Matrix realization of a homogeneous Hessian domain

07/11/2017
Auteurs : Hideyuki Ishi
Publication GSI2017
OAI : oai:www.see.asso.fr:17410:22644
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Extending previous results about matrix realization of a homogeneous cone by the author, we realize any homogeneous Hessian domain as a set of symmetric matrices with a speci c block decomposition.
A global potential function as well as a transitive affine group action preserving the Hessian structure is also expressed in terms of the matrix realization.

Matrix realization of a homogeneous Hessian domain

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application/pdf Matrix realization of a homogeneous Hessian domain Hideyuki Ishi
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Matrix realization of a homogeneous Hessian domain

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A global potential function as well as a transitive affine group action preserving the Hessian structure is also expressed in terms of the matrix realization.
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Matrix realization of a homogeneous Hessian domain Hideyuki Ishi12 1 Graduate School of Mathematics, Nagoya University, Furo-cho, Nagoya 464-8602, Japan hideyuki@math.nagoya-u.ac.jp 2 JST, PRESTO, 4-1-8, Honcho, Kawaguchi 332-0012, Japan Abstract. Extending previous results about matrix realization of a ho- mogeneous cone by the author, we realize any homogeneous Hessian do- main as a set of symmetric matrices with a specific block decomposition. A global potential function as well as a transitive affine group action preserving the Hessian structure is also expressed in terms of the matrix realization. Keywords: homogeneous Hessian domain, left-symmetric algebra, nor- mal Hessian algebra 1 Introduction A Riemannian manifold with a flat connection is called a Hessian manifold if the metric is locally expressed as the Hessian matrix of a smooth function with respect to affine coordinates. Such Hessian structure is very important in In- formation Geometry ([9]). A Hessian manifold is said to be homogeneous if its automorphism group acts on the manifold transitively, where the automorphism group is defined as the set of all diffeomorphisms preserving both the flat connec- tion and the metric. Shima [8] established a basic theory of homogeneous Hessian manifolds. He showed that the universal covering space of a homogeneous Hes- sian manifold is a convex domain equipped with a homogeneous Hessian metric. Furthermore, the convex domain is shown to be the direct product of an affine homogeneous convex domain (containing no straight line) and a vector space. A normal Hessian algebra, which is a left-symmetric algebra with a compatible in- ner product, plays an important role in Shima’s theory as a convenient algebraic tool. In this paper, combining the theory of normal Hessian algebras with the matrix realization method developed by the author ([3], [4], [5]), we realize any homogeneous Hessian domain as a set of symmetric matrices with a specific block decomposition (Theorem 4). A global potential function as well as a transitive affine group action preserving the Hessian structure is also expressed in terms of the matrix realization (see (8) and (9) respectively). The details with complete proofs of statements will be published elsewhere. This research was supported by JST PRESTO and JSPS KAKENHI Grant Number 16K05174. 2 Normal Hessian algebras Let V be a finite dimensional real vector space, D a domain in V , and g a Hessian metric on D. By definition, g is locally expressed as the Hessian matrix of a convex function. If there exists a smooth function φ : D → R whose Hessian matrix equals g at every point in D, we call φ a global potential of the Hessian metric g. The automorphism group Aut(D, g) of the Hessian domain (D, g) is defined by Aut(D, g) := { α ∈ Aff(V ) ; α(D) = D, α∗ g = g }. A Hessian domain (D, g) is said to be homogeneous if Aut(D, g) acts transitively on D. Let (D, g) be a homogeneous Hessian domain in what follows. Shima [8] showed that there exists a triangular solvable Lie subgroup H ⊂ Aut(D, g) acting simply transitively on D. Here the word triangular means that there exists a basis of V such that the linear part of every affine transformation h ∈ H is expressed as an upper triangular matrix with respect to the basis. Let us take and fix a point p0 ∈ D. Then we have a diffeomorphism H ∋ h 7→ h · p0 ∈ D. Differentiating the diffeomorphism, we have a linear isomorphism h ∋ X 7→ X(p0) ∈ V ≡ Tp0 D, where h is the Lie algebra of H, and h is identified with a set of affine vector fields. For v ∈ V , we denote by Xv a unique vector field belonging to h such that Xv(p0) = v, and by Lv the linear part of Xv. Then we have Xv(p) = Lv(p − p0) + v. Now we introduce a bilinear product △ on V by x△y := Lx y ∈ V (x, y ∈ V ). The algebra (V, △) is not commutative nor associative in general. Instead, we have the following equality [Lx, Ly] = Lx△y−y△x (x, y ∈ V ), which is equivalent to x△(y△z) − (x△y)△z = y△(x△z) − (y△x)△z (x, y, z ∈ V ). (1) The algebra satisfying the equality above is called a left-symmetric algebra (Koszul-Vinberg algebra [1] or pre-Lie algebra [7]). Since h is triangular, all eigenvalues of the left-multiplication operator Lx are real for every x ∈ V . A left-symmetric algebra is said to be normal if this eigenvalue condition of Lx is satisfied. By means of the identification V ≡ Tp0 D, we transfer an inner product (·|·) on V from the metric g on Tp0 D. Then we have (x|y△z) − (x△y|z) = (y|x△z) − (y△x|z) (x, y, z ∈ D). (2) Shima named a normal left-symmetric algebra with an inner product satisfying (2) a normal Hessian algebra, and we call the inner product a Hessian inner product on the algebra. We have seen that a homogeneous Hessian domain gives rise to a normal Hessian algebra. The converse is also true. When a normal Hessian algebra (V, △) is given, we take any point p0 ∈ V , and consider the set h of affine vector fields Xv (v ∈ V ) defined by Xv(p) := Lv(p − p0) + v. Thanks to the left-symmetry, h forms a Lie algebra. Let H be the Lie subgroup of Aff(V ) whose Lie algebra is h. It is shown that the H-orbit D = H · p0 through p0 is a convex domain in V . Let g be the H-invariant metric on D which coincides with the Hessian inner product on V at the tangent space Tp0 D ≡ V . Then (D, g) is a homogeneous Hessian domain. In this way, we have a one-to- one correspondence between homogeneous Hessian domains and normal Hessian algebras up to isomorphisms. A left-symmetric algebra is said to be compact if there exists a linear form ξ ∈ V ∗ such that (x|y)ξ := ξ(x△y) gives a positive inner product on V . In view of (1), we see that the inner product (·|·)ξ satisfies (2). Namely, a compact normal left-symmetric algebra (clan) is a normal Hessian algebra. The clans are studied by Koszul [6] and Vinberg [10]. Vinberg showed that clans are in one-to- one correspondence with affine homogeneous convex domains, and that a clan has a unit element if and only if the corresponding domain is a homogeneous cone. Using the latter fact, the author realized all homogeneous cones as well as clans with a unit element as the set of symmetric matrices with certain block decompositions in [5] (similar results are obtained by many researchers, e. g. [2], [11], [12]). We recall the results of [5] briefly in the next section. 3 Matrix realization of a clan with a unit element Let Vn be the vector space of real symmetric matrices of size n. We define a bilinear product △ on Vn by x△y := x ˇ y + y t (x ˇ ) (x, y ∈ Vn), where x ˇ is a lower triangular matrix defined by (x ˇ )ij :=      0 (i < j), xii/2 (i = j), xij (i > j). Then (Vn, △) is a clan with a unit element En. Let n = n1 + n2 + · · · + nr be a partition, and let Vlk ⊂ Mat(nl, nk; R) (1 ≤ k < l ≤ r) be vector spaces satisfying (V1) A ∈ Vlk, B ∈ Vki ⇒ AB ∈ Vli for 1 ≤ i < k < l ≤ r, (V2) A ∈ Vli, B ∈ Vki ⇒ A t B ∈ Vlk for 1 ≤ i < k < l ≤ r, (V3) A ∈ Vlk ⇒ A t A ∈ REnl for 1 ≤ k < l ≤ r. Let ZV be the subspace of Vn consisting of symmetric matrices x of the form x =      X11 t X21 · · · t Xr1 X21 X22 t Xr2 . . . ... Xr1 Xr2 Xrr      ( Xll = xllEnl , xll ∈ R (l = 1, . . . , r) Xlk ∈ Vlk (1 ≤ k < l ≤ r) ) . (3) Thanks to (V1)–(V3), ZV is a subalgebra of the clan (Vn, △). Theorem 1 ([5]) Every clan with a unit element is isomorphic to the algebra (ZV , △) with appropriate {Vlk}1≤k 0 (l = 1, . . . , r) } . Then hV forms a linear Lie algebra, and HV equals the corresponding Lie group exp hV . The domain ΩV is a homogeneous cone on which HV acts simply tran- sitively by the action ρ(T)x := Txt T (x ∈ ZV , T ∈ HV ). In fact, the transfor- mation group ρ(HV ) on ΩV coincides with the integration of the vector fields Xv (v ∈ ZV ) given by Xv(p) := v△(p − En) + v (p ∈ ΩV ). By (V3), we can define an inner product (·|·)Vlk on each Vlk (1 ≤ k < l ≤ r) in such a way that Xlk t Ylk + Ylk t Xlk = 2(Xlk|Ylk)Vlk Enl (Xlk, Ylk ∈ Vlk). For s = (s1, . . . , sr) ∈ Rr >0, define a linear form ξs on ZV by ξs(x) := r ∑ k=1 skxkk (x ∈ ZV ). Then (x|y)s := ξs(x△y) = r ∑ l=1 sl(xllyll + ∑ k