘ j is another torus in H and T ∘ S ∘ T coincides with the image of SU(2)m. H acts transitively on X.
[23] The preceding theory describes irreducible Hermitian symmetric spaces of tube type in terms of unital Jordan algebras.
In Loos (1977) general Hermitian symmetric spaces are described by a systematic extension of the above theory to Jordan pairs.
In the development of Koecher (1969) harvtxt error: no target: CITEREFKoecher1969 (help), however, irreducible Hermitian symmetric spaces not of tube type are described in terms of period two automorphisms of simple Euclidean Jordan algebras.
τ is assumed to satisfy the additional condition that the trace form on V defines an inner product.
For a in V, define Qτ(a) to be the restriction of Q(a) to V. For a pair (a,b) in V2, define Bτ(a,b) and Rτ(a,b) to be the restriction of B(a,b) and R(a,b) to V. Then V is simple if and only if the only subspaces invariant under all the operators Qτ(a) and Rτ(a,b) are (0) and V. The conditions for quasi-invertibility in A show that Bτ(a,b) is invertible if and only if B(a,b) is invertible.
The commuting self-adjoint operators L(x)L(y) with x, y odd powers of a act on F, so can be simultaneously diagonalized by an orthonormal basis ei.
Any two frames are related by an element in the subgroup of the structure group of Eτ preserving the trace form.
The same arguments show that the fixed point subgroup Hτ is generated by Kτ and the image of SU(2)m. It is a compact connected Lie group.
The Hermitian symmetric space of non-compact type have an unbounded realization, analogous the upper half-plane in C. Möbius transformations in PSL(2,C) corresponding to the Cayley transform and its inverse give biholomorphisms of the Riemann sphere exchanging the unit disk and the upper halfplane.
A partial Cayley transform can be defined in that case for any given maximal tripotent e = Σ εi ei in Eτ.
[26] For simple Euclidean Jordan algebras E with complexication A, the Hermitian symmetric spaces of compact type X can be described explicitly as follows, using Cartan's classification.
Indeed, this can be verified directly for diagonal, upper and lower unitriangular matrices which correspond to the operators W, Sc and Tb.
Indeed, this can be verified directly for diagonal, upper and lower unitriangular matrices which correspond to the operators W, Sc and Tb.
The Hermitian symmetric spaces of compact type X for simple Euclidean Jordan algebras E with period two automorphism can be described explicitly as follows, using Cartan's classification.