CC BY
49
MSC 32M15, 22E30
Representations of clans and homogeneous cones
© H. Ishi Nagoya University, Nagoya, Japan
We present a canonical way to construct an injective representation of a given clan (compact normal left symmetric algebra), which gives rise to the realization of a homogeneous cone in the space of real symmetric matrices by Rothaus [5] and Xu [8]
Keywords: homogeneous cones, clans, representations, real symmetric matrices
§ 1. Introduction
Since Vinberg [7] established the correspondence between homogeneous cones and clans (compact normal left symmetric algebra) with unit elements, the clans are convenient tools for the study of geometry and analysis on the homogeneous cones. The correspondence is in a sense similar to the one between Lie groups and Lie algebras, and also the one between symmetric cones and Euclidean Jordan algebras. In this article, we shall consider representations of clans and their relations to representations of homogeneous cones in the sense of Rothaus [5], After some preliminaries in § 2, we see in § 3 that a representation of a clan is at the same time a representation of a homogeneous eone(Proposition 1), while every representation of a homogeneous cone is obtained as a composition of a representation of the associated clan with some linear transformations (Theorem 2), In § 4, we construct representations R1,..., Rr of a given clan V in a canonical wav (Theorem 3), These Rk already appeared in [4, Section 4], and we refer some computational arguments about them to [4], Taking the direct sum of Ri,...,Rr, we have an injective representation R of the cl an V (Theorem 4), Then R gives a linear imbedding of the corresponding homogeneous cone into the space of real symmetric matrices, which coincides with the results by Rothaus [5] and Xu [8] eventually.
§ 2. Correspondence between clans and homogeneous cones
Let V be a finite dimensional real vector space, and Q C V an open convex cone containing no line. We assume that Q is homogeneous, that is, the group GL(Q) := {g e GL(V); gQ = Q} acts on Q transitively. By [7, Chapter 1, Theorem 1], there exists a connected split solvable Lie subgroup H C GL(Q) which acts on Q
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simply transitively. Such H is unique up to conjugacy in GL(Q), Let h C End(V) be the Lie algebra of H, and fa a point E in Q, Then we have the linear isomorphism h 9 L L • E e V obtained by differentiating the orbit map H 9 h h • E e Q. Thus, for an element x e V, there exists a unique Lx e h for whieh Lx • E = x. We define a bilinear multiplication A on V by xAy := Lx • y e V, x,y e V, Then the algebra (V, A) is a clan (compact normal left symmetric algebra) with a unit E
(Cl) Putting [xAyAz] := xA(yAz) — (xAy)Az for x,y, z e V, one has [xAyAz] = [yAxAz] (left symmetry).
(C2) The bilinear form (x|y) := tr LxAy (x,y e V) defines a positive definite inner V
(C3) For each x e V, the linear operator Lx on V has only real eigenvalues (normality).
In terms of the clan structure (V, A), the cone Q is described as Q = {(exp Lx) • E; x e V}, Actually, if we start from any clan (V, A) with a unit element E e V, we
V
of homogeneous cones and the class of clans with unit elements is one-to-one up to isomorphisms ([7, Chapter 2, Theorem 2]),
Let us give an example of a homogeneous cone. We denote by Sym(m, R) the space of real symmetric matrices of size m, and by S+ the subset of Sym(m, R) consisting of positive definite elements. Then S+ is a homogeneous cone, on which the group GL(m, R) acts transitively by g • x := gx*g, x e S+, g e GL(m,R)), Let Hm C GL(m, R) be the group of lower triangular matrices with positive diagonals. Then the Lie algebra hm of Hm is the vector space of lower triangular matrices. For x e Sym(m, R), we define x e hm by
i° (i<j),
(x)ij : < xii/2 (i j),
[xij (i>j),
and set x := tx. choose the unit matrix Im as element E, Then the clan structure on Sym(m, R) associated to the cone S+ is given by
xAy = xy + yx, x,y e Sym(r,R).
( V, A )
V to Sym(m, R) with so me m, that is, a linear map 0 : V ^ Sym(m, R) with the property
0(xAy) = 0(y) + 0(y) 0(x) (x, y e V). (2.1)
We require also 0(E) = Im if (V, A) has a unit element E, which we always assume unless otherwise stated in what follows.
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3. Representations of homogeneous cones
We recall the notion of representation of a homogeneous cone introduced by QV
0 : V ^ Sym(m, R) is said to be a representation of the homogeneous cone Q if the following two conditions are satisfied:
(Rl) 0(Q) C sm;
(R2) the group G^(Q) consisting of g e GL(Q) for which there exist g e GL(m, R), such that 0(g • y) = g 0(y) *g for all y e V, acts on Q transitively.
Proposition 1 Let Q C V be a homogeneous cone, and (V, A) the associated clan. If a linear 'map 0 : V ^ Sym(m, R) is a representation of the clan (V, A), then 0 is a representation of the homogeneous cone Q.
Proof. We see from the relation (1) that
0((expLx) • y) = (exp 0(x)) 0(y) (^exp0(x)) , xy G V (3.2)
Since h = {Lx; x G V} is split solvable, exp : h ^ H is a diifeomorphism. Thus, for any h G H, there exists x G V for which h = exp Lx. Putting h := exp 0(x) G Hm,
we have 0(h • y) = h0(y) *h (y G V), so that h G G^, Thus G^ contains H, which acts on Q transitively. On the other hand, we have 0(h • E) = h *h G S+. Therefore 0(Q) C Sm and Proposition 1 is verified. □
Proposition 1 means that a representation of clan is automatically a representation of homogeneous cone. On the contrary, a representation of homogeneous cone is not necessarily a representation of clan in general. All
Q
Q
and S+.
Theorem 2 Let 0 : V ^ Sym(m, R) be a representation of a homogeneous cone Q. Then there exist elements go G GL(Q), Ao G GL(m, R) and a representation
0o : V ^ Sym(m, R) of the clan (V, A) for which
0(x) = Ao 0o(go • x) Ao, x G V. (3.3)
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Proof. Let G$(Q, Rm) be the group given by
G$(Q, Rm) := {(g,g) e GL(Q) x GL(m, R); 0(g • y) = g0(y) *g for all y e V} .
By definition, we have a surjeetive homomorphism n : G$(Q, Rm) — G$(Q) mapping (g,g) to g. Let G be the identity eomponent of G$(Q,Rm), Since G is the identity component of an algebraic group, we have a decomposition G = HK with HnK = {e}, where H is a maximal connected split solvable subgroup and K is a maximal compact subgroup [6], Without loss of generality, we can assume that Ker n C K, Then n : H —— n(H) =: H$ C G$ is an isomorphism. In other words, we can define a representation a : H$ — GL(m, R) in such a wav that (h,a(h)) e H for h e H$. Since H$ is split solvable, the representation a is simultaneously triangularizable, that is, there exists A1 e GL(m, R) for which A1a(h)A-1 e Hm, h e H$. On the other hand, the groups H^d H are conjugate in GL(Q), so that we can take g1 e GL(Q) for which H$ = g1Hg-^, Noting that A1 0(g1 • E) tA1 e we take A2 e Hm for which A1 0(g1 • E) tA1 = A2*A2. Setting A3 := A-1A1 e GL(m,R), we define ao(h) := A3 a(g1hg-1) A-1 e GL(m, R), h e ^^d 0o(x) := A3 0(g1 •x) *A3 e Sym(m,R), x e V. Then we have ao(h) e Hm, h e ^^d 0o(E) = /m, For h e H
y e V
0o(h • y) = A3 0((g1hg—1) • (g1 • y)) *A3
= A3 a(g1hgi_1) 0(g1 • y) ta(g1hg-1) A3
= ao(h) 0o (y) *ao(h).
Differentiating this relation, we obtain
0o(xAy) = o(Lx)0o(y) + 0o(y)*ao(Lx), x,y e V,
where ao is the differential representation of ao. In particular, since 0o(E) = /m, we have
0o(x) = a o(Lx) + *a o(Lx).
Noting that ao(Lx) e hm we obtain ao(Lx) = 0o(x). Therefore we have
0o(xAy) = 0o(x) 0o(y) + 0o (y) 0o(x),
which means that 0o is a representation of the clan (V, A). Putting Ao := A-1 and go := g-1, we have (3). □
4. Canonical representation of a clan
Let V = Vkj be the normal decomposition of a clan V with respect to
primitive idempotents Ei,..., Er of V. Namely, we have E = E1 + ... + Er and
Vkj = \ x G V; EiAx = -(ôik + ôj)x, xAEi = ôjx for i = 1,..., r }>. (4,4)
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Then we have Vkk = REk, Moreover, the following multiplication relations hold:
Vifc AVkj C Vj-, if k = i, j, then V;kAVj = 0, (4,5)
V;k AVmk C V;m, Vm1 according to / > m or m > /.
For k = 1,..., r, we set Mk := Vk1 © • • • © Vk,k-1 © Vkk, By (5) we have
MkAM; C Mi, MiAMfc C Mi
for 1 ^ k < I ^ r, Therefore, if we set Ik := Mk © ... © Mr, k = 1,..., r, we have a two-sided ideal sequence
V = I1 DI2 D ... D Ir D Ir+1 := {0}.
V
defined in the axiom (C2), Putting nkj := dim Vkj, 1 ^ j < k ^ r, we take an orthonormal basis f7}, 1 ^ a ^ nkj-, of the subspace Vkj, Put mk := dimMk, k = 1,..., r. Noting that mk = nk1 + ... + nk,k-1 + 1, we define a basis {ePk)},
1 ^ p ^ mk, of Mk by
f , P = a + Ei<j 1 ^ a ^ nkj,
|V2|Efc|-1 Efc , p = mfc.
We remark that the basis {ePk)} is not orthonormal because ||e||mk = -\/2. In view of the natural isomorphism Mk 9 v — V := v + Ik+1 e Ik/Ik+1, we write Mk for the quotient space Ik/Ik+1, For x e V, let Rx be the right-multiplication operator on the clan V by x. We write Rk(x) (resp. Lk(x)) for the matrix of the linear operator on Mk induced by Rx (resp. Lx) with respect to the basis {ePk)}, 1 ^ p ^ mk, of Mk. It is shown in [4, Lemma 4,2] that Rk(x) is symmetric for x e V,
Theorem 3 The linear 'map Rk : V — Sym(mk, R) is a representation of the elan ( V, A ) k = 1 , . . . , r
Proof. We define a linear form Ek on V by
(^ ^ xiEi ^ ^ xji , Ek) : xk, xi e ^-, xji e V
i=1 i<j
By [4, Lemma 4,2] we have
Rk (x) = Lk (x) + *Lk (x) - (x , Ek) /mfc.
On the other hand, we see from (5) that Lk(x) is a lower triangular matrix for x e V, Thus we obtain
Rk(x) = Lk(x) - 1 (x , Ek) /mfc.
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On the other hand, we have by [4, (4,14)]
Rk (xAy) = Lfc(x)Rk(y) + (y)iLfc(x) - (x , Ek) Rfc(y).
Therefore we have
Rk(xAy) = Rk(x)Rk(y) + Rk(y)Rfc(x), x, y G V. Clearly Rk (E) = /TOfc, which completes the proof.
□
Noting that mi + ... + mr = dim V(=: n), we define a linear map R : V ^ Sym(n, R) by
R(x)
Ri (x)
V
R2(x)
\
Rr(x) J
x V.
Namely, R is the direct sum of the representations R1,..., Rr, so that R is also a
( V, A )
Theorem 4 Tfte representation R : V — Sym(n, R) of the cl an V is injective.
Proof. Assume R(x) = 0 with x = ^ 1<i<j<r x7j, x^j e Vjj. We see from [4, (4,11)] that Rk (x) = 0 yields xki = 0 for i = 1,..., k. Therefore x = 0, whence Theorem 4 follows, □
Theorem 4 together with Proposition 1 implies that any homogeneous cone Q is linearly imbedded into S+ with n = dim Q This description of the cone Q is already given by [5] and [8] (see also [3] and [4]),
Rk k = 1 , . . . , r V
Sym(r, R), Then the normal decomposition coincides with the natural decomposition by the entries. The inner product is given by (x|y) = dtr xy with d = (r + 1)/2. The ideal Ik, k = 1,..., r, equals the set {x e Sym(r, R); xjj = 0 for all i, j < k}. Noting that dimMk = k we take the basis of the module Mk as ePk) := d-1/2(Ekp + Epk),
1 < p < k, and ekk) := -\/2d-1/2Ek^, where Ejj is the (i, j)-matrix unit. Then we have
Rk(x) = (xjj)1<i,j<k e Sym(k,R), x = (xjj) e Sym(r,R).
It is easy to see directly that Rk is a representation of the clan Sym(r, R) in this case.
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References
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2, H, Ishi, Basic relative invariants associated to homogeneous cones and applications, J, Lie Theory, 2001, vol. 11, 155-171,
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5. O. S. Eothaus. The construction of homogeneous convex cones, Ann. Math., 1966, vol. 83, 358-376. Correction: ibid, 1968, vol. 87, 399.
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