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JORDAN TRIPLE SYSTEMS : PROPERTIES OF THE GENERIC MINIMAL POLYNOMIAL
Guy ROOS ESA CNRS 6086 ’’Groupes de Lie et geometrie” Departement de Mathematiques Universite de Poitiers 86022 POITIERS FRANCE
Abstract
Hermitian positive Jordan triple systems (HPJTS) correspond to bounded symmetric complex domains. We give a survey of the properties of HPJTS and especially of their generic minimal polynomial. As an application, we give a description of the canonical projective realization of the compactification. It turns out that for a natural normalization, the Euclidean volume of a bounded circled homogeneous complex domain is an integer which is equal to the projective degree of the above compactification. The properties of the generic minimal polynomial also lead to the concept of polynomial morphisms of JTS, for which some open problems are stated.
1 Hermitian positive Jordan triple systems
Definition 1 Let V be a finite dimensional complex vector space. A structure of Jordan triple system over V is a ternary product (the Jordan triple product) (x, y, z) {xyz} = D(x, y)z = Q(x, z)y which is complex bilinear and symmetric with respect to (x, z), complex antilinear with respect to y and satisfies the Jordan identity
{xy{uvw}} - {u(’{xyw}} = {{xiju}vu;} - {u{i-ry}№}.
A Jordan triple system is said hermitian positive if (u\v) = trD(u,u) is positive definite.
The quadratic representation is defined by
Q(x)y = ^{xy^}-
The following fundamental identity for the quadratic representation is a consequence of the Jordan identity :
Q{Q{x)y) = Q{x)Q{y)Q{x).
The Bergman operator B is defined by
B(x, y) = I - D(x,y) + Q{x)Q(y),
where I denotes the identity operator in V. It is also a consequence of the Jordan identity that the following fundamental identity holds for the Bergman operator :
Q(B(x, y)z) = B{x,y)Q{z)B{y,x).
Example 1 Type /Pi„ 1 < p < q ■ V7 = X u(C) (space of p x q matrices with complex entries),
endowed with the triple product
{xyz} = xlyz + ztyx.
Example 2 Type IIn : V = -4\(C) (space of n X n alternating matrices) with the same triple product.
™ 3 r„c , v = *(0 x „ ^ ^ ^ (4t _ ^
Example 4 T„c IV,„ : V = Cn Mh ,k q„aJ„Uc opcn,or d,f,eJ by
Q{x)y = q(x,y)x - q(x)y,
where q(x) = ^x-1 q(x,y) = 2Zxiyi.
Example 5 Tyne VI ■ 1/ — 7-/ iVO A
- c, «, 4<m*- ^ ,?cn;
Q(x)y = (x|y)i -Is
*"** •*' •** ™'™ •” *.W «- (,W a,
Example 6 T„e V ; V = A(e,«(Oc), <Ae „J5paee ofH^Oc) cons„t,ns ,» maMct, of lhe ,„m
m* the s.mc ,„dratlc atml„_ Hm a dMcs (4e Cayky ^ ^ ^
Actually, the p,ec„di„6 examples »xha« the llst of simp,e „ermitian poS]tjve ^ ^ ^
2 The generic minimal polynomial
2.1 The generic minimal polynomial of a unital Jordan algebra
Recall that a (finite dimensional) Jordan algebra ovpr C i= = 1
bilinear product (*, y) xy satlsfying the Jordan comPIex vector space A with a comrnutat,ve
x(x2y) = x\xy). '
element e. Then for eachVe^Md^^ the Jordan algebra A has a unit
way ; the map p i—> p(x) ;s t},e ai,rphr„ u x, possible to define p(x) in the usual
m°n,c j,„, ils\'„r cm T,J
is called regular if its rank is maximal • the rank of r 1 1 1S 1311 °f X' An eIement x G A
algebra A. The following result holds in a finite dimensionaUord^'T I! ^ ^ °f the J°rdan
generally m a power associative algebra) : ^ s ra Wlth unit element (and more
Proposition 1 There exist polynomials rrii m on A nf r r j
each regular element x G A, the minimal polynomial mx is ’equal to ^ h ^ Ulat’ for
mx = r--ml(x)Tr-' + ...+ (-.lymr{x)
Definition 2 The polynomial
m(T, x) = Tr - mi{x)Tr-1 + •.. + (_iymr{x)
is called the generic minimal polynomial of A at r Th» r r
the polynomial mr is called the determinant and denoted Z= dT ^ * Ca'1Cd ^
The following relations hold :
m(X,x) = det(Ae - *), det e = 1, det(/>(z)j,) = (det xf det y.
Here P denotes the quadratic operator of the Jordan algebra A :
p{x)y=2x{xy) - x^y. ■
2.2 The generic minimal polynomial of a Jordan algebra
Here we consider a Jordan algebra A over C, but we don’t further assume it has a unit element. Let A = C ® A the algebra obtained from A in the usual way : (A © x)(/j @y) — A/j © (Ay + pix + xy). It is easy to check that A is a Jordan algebra with unit element e = 1 © 0. The generic minimal polynomial in of 0 © x in A has the form m(T, 0 © x) = Tm(T, x). If A has already a unit element, then m(T, x) coincides with the previously defined generic minimal polynomial.
In the general case, the relation fh(T, 0©:r) = Tm(T,x) will be taken as^he definition of the generic minimal polynomial m(T, z).The generic norm is N(x) = m(l, x). One has det(A © —x) = Am(A, x) and det(l © —x) = N(x), where det denotes the determinant in A. An element x G A is said to be regular if 0 © x is regular in A, which is equivalent to say that (x, x2, . . ., xr+1) has maximal dimension r.
2.3 The generic minimal polynomial of a Jordan triple system
Let us consider now a Jordan triple system V. If y G V, a structure of Jordan algebra, denoted V^y\ is defined by taking on V the product oy defined by
x °y z =
Then is a Jordan algebra. We say that (x,y) G V x V is regular if V^ has maximal rank and if x is regular in The number r = max{p(y);y G V}, where p(y) denotes the rank of the Jordan algebra
Vr(y), is called the rank of the Jordan triple system V.
Theorem 1 Let V be a Jordan tuple system of rank r. There exist (real) polynomials on
VxV, homogeneous of respective bidegrees (1,1),..., (r, r), such that for each regular pair (x, y) G V x V, the minimal polynomial of x m is equal to
Tr - mi (x, y)TT~l +----h (-1 )rmr(x,y).
Here V denotes the space V with the conjugate complex structure.
Definition 3 The polynomial
m(T,x,y) = Tr - mi{x,y)Tr~l +--------h (-l)rmr(x\ y)
is called the generic minimal polynomial of V (at (x,y)). The (inhomogeneous) polynomial Ar : V x V —> C defined by
N(x,y) = m(l,x,y)
is called the generic norm.
Proposition 2 The following identities hold for the polynomials nrik (1 < k < r) :
mk(x, y) = mk(y, x), nik{Q{x)y, z) = mk(x, Q{y)z).
Hereunder are the minimal polynomials corresponding to some examples listed in the preceding section
1. Type Ip: m(T, x, y) — Det (TIp - xy"), where Det is the usual determinant of square matrices.
2. Type III„ : m(T, x, y) — Det (TJ„ - xy’).
3. Type IV„ : m(T,x,y)--T2-q(x,y) + q{x)q(y).
3 Inverse and quasi-inverse
3.1 Invertibility in a Jordan algebra with unit element
Let A be a Jordan algebra with unit element e. Recall that the quadratic operator P in A is defined by P(x)y = 2x(xy) — x2y. For a £ A, we denote by C[a] the unital subalgebra generated by a.
Theorem 2 For a £ A, the following properties are equivalent :
1. a is invertible in C[a] /
2. P(a) is invertible ;
3. e is in the range of P{a) ; ■
4. there exists b £ A such that P(a)b = a and P(a)b2 = e ;
5. det a / 0 .
If an element a £ A satisfies to these equivalent conditions, then it is called invertible in the Jordan algebra A ; if a is invertible, the element a-1 = P(a)-1a is called the inverse of a. Note that ab = e, but this last condition is in general not sufficient for a to be invertible in the above sense.
3.2 Quasi-invertibility in a Jordan algebra
Let A be a Jordan algebra and let A = C®A the corresponding unital algebra. For x £ A, we denote by L(x) the multiplication operator : L(x)y — xy.
Definition 4 An element x £ A is called quasi-invertible iff 1 ® — x is invertible in the unital Jordan algebra A. If x £ A is quasi-invertible, the quasi-inverse of x is the element z such that 1 ® z is the inverse of ] © — x in A.
Theorem 3 For x £ A, the following properties are equivalent :
1. x is quasi-invertible ; ~
2. there exisi.-i z £ Cq[x] (z is the value at x of a polynomial without constant coefficient) such that
z — x — xz ;
3. the linear operator I — 2L(x) + P(x) is invertible on A ;
4- 2x — x2 is in the range of I — 2L(x) + P(x) ;
5. there exists z £ A such that (/ — 2L(x) + P(x))z — x — x2 and (/ — 2L(x) + P(x))z2 = x2 ;
6. N(x) ^ 0.
If x is quasi-invertible, its quasi-inverse is given by
2 = (I — 2L(x) + P(x))~1x2; if x is close to 0, one also has 2 = X^=i xU■
3.3 Quasi-invertibility in a Jordan triple system
Let V be a Jordan triple system. For y £ V, recall that is the Jordan algebra obtained by taking on
V the product x oy z = ^{xyz}. The operators L and P for turn out to be Ly(x) = \D{x,y) and
Py(x) = Q(x)Q(y). As a consequence, I — 2Ly(x) + Py(x) — B(x, y), the Bergman operator.
Definition 5 A pair (x, y) £ V x V is called quasi-invertible iff x is quasi-invertible in the Jordan algebra ViyK The quasi-inverse xy is the quasi-inverse of x in V.
Theorem 4 For (x,y) £ V x V, the following conditions are equivalent :
1. the pair(x,y) is quasi-invertible ;
2. the Bergman operator B(x, y) is invertible ;
3. there exists z E V such that B(x, y)z = x - Q{x)y and B(x, y)Q{z)y = Q(x)y ;
4■ the element 2x — Q(x)y belongs to the range of B(x,y) ;
5. N(x,y)^0.
If (x,y) is quasi-invertible, the quasi-inverse xy is given by
xy = B{x,y)~l(x - Q(x)y).
If x is close to 0, then xy = Y^7=i x(-n’y\ where x("■») denotes the n-th power of x in the Jordan algebra
y(y)'
Quasi-invertibility has the following symmetry property :
Proposition 3 A pair (x, y) is quasi-invertible iff (y, x) is quasi-invertible ; then yx = y + Q(y)xy.
4 Spectral theory
^From now on, V is a hermitian positive Jordan triple system. Moreover, we will assume that V is simple, that is V is not the direct sum of two non trivial subsystems with component-wise triple product. Any hermitian positive Jordan triple system is in fact semi-simple, that is the direct sum of a finite family of simple subsystems.
An automorphism / : V —> V of the Jordan triple system V is a complex linear isomorphism preserving the triple product : f{u, v, w} = {fu, fv, fw}. The automorphims of V form a group, denoted Aut V, which is a compact Lie group ; we will denote by K its identity component.
Definition 6 An element c 6 V is called tripotent if c is an idempotent in V^c\ that is if {ccc} = 2c.
Proposition 4 If c is a tripotent, the operator D(c,c) annihilates the polynomial T(T - 1 )(T - 2).
Definition 7 Let e be a tripotent. The decomposition V = V0{c) © V\ (c) © lA(c), where Vj(c) is the
eigenspace V; (c) = {x £ V ; D(c,c)x — jx}, is called the Peirce decomposition of V (vvith respect to the
tripotent c).
Definition 8 Two tripotents c\ and c-> are called orthogonal if D(ci,c2) = 0.
If ci and c2 are orthogonal tripotents, then D(ci,c^ and D(c2,c2) commute and cx + c2 is also a tripotent.
Definition 9 A non zero tripotent c is primitive if it is not the sum of non zero orthogonal tripotents. A tripotent c is maximal if there is no non zero tripotent orthogonal to c. A frame of V is a maximal sequence (ci,. . ., cr) of pairwise orthogonal primitive tripotents.
Definition 10 Let c = (cj^, .... Cr) be a frame. For 0 < i < j < r, let
K;(c) = {x £ V ; D(ck,ck)x = + ^)x, 1 < k < r} :
the decomposition V = ©0<i<j<r ^;(c) IS called the simultaneous Peirce decomposition with respect to the frame c.
Theorem 5 I^et V be a simple hermitian positive Jordan triple system. Then there exist frames for V. Alt frames have the same number of elements, which is equal to the rank r of V. The subspaccs =■ Vij(c) of the simultaneous Peirce decomposition have the following properties : Vqq = 0 ; Va = Ce; (0 < i); all Vij’s (0 < i < j) have the same dimension ; allVoi’s (0 < i) have the same dimension. The group K acts transitively on the manifold T of frames of V.
Definition 11 The numerical invariants of V are
a = dimVij (0 < i < j), b = dim Vo» (0 < i).
The genus of V is the number defined by
<7 = 2 + a(i— 1) + b.
Theorem 6 Let V be a simple hermitian positive Jordan triple system. Then any x £ V can be written in a unique way
x — AiCj + A2C2 + ■ • • + ApCp,
where Ai > A2 > • ■ ■ > Ap > 0 and C\, c2 .. ■, cp are pairwise orthogonal tripotents. The element x is regular iff p = r ; then (ci, c2 . . . , cr) is a frame of V.
Definition 12 The decomposition x - AiCj. + A2c2 +----------b Apcp is called spectral decomposition of x.
The following proposition shows the relation between the generic minimal polynomial and the eigenvalues Ai, A2,.. •, Ar of a regular element x.
Proposition 5 Let V be a simple hermitian positive Jordan triple system. Let x = Aici+A2c2 + - • - + Arcr be the spectral decomposition of a regular element x. Then the generic minimal polynomial at (x,x) is
r
m(T,x,x) = ]J(T-A?).
1=1
The following identities hold :
dct B{x, y) = N(x, y)q, trD(x,y) = gmi(x,y).
5 Schmid decomposition
Let V be a simple hermitian positive Jordan triple system. We denote by V(V) the space of complex polynomials on V. The group K acts naturally on V(V).
Definition 13 For n = (n1,n2, • • ,nr) £ Nr with ri\ > 712 ^ ••• ^ rir > 0, let mu be the unique complex polynomial on V x V such that, for each regular x with spectral decomposition x = A^ +A2c2 +
■ • ■ + ATcT, the value of mn at (x,x) is
/ \ \ ^ \2rii \ 2n2 \2nr
For y £ V, we denote by mn,y the polynomial on V defined by
mn, y(x) = mn(x,y) and by 'Pn(V) the subspace ofV(V) spanned by the polynomials mn,y, y £ V.
k times
Remark 1 Let 1 < k < r ; let us denote by < k > the multi-integer < k >= (1,... 1, 0,. .., 0). Then m<jt> = rnk, the homogeneous component of bidegree (k,k) of the generic norm.
Theorem 7 (Schmid decomposition) The spaceV(V) admits the following decomposition into irreducible, pairwise inequivalent A'-modules :
V(V) = 0 7>n(V),
n>0
where n > 0 for n = (n1,n2, . . - ,nr) £ Nr means ni > n2 > ■ ■ ■ > nr > 0. Moreover, the polynomial mn is a reproducing kernel for Pn(V), which means there is an hermitian scalar product ( | ) on 'Pn(V) such that
mn(x,y) = (mn,x|™n,y)- ‘
Let us consider the (antilinear) isomorphism d : V —> V* defined by
■d(y)(x) = mi(x,y) = g-1 tr D(x,y).
Using this isomorphism, we get an antilinear isomorphism tp : V{V) —► O. where Q. V is the symmetric algebra of V. For n > 0, let Vn = <p{Pn(V)) and <7n : V —* Vn C 0|n| V defined by
(7n(y) = (p o mh'y (y £ V). The hermitian scalar product ( | ) is also transferred from Pn(V) to Vn ; the
following relation then holds :
mn(x,y) = (<Tn(*)kn(2/))-Note that (ro(x) = 1 and that cr<1> : V —► V is the identity map.
6 Compactification
6.1 Compactification of an hermitian positive Jordan triple system
The following description of the compactification has been given by 0. Loos. Let V be an HPJTS ; one can define an equivalence relation ~ on V x V by
(x, y) ~ (x',i/) ((x, y - j/')isquasi - invertiblefe x' = xy_y ).
Roughly speaking, this equivalence relation means xy = x,y'. Let us denote by [x, y] the equivalence class
of (x, y) and by X the quotient space (V x V)/
Theorem 8 There exists a unique structure of smooth algebraic variety on X such that, for each v 6 V,
Xv = {[x,v] ; x 6 V) is an open affine subvariety and [x, v] >-»• x is an isomorphism of Xv on V.
The injection x i—* [x,0] of V in A' will be called the canonical compactification of V.
6.2 Projective imbedding
Let V be a simple HPJTS of rank r. For 0 < k < r, we write Vk for V<k> and ak for a<k>. Let W be the subspace of 0* V defined by _
W = ©0<i:<r^
and a : V----->■ P(W) the map given by
<t(x) = [l,x,cr2(x),...,crr(x)].
The compactification described above has then the following projective realization.
Theorem 9 Let V be a simple HPJTS of rank r. The closure of a(V) is an algebraic submanifold X of P{\V), which is tsomorphic to X, and a : V —> X is isomorphic to the canonical compactification
V —*X.
^From now on, we will identify X with X.
6.3 Volume computations
Let a be the Kahler form on V associated to the hermitian inner product mi :
a = -^—ddrrii.
" /7T
We endow V with the volume form an (n — dimV7) ; the volume of the unit ball associated to mj is then equal to 1.
On W = ©o<i<rVr, we put the hermitian scalar product ( | ) which is the direct sum of the hermitian scalar products ( | ) on the Ws. Let F be defined on W by F(z) = (z | z) and let f3 be the associated Fubini-Study form on P{W) :
S3 — -~-dd log F.
For a smooth submanifold Z of pure dimension d in P{W), the degree of Z in P{W) is then given by
deg Z = Jjd.
In particular, the degree in P(W) of the compactification X of V is
degX = //?"=/
Jx Jv
as cr(V^) is an open dense subset of X. From the definition of cr and from the relations mk(x, x) — (ak(x) | trk(x)), it follows that
a*j3 = —53log(1 + mi(x,x)+- ■■ + mr(x,x)) - ^-dd\ogN(x,-x).
2ir ^
The following proposition allows us to compute <r*/?n (where n = dim 7).
Proposition 6 The identity
^J-d’B\ogN{x,-x)^ = det B{x, x)_1 an = N{x,x)~9an
holds in the hermitian positive Jordan triple system V.
Proposition 7 Let
<3> : T x {Aoo > Ae > • • • > Av > '} —> VV|} be the diffeomorphism defined by
T
$((ci, ■ ■ ■, cr), (Ai,..., Ar)) = 7: AjC};
i = i
here I is the manifold of frames of V and Vreg the open dense subset of regular elements of V. The
pull-back of the volume element an by <I> is
<b*an = 0 A JJ Aji+1 Yl (Aj - A^)° dAi A . . .dAr, j = i i
where a and b are the numerical invariants of V and 0 is a A-invariant volume form on V .
Definition 14 Let V be an IIPJTS. The bounded symmetric domain D associated to V' is the unit ball
of V for the spectral norm, that is the set of elements x £ V whose spectral decomposition is
x — AiCi -f A2C2 + • ■ ■ + ApCp
with 1 > Ai > A2 > ■ • • > Ap > 0. We define the real analytic maps 1? : V ► D and ip : D ► V,
inverse of each other, by
i?(x) = B(x, —x)~ll4x — ^ Ay(1 + Aj) l!~Cj if x = AjCi + A2c2 + • • • + G V and
ip(x) = B(x, x)~lt4x = ^2 l^ci
if x — A1C1 4- A2C2 + ■ ■ ■ + ApCp G D.
Theorem 10 The following relations hold in a simple HPJTS V of dimension n and genus g :
r (N{x,-x)san) = N{x,x)~s~san, i>* {N(x, x)san) = N(x,-x)-g~san.
Taking s = 0 in the last relation and integrating over V, we get
Theorem 11 The volume of the bounded symmetric domain D (with respect to the normalized form a )
is equal to the degree of the compactification X in P(W).
7 Open problems
1. If one looks over the examples given by the ^slfi^atl°^’y • homogeneous of degree kTand have a structure of Jordan triple system such that ? V J
is a
-»——*
CTfc (Q(x)y) = QtM*)) *k{y)>
„here ft ,s the quadratic operator fo, Vt. Understand this JTS structure on V,?
There are other polynomial morphisms of Jordan triple systems, for example the quadratic repre-
sentatlon Q-V —► EndR{V)
which is a quadratic morphism as shown by the fundamental identity
Q(Q{x)y) = Q{x)Q{y)Q(x)-
understand and ciassify l,omoge„eous polynomial morphisms between s.mple hermit,an positive
Jordan triple systems?
References
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metric domains, J. Fund. Anal., 88, 64-89, 199 .
,rT , KORANYI Alan, Analysis on Symmetric Cone, Clarendon I rcss, Orfoul
[2] FARAhl, Jacques, hULiA^n, »
l312s.^ ^ ^ °!Ca,'!m
Irvine, 1977. „ POOS
p q f'x s 1992in the Proceedings oj the Conference.
