On Automorphisms of Siegel Domains

The notion of "Siegel domains" was introduced by . It was then shown that every homogeneous bounded domain is holomorphically equivalent to a Siegel domain (of the second kind) determined uniquely up to an affine isomorphism ([15] . The purpose of the present note is to show that this classification can also be obtained immediately from my previous result on linear imbeddings of self-dual cones ([10a]).Our method is based on the following two observations: (I) There are natural equivalences between the three categories of (punctured) self-dual cones, the corresponding reductive Lie algebras (with fixed Cartan involutions), and formally real Jordan algebras ( § 1).(II) There is a natural bisection between the set of isomorphism classes of quasi-symmetric Siegel domains and that of equivalence classes of the pairs formed of a self-dual cone and a (linear) "representation" of it ( §3, Proposition 2).

Section: Lemma Let ψ: U -> U / Be a Linear Map With φ(E) -E F Thenmentioning

confidence: 99%

On classification of quasi-symmetric domains

Satake

1976

“…If 8=1, then this theorem was proved by Tanaka [14] and Murakami [8], Nakajima [18] calculated the dimensions of Q1/2 and gi of this theorem by using different methods.…”

Section: Proof First We Suppose That D(v F) Is Degenerate Then R(smentioning

confidence: 97%

“…This idea of determining gi is due to Murakami [8]. Therefore, for each BeH(p,R),BY belongs to So, BY must be of the form (3.4) for each BeH(p>R)> which implies that Y must be of the form (3.7).…”

Section: )mentioning

confidence: 99%

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Siegel domains over self-dual cones and their automorphisms

Tsuji

1974

The Lie algebra gr of all infinitesimal automorphisms of a Siegel domain in terms of polynomial vector fields was investigated by Kaup, Matsushima and Ochiai [6]. It was proved in [6] that gr is a graded Lie algebra; gr = g-1 + g-1/2 + g0 + g1/2 + g1 and the Lie subalgebra ga of all infinitesimal affine automorphisms is given by the graded subalgebra; ga = g-1 + g-1/2 + g0. Nakajima [9] proved without the assumption of homogeneity that the non-affine parts g1/2 and g1 can be determined from the affine part ga.

“…we refer to [26] and [36]. Proof, a) We choose a norm | | on U and a norm || || on V. The norm induced on End c U (resp.…”

Section: That Aut D(k S) Operates Transitively On D(k S) If and Onlmentioning

confidence: 99%

Homogeneous Siegel domains

Dorfmeister

1982

In 1935 E. Cartan classified all symmetric bounded domains [6]. At that time he proved that a bounded symmetric domain is homogeneous with respect to its group of holomorphic automorphisms. Thus the more general problem of investigating homogeneous bounded domains arose. It was known to E. Cartan that all homogeneous bounded domains of dimension <3 are symmetric [6] In this work we present a new method for the description of homogeneous Siegel domains. This method entails a classification of the domains considered and makes it possible to answer several of the open question. We are able to reproduce the known results and, in some cases, Received April 23, 1979. This article coincides up to minor changes with chapter I of the authors "Habilitationsschrift" [13] which has been accepted by the "Mathematisch-Naturwissenschaftliche Fakultat der Westfalischen Wilhelms-Universitat zu Mϋnster".https://www.cambridge.org/core/terms. https://doi