A necessary condition for discrete branching laws for Klein four symmetric pairs

2021 ◽  
Author(s):  
Haian He
Keyword(s):  
Simple Formula ◽  
Necessary Condition ◽  
Infinite Dimensional ◽  
Branching Laws

We show a necessary condition for Klein four symmetric pairs [Formula: see text] satisfying the condition (D.D.); that is, there exists at least one infinite-dimensional simple [Formula: see text]-module that is discretely decomposable as a [Formula: see text]-module. This work is a continuation of [A criterion for discrete branching laws for Klein four symmetric pairs and its application to [Formula: see text], Int. J. Math. 31(6) (2020) 2050049]. Moreover, we define associated Klein four symmetric pairs, and we may use these tools to compute that a class of Klein four symmetric pairs do not satisfy the condition (D.D.); for example, [Formula: see text].

scholarly journals A criterion for discrete branching laws for Klein four symmetric pairs and its application to E6(−14)

2020 ◽  
Vol 31 (06) ◽  
pp. 2050049
Author(s):  
Haian He
Keyword(s):  
Lie Group ◽  
Simple Formula ◽  
Complete Classification ◽  
Necessary Condition ◽  
Symmetric Pair ◽  
Branching Laws ◽  
Nonidentity Element

Let [Formula: see text] be a noncompact connected simple Lie group, and [Formula: see text] a Klein four-symmetric pair. In this paper, we show a necessary condition for the discrete decomposability of unitarizable simple [Formula: see text]-modules for Klein for symmetric pairs. Precisely, if certain conditions hold for [Formula: see text], there does not exist a unitarizable simple [Formula: see text]-module that is discretely decomposable as a [Formula: see text]-module. As an application, for [Formula: see text], we obtain a complete classification of Klein four symmetric pairs [Formula: see text], with [Formula: see text] noncompact, such that there exists at least one nontrivial unitarizable simple [Formula: see text]-module that is discretely decomposable as a [Formula: see text]-module and is also discretely decomposable as a [Formula: see text]-module for some nonidentity element [Formula: see text].

Elements of Spectral Theory for Generalized Derivations II : The Semifredholm Domain

1981 ◽  
Vol 33 (5) ◽  
pp. 1205-1231 ◽  
Author(s):  
Lawrence A. Fialkow
Keyword(s):  
Hilbert Spaces ◽  
Simple Formula ◽  
Decomposition Theorem ◽  
Linear Operators ◽  
Generalized Derivations ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Hilbert Space Operators ◽  
Algebraic Invariants ◽  
Fredholm Theorem

Let and denote infinite dimensional Hilbert spaces and let denote the space of all bounded linear operators from to . For A in and B in , let τAB denote the operator on defined by τAB(X) = AX – XB. The purpose of this note is to characterize the semi-Fredholm domain of τAB (Corollary 3.16). Section 3 also contains formulas for ind(τAB – λ). These results depend in part on a decomposition theorem for Hilbert space operators corresponding to certain “singular points” of the semi-Fredholm domain (Theorem 2.2). Section 4 contains a particularly simple formula for ind(τAB – λ) (in terms of spectral and algebraic invariants of A and B) for the case when τAB – λ is Fredholm (Theorem 4.2). This result is used to prove that (τBA) = –ind(τAB) (Corollary 4.3). We also prove that when A and B are bi-quasi-triangular, then the semi-Fredholm domain of τAB contains no points corresponding to nonzero indices.

scholarly journals THE CHSH-TYPE INEQUALITIES FOR INFINITE-DIMENSIONAL QUANTUM SYSTEMS

2013 ◽  
Vol 27 (21) ◽  
pp. 1350151
Author(s):  
YU GUO
Keyword(s):  
Pure State ◽  
Bell Inequalities ◽  
Quantum Systems ◽  
Necessary Condition ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Pure States ◽  
Sufficient And Necessary Condition

By establishing CHSH operators and CHSH-type inequalities, we show that any entangled pure state in infinite-dimensional systems is entangled in a 2⊗2 subspace. We find that, for infinite-dimensional systems, the corresponding properties are similar to that of the two-qubit case: (i) The CHSH-type inequalities provide a sufficient and necessary condition for separability of pure states; (ii) The CHSH operators satisfy the Cirel'son inequalities; (iii) Any state which violates one of these Bell inequalities is distillable.

scholarly journals Branching laws for small unitary representations of GL(n, ℂ)

2014 ◽  
Vol 25 (06) ◽  
pp. 1450052
Author(s):  
Jan Möllers ◽  
Benjamin Schwarz
Keyword(s):  
Parabolic Subgroup ◽  
Principal Series ◽  
Unitary Representations ◽  
Symmetric Pair ◽  
Maximal Parabolic Subgroup ◽  
Infinite Dimensional ◽  
Branching Laws ◽  
Series Representations ◽  
Unitary Principal Series ◽  
Kirillov Dimension

The unitary principal series representations of G = GL (n, ℂ) induced from a character of the maximal parabolic subgroup P = ( GL (1, ℂ) × GL (n - 1, ℂ)) ⋉ ℂn-1 attain the minimal Gelfand–Kirillov dimension among all infinite-dimensional unitary representations of G. We find the explicit branching laws for the restriction of these representations to all reductive subgroups H of G such that (G, H) forms a symmetric pair.

scholarly journals Tensor products of classifiable C∗-algebras

2016 ◽  
Vol 09 (03) ◽  
pp. 485-504
Author(s):  
Huaxin Lin ◽  
Wei Sun
Keyword(s):  
General Result ◽  
Simple Formula ◽  
Tensor Products ◽  
Infinite Dimensional ◽  
C Algebras ◽  
Infinite Type ◽  
Tracial Rank ◽  
Universal Coefficient Theorem ◽  
Unital Simple

Let [Formula: see text] be the class of all unital separable simple [Formula: see text]-algebras [Formula: see text] such that [Formula: see text] has tracial rank no more than one for all UHF-algebra [Formula: see text] of infinite type. It has been shown that all amenable [Formula: see text]-stable [Formula: see text]-algebras in [Formula: see text] which satisfy the Universal Coefficient Theorem can be classified up to isomorphism by the Elliott invariant. In this note, we show that [Formula: see text] if and only if [Formula: see text] has tracial rank no more than one for some unital simple infinite dimensional AF-algebra [Formula: see text] In fact, we show that [Formula: see text] if and only if [Formula: see text] for some unital simple AH-algebra [Formula: see text] We actually prove a more general result. Other results regarding the tensor products of [Formula: see text]-algebras in [Formula: see text] are also obtained.

scholarly journals Value Functions and Optimality Conditions for Nonconvex Variational Problems with an Infinite Horizon in Banach Spaces

2021 ◽  
Author(s):  
Hélène Frankowska ◽  
Nobusumi Sagara
Keyword(s):  
Infinite Horizon ◽  
Variational Problems ◽  
Necessary Condition ◽  
Value Functions ◽  
The Maximum Principle ◽  
Lipschitz Continuous ◽  
Infinite Dimensional ◽  
Set Valued Mapping ◽  
Nonconvex Variational Problems ◽  
Upper Estimate

We investigate the value function of an infinite horizon variational problem in the infinite-dimensional setting. First, we provide an upper estimate of its Dini–Hadamard subdifferential in terms of the Clarke subdifferential of the Lipschitz continuous integrand and the Clarke normal cone to the graph of the set-valued mapping describing dynamics. Second, we derive a necessary condition for optimality in the form of an adjoint inclusion that grasps a connection between the Euler–Lagrange condition and the maximum principle. The main results are applied to the derivation of the necessary optimality condition of the spatial Ramsey growth model.

scholarly journals Geodesic uniqueness and derivatives of Bers projection

2003 ◽  
Vol 67 (1) ◽  
pp. 145-162 ◽  
Author(s):  
Hui Guo
Keyword(s):  
Teichmüller Space ◽  
Base Point ◽  
Necessary Condition ◽  
Teichmuller Space ◽  
Infinite Dimensional ◽  
Derivatives Of ◽  
Sufficient And Necessary Condition

In this paper, we discover a sufficient and necessary condition under which two geodesic segments joining the base point and another point in an infinite-dimensional Teichmüller space are the same.

scholarly journals AUTOMATA COMPUTATION OF BRANCHING LAWS FOR ENDOMORPHISMS OF CUNTZ ALGEBRAS

2007 ◽  
Vol 17 (07) ◽  
pp. 1389-1409 ◽  
Author(s):  
KATSUNORI KAWAMURA
Keyword(s):  
Semigroup Theory ◽  
Automata Theory ◽  
De Bruijn Graph ◽  
Cuntz Algebra ◽  
C Algebra ◽  
Infinite Dimensional ◽  
C Algebras ◽  
Branching Laws ◽  
Branching Law ◽  
Mealy Machine

We study the representation theory of C*-algebras by using semigroup theory and automata theory. The Cuntz algebra [Formula: see text] is a finitely generated, infinite-dimensional, noncommutative C*-algebra. A certain class of cyclic representations of [Formula: see text] is characterized by words from the alphabet 1,…,N, which is called a cycle. A class of endomorphisms of [Formula: see text] is defined by polynomial functions in canonical generators and their conjugates. Such an endomorphism ρ of [Formula: see text] transforms a cycle π to π ◦ ρ which is a direct sum of cycles π1,…,πn unique up to unitary equivalence. The passage from π to π1,…,πn is called a branching law for ρ. In this article, we construct a Mealy machine from the endomorphism in order to compute its branching law. We show that the branching law is obtained as outputs from the machine for the input information of a given representation. Furthermore the actual computation of the branching law is executed by using a generalized de Bruijn graph associated with the Mealy machine.

Analytic characterizations of infinite dimensional distributions

2017 ◽  
Vol 20 (02) ◽  
pp. 1750007 ◽  
Author(s):  
Luigi Accardi ◽  
Un Cig Ji ◽  
Kimiaki Saitô
Keyword(s):  
White Noise ◽  
Characterization Theorem ◽  
Growth Conditions ◽  
Necessary Condition ◽  
Infinite Dimensional ◽  
S Transform ◽  
White Noise Functionals

We revisit the analytic characterization theorem for S-transform of infinite dimensional distributions. Then we prove that the nuclearity of the space of test white noise functionals is a necessary condition for the characterization of the S-transform in terms of analytic and growth conditions.

The Infancy of Solar-Type Stars and its Relevance for the Origin of Life

1997 ◽  
Vol 161 ◽  
pp. 267-282 ◽  
Author(s):  
Thierry Montmerle
Keyword(s):  
Main Sequence ◽  
Solar Nebula ◽  
Circumstellar Disks ◽  
T Tauri Stars ◽  
Necessary Condition ◽  
Sequence Evolution ◽  
T Tauri ◽  
Solar Type ◽  
Optical Domain ◽  
The Origin Of Life

AbstractFor life to develop, planets are a necessary condition. Likewise, for planets to form, stars must be surrounded by circumstellar disks, at least some time during their pre-main sequence evolution. Much progress has been made recently in the study of young solar-like stars. In the optical domain, these stars are known as «T Tauri stars». A significant number show IR excess, and other phenomena indirectly suggesting the presence of circumstellar disks. The current wisdom is that there is an evolutionary sequence from protostars to T Tauri stars. This sequence is characterized by the initial presence of disks, with lifetimes ~ 1-10 Myr after the intial collapse of a dense envelope having given birth to a star. While they are present, about 30% of the disks have masses larger than the minimum solar nebula. Their disappearance may correspond to the growth of dust grains, followed by planetesimal and planet formation, but this is not yet demonstrated.

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