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REAL FORMS OF JORDAN ALGEBIUS
J.B. Kayoya
Available at: http: 11ww. ictp. trieste. it/-pub- of f IC/2003/160
United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS
REAL FORMS OF JORDAN- ALGEBRAS
J.B. Kayoya' Department of Mathematics, University of Burundi, Faculty of Science, B.P. 2700, Bujumbura, urundi and The Abdus Salam International Centre for Theoretical Physics, 7 iete, Italy.
Abstract The problem of classification of simple Jordan algebras and Lie-Triple Systems in general had been deeply studied [1] 7 [8], [11], 13], 14], [15], 16], 17], and satisfying results had been obtained. So, no new result on classification is in this aper, but we think that the proofs of theorems 42 and 44 are new and can be of help to understand the structure of simple real Jordan algebras better. At the end of the paper, we remark that some symmetric spaces associated to these Jordan algebras can be viewed as real forms of a cmplex symmetric space.
MIRAMARE - TRIE13TE December 2003
'jkayoya0yahoo.fr 1. INTRODUCTION
Considering the simplicity and the rank of their maximal Euclidean subalgebras, we classify simple real Jordan algebras. Thus, theorems 41 42 and 44 on this classification can improve a better understanding of the structure of real simple Jordan algebras. At first, we show that simple real Jordan algebras are real forms of simple complex Jordan algebra or simple complex Jordan algebras viewed as real ones.
Since the classification of simple complex Jordan algebras are well known and are in one to one correspondence with their Euclidian real form up to isomorphism [11], the principle of classification will be as follows: we will choose a real Euclidean form W of the simple complex Jordan algebra V-. If a is an involution of W, with W and W being the eigenspaces corre- sponding to the eigenvalues 1 and - of a, then = W + iW- is a real form of VC called the modification of W by a, and all the real forms of are obtained in this way. Moreover two modifications and V6 are isomorphic if a and are conjugate under an automorphism of W. So, the problem of classification of the real forms of is solved by the determination of the conjugacy classes of involutive automorphisms of W. This determination is the aim of our paper which is organized in three parts: In the first part, we give some definitions and properties of Jordan algebras, Hurwitz algebras and Hermitian algebras. In the second part, we explain the principle of classification, and in the last part, we consider a real Euclidian form W for each class of simple complex Jordan , determining all the classes of involutive automorphisms of W and give the associated modifications. As announced in 13), three types of modifications are obtained: Type I: Obtained if the subalgebra W+ corresponding to a is not simple. Type II: Obtained if the subalgebra W+ corresponding to a is simple and the ranks of W+ and W axe the same. Type III: Obtained if the subalgebra W+ corresponding to a is simple and the ranks of W+ and W axe different. In [10], the simple complex Jordan algebras viewed as real ones has been classified as Type IV. At the end of this paper, lists of some symmetric spaces associated to these types of Jordan algebras are given. Some of them can be viewed as real forms of a same complex symmetric space.
2. JORDAN ALGEBRAS AND HURWITZ ALGEBRAS
2.1. Jordan algebras. If F is the field of real numbers R or complex numbers , a Jordan algebra V over F is a -commutative algebra such that:
X(X 2Y = X (XY); X, G (2.1) 2 If we denote by L(x) the linear mapping defined by
L (x) y = xy; Vx, y CzV, (2.2)
then the relation 2-1) can be written as
[L(x), L (X2) = 0, VX V (2.3) Let us define on V the bilinear form
B (x, y = TrL(xy) Vx, E V (2.4) The algebra V is Euclidean if B is positive definite, and is semisimple if B is non degenerated. The algebra V is simple if it does't contain any non trivial ideal. One can show that simple algebras are always semisimple and that semisimple algebras axe direct sums of simple ideals [ I. In a semisimple Jordan algebra, there is always a unit eement e [1]. For an element x E V if we consider the number
m(x = (inf k > e, x, x 2 x kare linear dependently (2.5) then the rank of V is defined by
r supfm(x), x E Vj- (2-6)
An element c of V is an idempotent if it verifies c 2 = c. Two idempotents a and b are orthogonal if ab = 0. The idempotent c is primitive if it cannot be written as c = a b, where a and b are nonzero orthogonal idempotents. If c is an idempotent o V, one can show that the eigenvalues of L(c) axe 1, 21 and 0 [1 1]. Then if V(c, a) is the eigen space of V corresponding to eigenvalue a of L(c), we have the following decomposition I = V(C' 1) (DV(C' ) E) V(c, 0). (2.7) 2 This decomposition is called the Peirce decomposition associated to the idempotent c. One can easily verify that the eigenspaces Vc, 1) and , 0) are Jordan subalgebras of V [111. The idempotent c is said to be absolute if V(c, 1) is one dmensional. We say that a system of idempotents f cl, ... , k I is complete if cl + ... + k = e. A Jordan frame is complete system {cl,..., cj} of orthogonal primitive idempotents. If the length I of this frame and the rank r of V are the same, we say that V is reduced; in this case each idempotent of the Jordan frame is absolute. Given a Jordan frame f cl,..., c , the Peirce decomposition 2.7) can be generalized as follows:
I I V 1: Vi E)E Vj, (2.8) i=1 i n(ab = n(a)n(b), Va, b E A. (2.9) If A is real and n positive definite, we say that A is Euclidean. In this case, we can associate to n the bilinear form: n(a, b = [n(a + b - n(a)n(b)], Va, b E A. (2.10) 2 The conjugation in A is defined by = n (a, 1 - a, Va, E A. (2.11) If v is an element of A and an nonzero number of F such that n(v = -,4, then the Cayley-Dickson extension 11] denoted A(M) is the algebra A x A = A A, with the follow- ing identifications (a, 0) -- + a; (0, b) - b. (2.12) The product in A(p) is defined by: (a, vbl)(a2 + v2 = a1a2 + M6 b + v(bcC2 + 2al). (2.13) The norm and the conjugation of an element x = a vb of A(p) are given by: n(a + vb = n(a - n(b)); = a - vb (2.14) Rom the Hurwitz theorem 19], proposition 25.1, we know that up to isomorphism, the only Hurwitz algebra are B C, H and , where R and are the fields of real number and complex number, H and are the algebra of quaternion number and octonion number [18]. 2.3. The algebras Herm(r, A). Let A be an Euclidean Hurwitz algebra, and M(r, A) the algebra of matrix of order r with coefficients in A. We can notice that when A = IR,C, or the algebra M(r, A) is associative; but the algebra M(rG) is not associative. The hermitian matrix in M(r, A) is defined as: Herm (r, A) x E M (r, A) xj = j (2.15) If j is an element in A such that 3 2 -1, we can write each element z of the Cayley-Dickson extension A-l) as: z x jy, xy E A. (2.16) Using this relation, we can write each matrix Z of M (r, A(- 1) as Z = X + iY, X, Y E M(r, A). (2.17) 4 If we pose 0 (2.18) J= ( , I, ) where 1, is the identity matrix of order r, we can consic M (r, A(- 1)) as a subspace of M (2r, A) in this way M (r, A(- 1)) W E M (2r, A) JW = VJJ. (2.19) This means that the matrix Z of 2.17) is written as z X Y (2.20) Using this identification and the conjugation 2.14), one can verify that the matrix Z is A-l)- Hermitian when X is A-Hermitian and Y is skew-syrnmetric. So, we obtain this inclusion relation Herm(r, A(- 1)) Herrr]L(2r, A). (2.21) Now, if we consider on Herm (m, A) the Jordan product 1 = X + X], (2.22) 2 one can show [11] that V, Herm(r, A) is a Jordan algebra if A C or But if A = we have a Jordan structure only if r < 3. 3. PRINCIPLES OF CLASSIFICATION If V is a complex Jordan algebra, then V can be considered as a real algebra with a complex structure: it means a real linear isomorphism J of V such that: j2 = -I, and JL(x = L(x)J, Vx E V. If V is a real algebra, the complexified algebra of V is the vector space V V with the product: (U1,V1)(U2,V2 = (UIU2 - VIV2011V2 + VlU2)- ,rhe corresponding complex structure is J(u'V = -V'u)- Lemma 31. If V is a Jordan algebra with a complex tructure J then is isomorphic to V x V, where V is the algebra V with the complex structure -J. Proof. In fact, the mapping from VC to V x V defined by: 4) (u, V) = ( + JV, u - JV) is a one to one homomorphism algebra and 4)J(U, V = V, U = (J( + JV) Ju JV) = J-P(U, V) 5 Lemma 32. Let V a simple real Jordan algebra, if VC is not smple then V has a complex structure. Proof. We identify V with the subalgebra of V- defined by 1(u,0 : E VI, and we consider a non trivial ideal W of . Since V is simple, vnw= 0, because the other casevnw=V implies , JV C W and V + JV = = W which is impossible. From this, we deduce that the set Wo = {u E V 3v E Vl(uv) E WI is not empty, so there exists a R-linear isomorphism T of V such that W = (uTu) : u E }. Now, if w = u, Tu) is an element of W then Jw = -Tu, u) and so T 2 = -I. Moreover if v E , then vw E W and (uv, T(u)v) E W. This implies that T(uv = T(u)T(v) which means that T defines a complex structure on V. Proposition 33. If V is a simple real Jordan algebra, then V is a real form of a simple complex Jordan algebra, or a simple complex Jordan algebra viewed as a real one. Proof. Let be the complexification of V. If is simple, then V is real form of a simple complex Jordan algebra. If is not simple, from lemma 32), we can define a complex structure on V. 0 By this proposition, we can say that there are two classes of simple real Jordan algebras, the first class contains all simple complex Jordan algebras viewed as real ones and the second class contains the real forms of these simple complex Jordan algebras. In the second class, we find at first the family of real Euclidean forms which are in a one to one correspondence with the simple complex algebras up to isomorphism [11]. And in the second class, we find the modifications of the Euclidean real forms. Proposition 34. If V Z's a real form of a simple complex Jordan algebra , then there exists an involution a of V such that the eigen spaces V+ = x E V a(x = x, and V = x E V a(x = xJ define an Euclidean real form W = V+ + iV- of V'C. One can find the proof of the proposition in Helwig[8] or in 13] proposition 22.5. Proposition 35. Two real forms V, and V2 of V'C a-re isomorphic if they are conjugated under an automorphism of VC. 6 Proof. When is an isomorphism of V to V2, by C- linear extension, one can define an auto- morphism 41Dof by 41)(z = II(x + iy = Ox) + io(y), x, y E V. On the other hand, if D is an automorphism of Vr- such that 1(V = V2, then the restriction of to V, gives an isomorphism of V, onto V2. 0 Proposition 36. (i) Let W be an Euclidean real orm of , and c a non trivial involution Of W if W+ Ix E W : a(x = x}; and W = x E W : a(x = -x}, are the eigen spaces corresponding to a, then V,, = W+ + iW- is a real form of VIC. (ii) Each non Euclidean real form of VC is obtained in tis way. Proof. (i) By linear extension of a, we obtain a conjugation of defined by: &(x + iy = a(x) - ia(y), x, y E W ,and so the invariant subalgebra of under Vc = Iz E V: &(z = z} is a real form of . (ii) Suppose now that V is a non Euclidean real form of associated to the conjugation . From proposition 3.4), one deduces that there exists a Euclidean real form U of whose conjugation v commutes with r. This implies that the real form U is invariant under and if U = x E U : -r(x = x}; and U = x E U : r(x = -x}, then V = U = + i-. Since W and U are real Eclidian forms of VC, there exists an automorphism 4D of such that W = (U). Consequently, we can define the conjugation i = (D o o 4)-', and verify that W is invariant by This means that the restriction of # on W defines an involutions such that V = W+ + iW- Wp. 0 To complete our principles, we want to determine when two involutions a and,6 of a Euclidean Jordan algebra W give two isomorphic modifications W,, and W6. Lernma 37. Let V be a finite dimensional algebra over R or , and let ce be a positive definite automorphism of V, then a = exp(D), where D is a derivation. Proof. One can choose an orthonormal basis vi of eigen vctors of a: a(vi = Aivi; Ai > , 7 let Cijk be the structure constants of V with respect to this basis, then we have: n Vi v =E Ck Vk k=1 Since a is an automorphism, we obtain from the two last equalities that: CijkAiAj = rjkAk, Consequently, we have for each real t: CijkA!Ajt CijkAtk- Then, we obtain a one parameter subgroup of automorphisms at defined by n atv = j:(vjvj)A!. k=1 There exists a derivation D such that at = exp(tD) and a = a, = exp(D). A Cartan involution on a semisimple real Jordan algebra V, is an involution a such that the bilinear form B.,, x, y = B (a (x), y) is positive definite, where the bilinear form B is given by 2.4). Lemma 38. If a and,6 are two Cartan involutions of V, there exists an automorphism of V such that O a O 01 Proof. The automorphism = a8 is positive definite relatively to B,,. In fact B,, (J(x), y = B (aJ(x), y = Bp (x, y). By lemma 3.7), = exp(D), where D is a derivation. It follows that exp(tD)aexp(tD = a, exp(-tD),6exp(-tD) and aexp(tD = Jflexp(tD = Jexp(-tD),6 = exp((l - t)D]# For t we put exp( 1 D), then ao 0,6, and 2 2 a Proposition 39. Let W be an Euclidean real form of a simple complex Jordan algebra a and be two involutions of W with W,, and W6 the corresponding modifications. If Wa and W6 are isomorphic, then there exists an automorphism of W such that a V1 8 Proof. By proposition 3.5), there exists an automorphism -(b of such that: WO = I'D (W.) If we denote by and the conjugation of V- corresponding to a and , and -y the conjugation corresponding to W, and if we put U =V'(W), then Uis a real Euclidian form of and = O O 4 is the conjugation of corresponding to U. Since -y and are two Cartan involutions, from lemma 3.8), there exists an automorphism of which leaves V ivariant = T O O T1. If we put = IF, then leaves W invariant and such that a -1 F1 Corollary 310. If a and are two involutions of W nd if W = W+ W = V + V- are the decompositions corresponding to a and , then a and are conjugated if the subalgebras W+ and V are isomorphic. Proof. If is the isomorphism of W+ to V, we can consider that is a restriction of an auto- morphism of W such 4D(W- = _ And by - linear extension, we obtain an automorphism of VC defined by D(W + W- = 4(V+) + T.(V-). From proposition 3.9), we conclude that a and,8 axe conjugated. 0 Let us finish this section with a summary of our classification's principles. Real simple Jordan algebras are real forms of simple complex Jordan algebras or simple complex Jordan algebras Viewed as real oes. Up to isomorphism, there is a one to one correspondence between simple complex Jordan algebras and their Euclidean reals forms (see the table below). The others real forms are obtained as modifications of a certain Eclidean real form W associated to an involution of W. Two modifications of W axe isomorphic if their corresponding involutions are conjugated under an automorphism of W. X U'+, Sym(r, C I M(r, C) Skev;-(2r, C) Hem(3, 0) C R x R` Sym(r, R I Herm(r, C I Herrn(r, R I Herm(3, 0) The Jordan product for the Jordan algebras R x W+' is defined by (A, U) p, V) = (,\A + u.v), ),V + P), (3.1) where (.) is the scalar product in W-'. For the Jordan algebra of skew-symmetric matrix Skew(2r, C), the Jordan product is * [XJY + Yjx], 2 9 where J is given by 218). For the other cases, the product is given by 222). In the next section, we give all the involutions up to conjugacy for each Euclidean real form. 4. INVOLUTIONS. 4.1. Real form of rank two. We recall that an Euclidean Jordan algebra of rank two is isomorphic to W = R x R-l with the product 31). One verifies that the unit element of W is given in the natural basis by e = 1, . . . , 0). If a is an automorphism of W, then a(e = e and a(e' = e, where e = Rn-l is the orthogonal of e with respect the bilinear form 2.4). So, we obtain that a can be written as: a(A, u = (A, g (u)); g G 0 (n - ), (4.1) and conclude that each class of ivolutive automorphism of W is given by the diagonal matrix: IPq IP 0 ; p + q = n, q = 1,...,n - (4.2) 0 -1 ) 4.2. Real forms of rank more than two. In this case, the real Euclidean forms are isomorphic to: W = Herm(r, A); A = , C, H, 0, where r 3 if A = . If a is an involutive automorphism of W, the eigen spaces corresponding to a will be denoted by: W = x E W a(x = x, and W = x E W a(x = xl Theorem 41. If a is an involution of W = Herm(r, A) and if W+ is not simple, then a is conjugated to the automorphism O(X = IpqXIpq; p + q = r, where Ipq i given by equation 42) Proof. If a is an involution of W and if W+ is not simple, from lemma 3 of [8] and lemma 23 of 91 there exists an idempotent c in W, such that W = W(c' 1 + W(c' 0); W = W(c' 1 ),and a = P(w = 2L(W)2 - L(w2), 2 where w = - e, e being the unit element of W. Let fcl,..., c} be a Jordan frame of W such that c = cl + ... + cp, then u = cl + ... + cp - cp+1 - . - c,. If we consider the diagonal matrix ei whose diagonal elements are zero except in position i where we have 1, then el,..., eJ is a Jordan frame of W. Since Jordan frames of Euclidean Jordan algebras are conjugate, the element Ipq = el -- + ep - ep+l - . - e, is conjugated to w by an automorphism of W. Consequently, a is conjugated to the involution defined by: ,6(X = IpqXlpq- 10 0 Theorem 42. If a is an involution of W = Herm(rA) and if W+ is simple and of rank r, then a is conjugated to an involution acting on the matrix's coefficients of W by an involution of the Hurwitz algebra A Proof. If W+ and W axe of rank r, we can consider teir Peirce decomposition in the same Jordan frame Ici, ... , cr , we obtain r r = i d = 4(c + cj), i j; eeik = eik, i, j, k all different, so, we can define on (W+)ij and Wij the following product which is independent of k: u x v = eikU) (ekjv), k :i, j, equipped with this product, W+)ij and Wij axe Euclidean Hurwitz algebras. Since the element eij is in (W+)ij, the restriction ao of a to Wij is an involution of the Hurwitz algebra Wij, x), and the invariant subalgebra of ao is (W+)ij. From proposition V.1.5 of [11], we know that Wij is isomorphic to A = C, H, and 0, and from 19] the-re exists up to isomorphism a unique involutive automorphism of A. We conclude that a is conjugated with an involution acting on the matrix's coefficients of Herm(rA) : a (x = (ao xij)), Vx E W 0 After the involutions introduced in theorems 41 and 42, we remaxk that there is only one case not yet described: W is simple and of rank ro : r. In that case, we know from theorem 1.6.1 of 2] that ro = 2 . By the classification of Euclidean algebras and the fact that W is a Jordan algebra of matrix elements, we find that W is isomorphic to Herm(ro, AO). We axe going to precise the Hurwitz algebra AO and the corresponding involution. Lemma 43. Let a be an involution of W = Herm(r, A) such that W is isomorphic to Herm(ro, AO) with ro = 2, then AO = A- 1). Proof. By absurdity, we suppose that AO C A. In this case, the subalgebra W+ can be imbedded in the next subalgebra of W W0=1 Z 0 Z E Herm(ro, A) 0 0 Or, evidently Wo contains primitives idempotents of W. This is impossible because Wo and W+ are two isomorphic subalgebras of W and W+ doesn't contain any primitive idempotent of W (see theorem 16.1 of 2]). So, we have a strict inclusion A C AO Then, if we use the inclusion relation 2.21) and the fact that W+ is maximal in V w find that Ao = A-1). 0 11 Theorem 44. Let a be an involution of W = Herm(r, A) such that W Z's isomorphic to Herm(ro, Ao), then the involution a is conjugated to the involution defined by O(Z = JJ-1 where J 0 I10 ro = r -110 0 ) 2 Proof. We consider the involution O(Z) J2J-1 on the Jordan algebra W = Herm(2ro, A), and remark that Herm(roAo) JZ G W :8(Z = J2J-11. By corollary 3.10), we conclude that ce and axe conjugate. 4.3. Classification. Our conclusion in this paper is that the classification of real forms of simple complex Jordan given in 13] can be obtained by considering only the simplicity and the rank of the subalgebra W : Type 1. Obtained if W+ is not simple. If r > 3 W = Herm(rA), with A = RCH and 0. By theorem 41, the corresponding involutions are defined by a = IpqXIpq; p + q We remark that W = W(c, 1) + W(c, 0) with c = el ... + ep. The corresponding modifications V,, are isomorphic to the space W with the Jordan product XY = 1(XIpqY + YlpqX), 2 These algebras are alled mutations of W [8) and will be denoted by Herm(p, q; A). If r = 2 W = R x Rn-1, these involutions are defined by a(A, x = k Ipq (X)); with q = n - 2. In the natural basis lei, .., en} of R, we can choose a Jordan frame le = 2 (el + ej), C2 i (el - ej I of W such that W = W (el, 1) + W (l, 0) --- Re, RC2 Type II. Obtained if W+ is simple and of rank ro = r. If r > 3 by theorem 42 these involutions axe given by a(x = (ao(xj)), where ao is an involution of A. Nontrivial involutions are defined for A = , H and . If we pose A+ = u E A: ao(u = ul, A = u E A: ao(u = -u}, and Al = A-F + iA-, then the modifications of Type II are isomorphic Herm(r, A$), with the Jordan product 2.22). Let us notice that as Cayley-Dickson extension, the algebras 0 = R(l ), H` = (l), and V = H(1), and IR,C, H and axe the only real Hurwitz algebras up to isomorphism 19]. One can prove 13] that the Jordan algebra Herm(r, 0) is isomorphic to M(r, R) and Herm(r, H`) is isomorphic 12 to the Jordan algebra of skew-symmetric matrix Skew(2r, R) with the product XY 1 (XJ + YjX)1 2 where J is given by 2.18). If r = 2 these involutions axe defined by a(A, X) = (A, Ipq (X)); with < n - 3. Type III. Obtained if W+ is simple and of rank ro different of r. if r > 3 by theorem 43 W is isomorphic to Herm(ro, mathbbAO) with ro 2 and AO A(-1). Consequently the modifications of Type III are nly defined if the rank r is even. The condition AO = A- 1) implies that these modifications axe also not defined if AO = . Also they axe not defined if Ao because exceptional Jordan algebras can't be realized as subalgebras of special Jordan algebras (see 12], theorem 11, p. 389). One can also verify that the application ci(x = JtJ-1, defines an automorphism of Jordan algebras only if A = R or C. For theses cases, 'by simple calculation, we obtain that V. = Sym(2roc nM (ro, KI; if A = R = M(ro, M; if A= C. If r = 2 this involution is given by (,\, x) -x). We remaxk that in that case, W Re and is of rank 1. The following table summarizes the results of the classification. W Type I Type 1:1 Type III(r = 2ro) Sym(r, R) W+- Sym(p, R)e Sym(p, R) X Herm(ro, C) W- M(p x q,R)) X = SYM(ro C) V. SYM(P, q; R) X Sym(2ro, c n M(ro, Herm Fr,_i7__W+_ Herm(pU)OHerm(pC) By _M7r_1RT__ -HerMTr_0_1_W W- -- M( X , Q) iskew(i,,R) iHerm(ro, W. f---Herm(p, q; C) M(r, R) M(ro, H mFr, _H7 W+ Herm(p, 1) EJ)Herm(r, H) r, C) X W- M(p x q,R)) L skew(r, C) X W. Herm(p, ;1) Herm(r, r) x Hem(3, 0) W+ Herm(p, 0) E)Herm(p, 0) Herm(3, M x W- M(P Xq1o)) -_M(6 x 2 C) X V. Herm(p, 0) EDHerm(p, 0) Herm(3, 1r) X W+ Rci EDRC2 e RP 9 Re W- _ R' -2 ce R9 2 R-1 V. -_R2,. -2 = RP q a RI Let us finish this section by some definitions and remarks. If ro is the length of a Jordan frame of V,,, then ,, is said to be split if r = ro. In the other case, ,, is not split. We can see that Types I and II axe split while Types III and IV axe not split. The conformal group Conf (V) of a Jordan algebra V, is te group of rational transformations on V generated by the translations, the stucture group Str(V) and the inversion map j(x = x- Let us consider a closed subgroup of Conf (V,,,) defined bY G 1 E Conf (V.) (-a) g o (-a) = g. If g is the Lie algebra of G and if fcl,..., c o} is a Jordan frame of ,,, the subspace a defined by ro a ti L (ci), t E RI 13 is a cartan subalgebra of g. Using the root's table in [11] p.212, the root system of the pair (g a) can easily be determined. These root systems are in connection with the classification of simple real Jordan algebras by the following table which can be found in [10] and 20]. V. I Type I Type 11 Type III TypeIV A(g, a) I A,-, D, C,,, C,,, 5. SYMMETRIC SPACES ASSOCIATED WITH REAL SIMPLE JORDAN ALGEBRAS We present in the first part three hermitian symmetric spaces Ml, M2 and M3 associated with an Euclidean Jordan algebra as real forms a complex symmetric space M. In the second paxt, we give the equivalent of Ml, M2 and M3 when the Euclidean Jordan algebra is replaced by a Jordan algebra of Type I, II, III or IV. Let us announce that complete description of these spaces can be found in 2],[3],[4),[5],[6), [10) and 20). 5.1. Real forms of ermitian symmetric spaces. We begin with the Koecher-Vinberg the- orem associating the symmetric cone with the Euclidean Jordan algebra W and define the tube T = W M. It is well known [11] that To is a hermitian symmetric space isomorphic to GIK, where G is the connected component of the conformal group Conf (W) and K is the maximal compact subgroup of G. Two other hermitian symmetric spaces can be defined in the complexified symmetric space GCIK'C. The first one is the compact hermitian space UIK, where U is the compact real form of G-. This last space is isomorphic to the conformal compactifica- tion G(-'IPC of WC, here PC is the maximal parabolic subgroup of G. The second hermitian symmetric space is the ordered Makaxevic space GIH, where H is the stabilizer of ie in G, e being the unit element of W. Since G and U are real forms of GC, while K and H axe real forms of K'C, we remark that the symmetric spaces GIK, UIK and GIH are real forms of the same complex symmetric space GIKC. According to the classification of simple complex Jordan algebras, we obtain the following table of real forms of the hermitian symmetric space GIK' : WL M = GIKU Ml GIK M = UIK Ms = GIH Sy-T11-C) Sp(r, b) Sp(r, R) SP(r) --- TpT-, RT-- GL(rQ U(r) U(r) GL rR) M7r-C) I SL(2rC) SU(r, r) SU(2r) SU (r, r) S(GL(rC) x G(?,C)) S(U(r) x S(U(r) x U(r)) GL(r,!Q Skew(2r, C) SO(4r, C) so- ( SO(4r) SU'(4r) GL(2r, Q SU(2r) x SO(2) SU(2r) x SO(2) SU'(2r) x R C, SO(2 + n, C) SO 2,n SO(2-:") SO 2,n SO(nQ x C SO(n) x T SO(n) x T SO(1, n - ) x R Hem(3, u) E7 (C) E-r(-25) E7 E7(-25) E6(c) X C* E6 x SO(2) E6 x SO(2) E6(-26) SO(2) 5.2. Symmetric spaces associated to modifications of Euclidean Jordan algebra. We give the equivalent of the spaces Ml, M2 and M3 when W is replaced by another real simple Jordan algebra V not necessarily Euclidean. The construction of the spaces M, and M3 can be found in [10]. Let us give it briefly. One can consider a simple real Jordan algebra V with an involution a and the decomposition V = V+ V- The pair (V, a) defines a Euclidean Jordan algebra W dual of V by W = V+ + i-. Let be the symmetric cone associated to W. If 14 Q = Q n V, then the subset of V defined by + K is riemannian symmetric tube. By Cayley transformation, we give this in his bounded realization as Ml = GIK where G = g E Conf (V) : ja o g o ja = glo, IC = (Str(v n ov))0. The ordered Makarevic space M3 is defined in the Euclidean Jordan algebra W dual of (V, a). There also exists an involution of W such that V = V+ + iW-. The space M3 is defined by the precedent group G acting on W: G = g E Conf (W) : (-O o g o -O = glo. Then, the ordered Makaxevic space is defined by M = G'.e = GIH, where H is the stabilizer in G of the identity e of W Below is the list of riemannian Makaxevic spaces and ordered Makarevic spaces associated to real simple Jordan algebras. TypeIV Type I Type II Type III ( = 2ro Sym(r, Q -gy--Fr, RT- X Sym (2ro, R n M (ro, H) G Sp(r, R) SL(rR) x R X Sp(ro, Q K U(r) SO(r) X Sp(ro) H SL(rR) x R so X Sp ro, R) V M(r, C) He M(r, RT- M(ro, H) G SU(r, r) SL(rC) R SO(r, r) Sp(ro, ro) K S(U(r) x U(r)) Sp(r) SO(r) x SO(r) SP(ro) x Sp(roR) H SL(r, C) x R SU(p, q) SP(r, C) Herm(r, 1P S% r, R) Skew(2r, ) 3k 2 X * SO* (4r) SL(r, H) x R SO(2r, Q X K U(2r) SP(r) SO(2r) X H SL (r I) XR+ Sp(p, q)) SO'(2r) X V Hem(3, 0') Hem(3, 0) Herm(3, 0) X G E7(-25) EG(-26) x R SL 4, M X K E6 x SO(2) F4 Sp(4) X H EG(-26) X R FV-20) Sp(3, 1) X c X C-1 Rz,-2 Rpq Rn SOO 2, n) SOo(1, n - ) x R SOO (1, p) x SO (1, q) SO(1, n) SO(2) x SO(n) SO(n - ) SO(p) SO(q) SO(n) H SO(1, n - ) x R SO(i - - 2 -- 1. p - ) xSO(1, q - ) SO(1, n - ) The construction of the symmetric space M2 is given in 20]. It is the conformal compacti- fication conf (V)I.P of the Jordan algebra V, P is the opposite conjugate of the maximal par- abolic subgroup P. If U is the maximal compact subgroup of conf (V), it is easy to see that conf (V)IP -- UIK, where K = Str(V) n (V))O. In the following table is the list of these spaces according to the classification of simple real Jordan algebras. TypeIV Type I Typen Type III Sym(r, C -gy--Fr, R) X Sym(2ro, R n M(ro, M U Sp(r) SU(r) X U(2ro) K U(r) f ON X SA O) * M(r, C) Herm(r, U) M(r, R) M(r, M * SU(2r) S(U(r) x U(r) SO(2r) Sp(2r) K S(U(r) x U(r)) Sp(m) SO(r) x SO(r) - SP(1) x Sp(r) V Skew(2r, ) Herm(r, Skew(2r, R) X U SO(4r.) SU(2ro) SO(2ro) x S0(2ro) X K IU(2ro) Sp(ro) SO(2m) X V Hem(3, 0') Hem(3, 0) Herm(3, 0.) X U E7 Ee x SO(2) SU(8) X K E6 x SO(2) F4 Sp(4) X C X C-' R x R-1 RP.q R' U SO(n 2 SO(n) x SO(2) SO(p + 1) x SO(q + 1) SO(n + 1) SO(n) x SO(2 I SO(n - ) SO(p) x SO(q) _ SO(n) 15 Acknowledgments. The author would like to thank Professor Jacques Faraut for many useful discussions and the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy, for financial support. 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