The maximal subgroups of the classical groups

Dynkin, E. B.

doi:10.1090/trans2/006/03

Cited by 312 publications

(433 citation statements)

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“…In particular, among distance spheres on P 2 Ca, one (whose radius is 2/3 the distance to the cut locus) is normal homogeneous and all others are not naturally reductive. But, as follows from the present paper, spin (9) ls not Einstein. Hence the Spin (9) invariant Einstein metric on S 15 is not naturally reductive.…”

Section: F=e6©a2supporting

confidence: 54%

On normal homogeneous Einstein manifolds

Wang¹,

Ziller²

1985

Ann. Sci. École Norm. Sup.

108

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Section: F=e6©a2supporting

confidence: 54%

On normal homogeneous Einstein manifolds

Wang¹,

Ziller²

1985

Ann. Sci. École Norm. Sup.

108

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“…For N ≤ 3 one observes from explicit tables of group dimensions [58] that the same reasoning applies and R again equals the adjoint representation. The maximality of PSU(2 N ) in SO(4 N − 1) now directly follows from Dynkin's theorem [59] and the fact that PSU(2 N ) is simple and its adjoint representation is faithful and irreducible (acting on the fundamental representation space of SO (4 N − 1)). …”

Section: B23 Pentagon Conservation Equations For N = 2 Qubitsmentioning

confidence: 93%

Quantum theory from questions

2017

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We reconstruct the explicit formalism of qubit quantum theory from elementary rules on an observer's information acquisition. Our approach is purely operational: we consider an observer O interrogating a system S with binary questions and define S's state as O's 'catalogue of knowledge' about S. From the rules we derive the state spaces for N elementary systems and show that (a) they coincide with the set of density matrices over an N -qubit Hilbert space C 2 N ; (b) states evolve unitarily under the group PSU(2 N ) according to the von Neumann evolution equation; and (c) O's binary questions correspond to projective Pauli operator measurements with outcome probabilities given by the Born rule. As a by-product, this results in a propositional formulation of quantum theory. Aside from offering an informational explanation for the theory's architecture, the reconstruction also unravels new structural insights. We show that, in a derived quadratic information measure, (d) qubits satisfy inequalities which bound the information content in any set of mutually complementary questions to 1 bit; and (e) maximal sets of mutually complementary questions for one and two qubits must carry precisely 1 bit of information in pure states. The latter relations constitute conserved informational charges which define the unitary groups and, together with their conservation conditions, the sets of pure quantum states. These results highlight information as a 'charge of quantum theory' and the benefits of this informational approach. This work emphasizes the sufficiency of restricting to an observer's information to reconstruct the theory and completes the quantum reconstruction initiated in [1].

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“…We will prove two lemmas first. Although parts of the results of these lemmas are contained in the classical work of Dynkin [4], we shall give self-contained, geometric proofs that lead themselves directly to the proof of Theorem 1. Proof.…”

Section: Let G Be a Simply-connected Simple Algebraic Group Then Thmentioning

confidence: 99%

“…In either case, P 4 is a G-homogeneous space and does not have any G-stable subset other than itself, a contradiction. Therefore G must be SL (4) (4). We denote the P 3 equipped with an 5X(4)-action induced by β + (resp, /L) by P 3 + (resp.…”

Section: Let G Be a Simply-connected Simple Algebraic Group Which Ismentioning

confidence: 99%

“…For the precise statement, see Theorem 1 in the main text. For the proof of this theorem, we need a classification of closed subgroups of simple algebraic groups of codimension 1 and 2, which could be derived easily from the classical work of Dynkin [4]. However, we shall give a geometric proof independent of [4] which leads up directly to the proof of Theorem 1.…”

Section: § 1 Introductionmentioning

confidence: 99%

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Regular actions of simple algebraic groups on projective threefolds

Nakano¹

1989

Nagoya Mathematical Journal

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The purpose of this note is to study regular actions of simple algebraic groups on projective threefolds as an application of the theory of algebraic threefolds, especially Mori Theory and the theory of Fano threefolds (cf. Mori [11], Iskovskih [7, 8]). The motivation for this study is as follows. In a series of papers, Umemura, in part jointly with Mukai, has classified maximal connected algebraic subgroups of the Cremona group of three variables and also constructed minimal rational threefolds which correspond to such subgroups (cf. Umemura [16-19], Mukai-Ume-mura [12]). In particular, Umemura and Mukai studied in [12] the SL(2, C)-equivariant smooth projectivization of SL(2, C)/G, where G is a binary icosahedral or octahedral subgroup of SL(2, C). The study of equivariant smooth projectivization of SL(2, C)/G for any finite subgroup G has been completed along their lines in Nakano [14]. The main trick of these studies is the investigation of equivariant contraction maps of extremal rays in the context of Mori Theory [11]. In this note, we apply a similar idea to projective threefolds with a regular action of a simple algebraic group and determine which simple algebraic groups can act regularly and nontrivially on projective threefolds and in which fashion. We also need some standard (but difficult) facts from the theory of Fano threefolds. For the precise statement, see Theorem 1 in the main text. For the proof of this theorem, we need a classification of closed subgroups of simple algebraic groups of codimension 1 and 2, which could be derived easily from the classical work of Dynkin [4]. However, we shall give a geometric proof independent of [4] which leads up directly to the proof of Theorem 1. On the whole, we shall establish by geometric methods the scarcity of closed subgroups of small codimension in simple algebraic groups, which is implied in Dynkin [4].

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The maximal subgroups of the classical groups

Cited by 312 publications

References 0 publications

On normal homogeneous Einstein manifolds

On normal homogeneous Einstein manifolds

Quantum theory from questions

Regular actions of simple algebraic groups on projective threefolds

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