Complexifications of symmetric spaces and Jordan theory

Abstract

Abstract. Generalizing Hermitian and pseudo-Hermitian spaces, we define twisted complex symmetric spaces, and we show that they correspond to an algebraic object called Hermitian Jordan triple products. The main topic of this work is to investigate the class of real forms of twisted complex symmetric spaces, called the category of symmetric spaces with twist. We show that this category is equivalent to the category of all real Jordan triple systems, and we can use a work of B.O. Makarevič in order to classify the irreducible spaces. The classification shows that most irreducible symmetric spaces have exactly one twisted complexification. This leads to open problems concerning the relation of Jordan and Lie triple systems. Introduction Straight and twisted complexifications.The Euclidean vector space (R n , i x i y i ) can be "complexified" in two ways: first, we have the extension (C n , i z i w i ) of the scalar product to a complex bilinear form on C n , and second, we have the "Hermitification" (C n , i z i w i ) which yields a scalar product on C n . Of course, this works for any bilinear form b on R n ; let us call b C (z, w) its "straight" (complex bilinear) and b C (z, w) its "twisted" complexification.In this paper we investigate an analogue of this construction in a situation which geometrically is much less trivial due to the presence of curvature. A natural class of spaces to look at here is the class of symmetric spaces since these are characterized by one fundamental invariant, namely by the curvature tensor R itself which satisfies the algebraic identities of a Lie triple system (LTS; cf. Def. 1.1.1). It is almost trivial that any such space admits (locally) a straight complexification given by the C-trilinear extension R C of R; from a group theoretic point of view this can be interpreted as the complexification of the homogeneous symmetric space M = G/H locally by the complex symmetric space for the curvature R, the invariant almost complex structure J and vector fields X, Y, Z. We consider this relation as fundamental and take it as the defining relation of a twisted complex symmetric space. A twisted complexification is then a (local) imbedding of M as a real form of a twisted complex symmetric space. For instance, the unit disc SU(1, 1)/ SO(2) is a twisted complexification of its real form M = ] − 1, 1[ ∼ = R; the complex projective space U(n + 1)/(U(n) × U(1)) is a twisted complexification of the real projective space M = O(n + 1)/(O(n) × O(1)); the variety Gl(2n, R)/ Gl(n, C) of complex structures on R 2n is a twisted complexification of the symmetric space of group type M = Gl(n, R) (cf. Ex. 1.3.7).We do not claim that any symmetric space admits a twisted complexification and, if it does, that it is unique. In fact, there is a very subtle curvature obstruction conveniently described in terms of Jordan theory: it would be tempting to define the twisted complexification of a LTS R by R C (X, Y )Z, but this does not work. However, we will see that in the Jordan category this idea works perfec...

Canonical fibrations of contact metric (κ,μ)-spaces

“…Next we recall the notions of straight and twisted complexifications of a Lie Triple System (LTS). For more details we refer the reader to [1] and [2]. Given a Lie triple system (m, [ , , ]) we shall write as usual…”

“…and the Ad(SO(2) × SO(n))-invariant metric G i on b i : G i ((v w), (u z)) =< v, u > + < w, z >,determine an invariant Hermitian symmetric structure (J i , ḡi ) on B i ; here <, > denotes the standard inner product on R n and (v w) denotes the matrix i = 2. Observe that the decomposition of g i(18)g i = so(n) ⊕ m i , m i := Rξ ⊕ b i , ξ := ∈ SO(n), X = sξ + (v w) ∈ m i…”

STEIN EMBEDDING THEOREM FOR $\mathbb{B}$-MANIFOLDS

“…The aim of this paper is to provide an intrinsic characterization of the smooth manifolds endowed with two equidimensional supplementary foliations which admit a closed embedding into an affine space Such a class of manifolds appears in a natural way in different geometric settings, and they are designated by distinct names in the specialized literature as paracomplex manifolds, hyperbolic complex manifolds, etc. (see, among others, the survey paper [2] and the references mentioned therein, and for the corresponding homogeneous and symmetric spaces see also [1,4,9,10,12]). In studying these manifolds, we obtain an interesting relationship between B-holomorphy and foliations, B being the quadratic algebra of double numbers (see § 2).…”