clans defined by representations of euclidean jordan ... - J-Stage

Kyushu J. Math. 67 (2013), 163–202 doi:10.2206/kyushujm.67.163

CLANS DEFINED BY REPRESENTATIONS OF EUCLIDEAN JORDAN ALGEBRAS AND THE ASSOCIATED BASIC RELATIVE INVARIANTS Hideto NAKASHIMA and Takaaki NOMURA (Received 15 March 2012 and revised 4 June 2012)

Abstract. Starting with a representation ϕ of a Euclidean Jordan algebra V by selfadjoint operators on a real Euclidean vector space E, we introduce a clan structure in VE := E ⊕ V . By the adjunction of a unit element to VE , we obtain a clan VE0 with unit element. By computing the determinant of the right multiplication operators of VE0 , we get an explicit expression of the basic relative invariants of VE0 in terms of the Jordan algebra principal minors of V and the quadratic map associated with ϕ. For the dual clan of VE0 , we also obtain an explicit expression of the basic relative invariants in a parallel way.

0. Introduction Vinberg [15] established a one-to-one correspondence, up to isomorphisms, between homogeneous convex domains in Euclidean spaces and certain non-associative algebras called clans. Homogeneous open convex cones containing no entire line (homogeneous cones for short in what follows) correspond to clans with unit element. Among homogeneous cones, selfdual cones (called symmetric cones) form a nice class, and have been extensively studied from various points of view as presented in the book by Faraut and Korányi [7]. Jordan algebras serve efficiently as an algebraic tool there. In this way the ambient vector space V of a symmetric cone has two algebraic structures, clan and Jordan. At the very beginning of this work, we were interested in the interplay of these two structures. However, with the progress of the work we recognize that this interplay is just a special case (zero representation case) of clans obtained by selfadjoint representations of the Jordan algebra V . The key is that any selfadjoint Jordan algebra representation ϕ of V is automatically a representation of the clan V in the sense of Ishi [10] (see Proposition 3.3 of this paper). Then ϕ is a representation of in the sense of Rothaus [14] (see [10]), so that it is a J -morphism of in the sense of Dorfmeister [6]. Thus, our construction of a homogeneous cone from a Jordan algebra representation could be included in a more general scheme developed by Rothaus [14] or Dorfmeister [5, 6]. However, we would like to emphasize that since Jordan algebra representations are well-studied, we are able to make everything explicit. In particular, our objective is to obtain an explicit expression of the basic relative invariants in terms of the ingredients of the original Jordan algebra V and its representation ϕ. 2010 Mathematics Subject Classification: Primary 17C50; Secondary 16W10, 43A85. Keywords: basic relative invariants; clans; homogeneous cones; Jordan algebras. c 2013 Faculty of Mathematics, Kyushu University

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H. Nakashima and T. Nomura

Moreover, the homogeneous cones and their dual cones that we obtain are quite interesting in themselves. For example, the degrees of the basic relative invariants associated with the resulting dual cone are always 1, 2, . . . , r (r is the rank of the cone), whatever the representations ϕ. Now we describe the body of this paper in more detail. Let V be a simple Euclidean Jordan algebra of rank r with an inner product x | y = tr(xy) defined by the trace function tr(x) of V . We fix a Jordan frame c1 , . . . , cr of V , so that we have c1 + · · · + cr = e0 , where e0 is the unit element of V . Let be the symmetric cone of V . We consider a representation ϕ of V by selfadjoint operators on a real Euclidean vector space E with inner product · | ·E . If ϕ is a zero representation, then we are led to consider V as a clan (V , ), the product of which comes through an Iwasawa solvable subgroup of the linear automorphism group G() of . Note here that in this case G() is a reductive Lie group. Suppose next that ϕ is non-trivial. Then our task is to introduce a clan structure in the vector space VE := E ⊕ V . By using the Jordan frame, we can define the lower triangular part ϕ(x) of the selfadjoint operator ϕ(x) for x ∈ V (see (3.3)). The operators ϕ(x) actually represent a triangular action of the clan (V , ) on E. On the other hand, we have a symmetric bilinear map Q : E × E → V associated with ϕ defined by ϕ(x)ξ | ηE = Q(ξ, η) | x (x ∈ V , ξ, η ∈ E). We note that Q is -positive, that is, we have Q(ξ, ξ ) ∈ \ {0} for any 0 = ξ ∈ E. Using these materials, and recalling that the space V , originally a Jordan algebra, is now considered as a clan (V , ) defined in a canonical way as mentioned above, we introduce a product in VE by (ξ + x) (η + y) := ϕ(x)η + (Q(ξ, η) + x y)

(ξ, η ∈ E, x, y ∈ V ).

We see in Theorem 3.2 that the algebra (VE , ) is indeed a clan. Since ϕ is supposed to be non-trivial, VE does not have a unit element. The corresponding homogeneous convex domain is the real Siegel domain D(, Q) defined by the data and Q (see (3.9)). We now make an adjunction of a unit element e to VE , and get a clan VE0 := Re ⊕ VE to which corresponds a homogeneous cone 0 . Noting the fact that the unit element e0 of the Jordan algebra V is also a unit element of the clan (V , ), we put u := e − e0 . Then we also have VE0 = Ru ⊕ VE , and we will write general elements of VE0 as λu + ξ + x with λ ∈ R, ξ ∈ E and x ∈ V . The Siegel domain D(, Q) appears as the cross-section of 0 with the hyperplane u + VE . Since the irreducible factors of the determinant of the right multiplication operators are the basic relative invariants by Ishi and Nomura [11], we calculate Det R 0 (v) (v ∈ VE0 ) in Proposition 4.1 for the right multiplication operators R 0 (v) of VE0 . This is a preliminary step to obtain an explicit expression of the basic relative invariants of VE0 . To go further, we need to separate the cases according to the classification of simple Euclidean Jordan algebras (see [7], for example). Clearly the exceptional Jordan algebra Herm(3, O) does not have a non-trivial representation. Thus, we have two cases: (i) rank-two case, where V ∼ = R ⊕ W and W is a real vector space (Lorentzian type); (ii) the case Herm(r, K), where r ≥ 3 and K = R, C or H (Hermitian type). We first describe the results for the Hermitian case. Let V = Herm(r, K) and (ϕ, E) be a selfadjoint representation of the Jordan algebra V . By Clerc [3] we necessarily have

Clans defined by Jordan algebras

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E = Mat(r × p, K) with the canonical real inner product ξ | ηE = Re Tr(ξ η∗ ), and ϕ(x) is simply the left multiplication by x to ξ ∈ E. We see in Proposition 5.2 that the clan VE0 is isomorphic to a subclan X of Herm(r + p, K) defined by λIp ξ ∗ ; λ ∈ R, ξ ∈ E, x ∈ V . X := ξ x Our explicit expression of the basic relative invariants of the clan VE0 requires some care if p < r. In fact, the inequality p ≥ r corresponds exactly to the regularity condition of the representation ϕ (see Clerc [3]). Let j (x) (j = 1, . . . , r) be the left upper corner principal minors of the matrix x ∈ V . These are the basic relative invariants of V . Then we show in Theorem 5.4 that the basic relative invariants 00 (v), 01 (v), . . . , 0r (v) (v = λu + ξ + x ∈ VE0 ) are described as ⎧ 00 (v) = λ, ⎪ ⎪ ⎪ ⎪ ⎧ ⎪ 1 ∗ ⎪ ⎪ ⎪ ⎪ ⎪ j λx − ξ ξ (j = 1, . . . , min(p, r)), ⎪ ⎪ ⎪ ⎪ 2 ⎨ ⎪ ⎪ ⎪ ⎨ ⎞ (0.1) ⎛ 0 ⎪ 1 ∗ (v) = ⎪ ⎪ j λI ξ √ ⎪ ⎪ ⎜ p ⎪ ⎪ ⎪ ⎪ (p+j ) ⎜ 2 ⎟ ⎟ (j = p + 1, . . . , r; when p < r), ⎪ ⎪ ⎪ ⎪det ⎠ ⎝ 1 ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ x √ ξ 2 where for any Hermitian matrix X, det(p+j ) X stands for the determinant of the matrix formed by the left upper corner of X of size p + j , and if K = H, then the determinant is taken as the Jordan algebra determinant function. Note further that if p ≥ r we do not need the last expression in (0.1), and that if p < r the polynomials j (λx − 12 ξ ξ ∗ ) for p + 1 ≤ j ≤ r are not irreducible (cf. Corollary 5.5). To describe the results for the case of Lorentzian type, let W be an n-dimensional real vector space and B a positive-definite symmetric bilinear form on W . We put V := R ⊕ W . We know that this algebra has a canonical injection into the Clifford algebra Cl(W ) generated by W with the condition w · w = B(w, w) (see [2], for example). Any selfadjoint representation ϕ of the Jordan algebra V comes from a representation of Cl(W ) and vice versa (cf. Clerc [3]). In this case also, our expression of the basic relative invariants of the clan VE0 depends on whether the representation ϕ is regular or not. By Clerc [3, Théorème 3], ϕ is not regular if and only if ϕ is irreducible and n = 2, 3, 5, 9. Let 1 (x) and 2 (x) (x ∈ V ) be the Jordan algebra principal minors of V . In Theorem 5.8 we give an explicit expression of the basic relative invariants 00 (v), 01 (v) and 02 (v) (v = λu + ξ + x) of VE0 as follows: ⎧ ⎪ 00 (v) = λ, ⎪ ⎪ ⎪ ⎨0 (v) = (λx − 1 Q(ξ, ξ )), 1 2 1 ⎪ (x) − x , Q(ξ, ξ ) 1,n (ϕ is not regular), λ ⎪ 2 0 ⎪ ⎪ ⎩2 (v) = 1 2 (λx − 2 Q(ξ, ξ )) (otherwise), where · , · 1,n stands for the Lorentz metric. We further take a closer look at the nonregular case by realizing V as Herm(2, Kn−1 ), where K1 := R, K2 := C, K4 := H and

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K8 := O. Although we have two inequivalent irreducible representations ϕ if n = 3, 5, 9, the resulting clans VE0 , including the case n = 2, are isomorphic to Herm(3, Kn−1 ). Then the above expression for 02 (v) is equal to the determinant function of the Jordan algebra Herm(3, Kn−1 ) up to some minor adjustments like √1 ξ in (0.1). Although the isomorphism 2 V0 ∼ = Herm(3, Kn−1 ) can be detected in view of Vinberg [16, Proposition 3] by just noting E

the equidimensionality of the off-diagonals, we prefer in this paper an explicit map that gives an actual isomorphism. ∗ Our final objective is to realize the dual cone (0 ) of 0 , and to get an explicit expression of the associated basic relative invariants. By fixing an appropriate inner product ∗ in VE0 , the clan product of VE0 associated with (0 ) is given by using the transpose tL0v (v ∈ VE0 ) of the left multiplication operators L0v of VE0 as v v = tL0v v (v, v ∈ VE0 ). Then, ∗ we show in Proposition 6.1 that the cone (0 ) is described as

∗

(0 ) = {v = λu + ξ + x ∈ VE0 ; x ∈ and λ > 12 ϕ(x)−1 ξ | ξ E }. This description corresponds to what Rothaus calls in [14] the extension of by the representation ϕ. For the Hermitian case V = Herm(r, K), we prove in Theorem 6.3 that ∗ (0 ) is linearly isomorphic to the following cone : μ ∈ R, η ∈ Krp , y ∈ V , μ η∗ ∈ Herm(rp + 1, K); := Y = . (0.2) η y ⊗ Ip Y is positive-definite Here note that Krp is the space of K-column vectors of size rp, and not considered as a space of matrices. Let ∗j (y) (j = 1, . . . , r) be the principal minors from the right lower corner of the matrix y ∈ V . We denote by coy the cofactor matrix of y. Thus, coy = (Det y)y −1 if y is invertible in the case where K = R or C. For K = H, we consider coy in the Jordan algebra sense (cf. [7, Proposition II.2.4]). Then the basic relative invariants Pj (Y ) (k = 1, . . . , r, r + 1) associated with is given by (Corollary 6.4) (j = 1, . . . , r), ∗j (y) Pj (Y ) := ∗ co μ det y − η ( y ⊗ Ip )η (j = r + 1). We would like to emphasize that deg Pj = j for any j = 1, . . . , r, r + 1, so that these cones generalize the non-symmetric cone of rank three that appeared in [11, Section 3] with the basic relative invariants of degrees one, two and three. The Lorentzian case will be treated separately, and we get Theorem 6.6 which says ∗ that the basic relative invariants associated with (0 ) are given by the polynomials Pj (v) 0 (v ∈ VE ) defined by (j = 1, 2), ∗j (x) Pj (λu + ξ + x) = 1 λ det x − 2 ϕ( x )ξ | ξ E (j = 3), where ∗1 (x), ∗2 (x) are the Jordan algebra principal minors of V dual to 1 (x), 2 (x), and x → x is the restriction to V of the canonical automorphism of the Clifford algebra Cl(W ) extending the isometry w → −w of W (cf. [8], for example). Here also we emphasize that ∗ deg Pj = j for any j = 1, 2, 3, although (0 ) is not symmetric in general. Low-dimensional Lorentzian cases V = R ⊕ W with dim W = 4, 6, 7 and 8 for irreducible ϕ will be further


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observed by using explicit realizations of W as real subspaces of H or O. Then ϕ is realized as in Clerc [4, Section 6], although we will use left multiplication operators. The organization of this paper is as follows. Section 1 collects definitions and basic theorems on which this paper is grounded. In Section 2 we study a canonical clan structure (V , ) that is introduced in a simple Euclidean Jordan algebra V . We investigate the inductive structure of the right multiplication operators R(v) (v ∈ V ) in (V , ). This is our Theorem 2.7. As a result we obtain the irreducible factorization of Det R(v) in Theorem 2.9. This section gives details of the results announced in the paper [13] by the second author. Section 3 is devoted to describing the clan VE = E ⊕ V defined by a selfadjoint representation (ϕ, E) of V . In Section 4, we compute the determinant of the right multiplication operators in VE0 = Re ⊕ VE obtained from VE by the adjunction of a unit element e. In Section 5, we give an explicit expression to the basic relative invariants of VE0 . The final section, Section 6, deals with the dual clan of VE0 and we obtain an explicit expression of the basic relative invariants of this clan. The contents of this paper extend significantly the first author’s master thesis submitted to Kyushu University in February 2011. Notation For the determinant (respectively the trace) of operators and real or complex matrices we use the uppercase version Det (respectively Tr). For the determinant function and the trace function of a Jordan algebra we use det and tr as in the book [7]. Thus, when a Hermitian matrix X might not be real or complex, we write det X to avoid any misuse of the multiplicative property. 1. Preliminaries 1.1. Clans We begin this paper with the definition of clans following Vinberg [15], where it is shown that the correspondence is one-to-one up to isomorphisms between the class of homogeneous convex domains and the class of clans. Let V be a finite-dimensional real vector space with a bilinear product . We assume neither the associativity of the product nor the existence of unit element. For x ∈ V , we denote by Lx the left multiplication operator Lx y = x y (y ∈ V ). The pair (V , ) (or simply V ) is called a clan if the following three conditions are satisfied: (1) (V , ) is left-symmetric: Lx Ly − Ly Lx = Lx y−y x for all x, y ∈ V ; (2) there exists s ∈ V ∗ such that s(x y) defines an inner product in V ; (3) for each x ∈ V , the operator Lx has only real eigenvalues. Linear forms s with the property (2) are said to be admissible. Let V be a clan. By Vinberg [15, p. 369], V has a principal idempotent c by which V can be decomposed as V = V(1) ⊕ V(1/2), where V(1) := {x ∈ V ; Lc x = x},

V(1/2) := {x ∈ V ; Lc x = 12 x}.

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Denoting by Rx the right multiplication operator Rx y = y x (y ∈ V ), we also have V(1) = {x ∈ V ; Rc x = x},

V(1/2) = {x ∈ V ; Rc x = 0}.

We note here that if V has a unit element e, then c = e and evidently we have V(1) = V and V1/2 = {0}. The following multiplication rules hold: V(1) V(1) ⊂ V(1),

V(1) V(1/2) ⊂ V(1/2),

V(1/2) V(1) = {0},

V(1/2) V(1/2) ⊂ V(1) .

(1.1)

Clearly V(1) itself is a clan with unit element c. Let r be the rank of the clan V(1) and let c1 , . . . , cr be a complete set of primitive idempotents in V(1) , so that we have c1 + · · · + cr = c. Then, by [15, Proposition 8, p. 374] we have a normal decomposition (1.2) below of V(1) after relabeling c1 , . . . , cr if necessary. Let Vjj be the one-dimensional subspace Rcj , and for k > j put Vkj := {x ∈ V(1); Lci x = 12 (δij + δik )x, Rci x = δij x Then we have

V(1) =

for i = 1, . . . , r}.

Vkj ,

(1.2)

1≤j ≤k≤r

where the multiplication rules are Vj i Vlk = {0} (if i = k, l), Vj i Vki ⊂ Vjk or Vkj

Vkj Vj i ⊂ Vki ,

(according to j ≥ k or j ≤ k).

(1.3)

Moreover, the space V(1/2) is decomposed as V(1/2) =

r

Vk0

k=1

with Vk0 := {x ∈ V(1/2); Lci x = 12 δik x for i = 1, . . . , r}. 1.2. Homogeneous convex cones Let V be a finite-dimensional real vector space and an open convex cone in V containing no entire line. Let G() be the linear automorphism group of : G() := {g ∈ GL(V ); g() = }. As a closed subgroup of GL(V ), the group G() is a Lie group. We assume that is homogeneous, that is, G() acts on transitively. We know by Vinberg [15, Theorem 1, p. 359] that there exists a split solvable Lie subgroup H of G() acting on simply transitively. A function f on is said to be relatively invariant under the action of H if there exists a one-dimensional representation χ of H with which we have f (hx) = χ(h)f (x) for all h ∈ H and x ∈ .


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T HEOREM 1.1. (Ishi [9]) Let r be the rank of the cone . Then there exist irreducible relatively H -invariant polynomial functions 1 , . . . , r , by which any relatively H invariant polynomial function p(x) on V is written as p(x) = const · 1 (x)m1 · · · r (x)mr

((m1 , . . . , mr ) ∈ Zr≥0 ).

Moreover, one has = {x ∈ V ; 1 (x) > 0, . . . , r (x) > 0}. The polynomials 1 (x), . . . , r (x) are called the basic relative invariants of the cone . Now let (V , ) be the clan associated with the cone . We also call 1 (x), . . . , r (x) the basic relative invariants of the clan V . Since V x → Det Rx is a relatively H -invariant polynomial function (see the proof of [12, Lemma 2.7]), it is written as in Theorem 1.1. Moreover the following stronger fact is true. T HEOREM 1.2. (Ishi and Nomura [11]) The polynomials 1 (x), . . . , r (x) are the irreducible factors of Det Rx . 2. Jordan algebras and the associated clan structure Let V be a real vector space equipped with a bilinear product ◦. Then V is called a Jordan algebra if the following two conditions hold: x ◦ y = y ◦ x,

x 2 ◦ (x ◦ y) = x ◦ (x 2 ◦ y) (x, y ∈ V ).

We note that the product ◦ need not be associative. We often drop the symbol ◦ for simplicity when there is no risk of confusion. Our reference for Jordan algebras is the book by Faraut and Korányi [7]. A Jordan algebra V with unit element is said to be Euclidean if there exists a positive-definite symmetric bilinear form on V which is associative, that is to say, if there exists an inner product in V with respect to which the Jordan multiplication operators M(x) : y → xy are selfadjoint for all x ∈ V . A Jordan algebra V is said to be simple if V has only trivial ideals. It is known that Euclidean Jordan algebras correspond in a one-to-one way to symmetric cones up to isomorphisms. Since symmetric cones are a special kind of homogeneous open convex cones, we can introduce a clan structure canonically in Euclidean Jordan algebras as described below. Let V be a simple Euclidean Jordan algebra of rank r with unit element e0 . We equip V with the trace inner product x | y := tr(xy), where tr is the trace function on the Jordan algebra V . Let c1 , . . . , cr be a Jordan frame, so that we have c1 + · · · + cr = e0 . The Jordan frame c1 , . . . , cr yields an orthogonal decomposition† Vkj , (2.1) V = 1≤j ≤k≤r

where Vjj = Rcj (j = 1, . . . , r) and Vkj = {x ∈ V ; M(ci )x = 12 (δij + δik )x for i = 1, . . . , r}

(1 ≤ j < k ≤ r).

†Our Vkj are the Vjk in [7]. We have changed the notation in order to be consistent with the normal decomposition (1.2).

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The decomposition (2.1) is called the Peirce decomposition relative to the Jordan frame c1 , . . . , cr . Let := Int{x 2 ; x ∈ V } be the symmetric cone of the Euclidean Jordan algebra V and G() the linear automorphism group of the cone . We know that G() is reductive in this case. Let G be the identity component of G(), and g its Lie algebra. Let k be the derivation algebra Der(V ) of the Jordan algebra V and put p := {M(x); x ∈ V }. Then g = k + p is a Cartan decomposition of g with the usual Cartan involution θ X = − tX, where tX denotes the transpose of the operator X relative to the trace inner product · | · . Put a := RM(c1 ) ⊕ · · · ⊕ RM(cr ). Then a is a maximal abelian subspace of p. Let α1 , . . . , αr be the basis of a∗ dual to M(c1 ), . . . , M(cr ). We know that the positive a-roots are 1 2 (αk − αj ) (j < k) and that the corresponding root spaces nkj are described as nkj := {z 2 cj ; z ∈ Vkj },

(2.2)

where a 2 b := M(ab) + [M(a), M(b)]. Summing up all of the nkj as n := j