scholarly journals NONRELATIVISTIC HOLOGRAPHY — A GROUP-THEORETICAL PERSPECTIVE

2014 ◽  
Vol 29 (03n04) ◽  
pp. 1430001 ◽  
Author(s):  
V. K. DOBREV
Keyword(s):  
Representation Theory ◽  
Theoretical Perspective ◽  
Explicit Construction ◽  
Intertwining Operators ◽  
Difference Operators ◽  
Semisimple Lie Groups ◽  
Boundary Operators ◽  
Boundary Fields ◽  
Theoretical Results ◽  
Schrödinger Algebra

We give a review of some group-theoretical results related to nonrelativistic holography. Our main playgrounds are the Schrödinger equation and the Schrödinger algebra. We first recall the interpretation of nonrelativistic holography as equivalence between representations of the Schrödinger algebra describing bulk fields and boundary fields. One important result is the explicit construction of the boundary-to-bulk operators in the framework of representation theory, and that these operators and the bulk-to-boundary operators are intertwining operators. Further, we recall the fact that there is a hierarchy of equations on the boundary, invariant with respect to Schrödinger algebra. We also review the explicit construction of an analogous hierarchy of invariant equations in the bulk, and that the two hierarchies are equivalent via the bulk-to-boundary intertwining operators. The derivation of these hierarchies uses a mechanism introduced first for semisimple Lie groups and adapted to the nonsemisimple Schrödinger algebra. These require development of the representation theory of the Schrödinger algebra which is reviewed in some detail. We also recall the q-deformation of the Schrödinger algebra. Finally, the realization of the Schrödinger algebra via difference operators is reviewed.

On a construction of unitary cocycles and the representation theory of amenable groups

1983 ◽  
Vol 3 (1) ◽  
pp. 129-133 ◽  
Author(s):  
Colin E. Sutherland
Keyword(s):  
Representation Theory ◽  
Probability Space ◽  
Amenable Group ◽  
Unitary Representations ◽  
Intertwining Operators ◽  
Type I ◽  
Amenable Groups

AbstractIf K is a countable amenable group acting freely and ergodically on a probability space (Γ, μ), and G is an arbitrary countable amenable group, we construct an injection of the space of unitary representations of G into the space of unitary 1-cocyles for K on (Γ, μ); this injection preserves intertwining operators. We apply this to show that for many of the standard non-type-I amenable groups H, the representation theory of H contains that of every countable amenable group.

scholarly journals LA FORMACIÓN DIDÁCTICO-MATEMÁTICA DE DOCENTES: RESULTADOS TEÓRICOS

PARADIGMA ◽  
2020 ◽  
pp. 271-288
Author(s):  
Nancy Montes de Oca Recio
Keyword(s):  
Human Development ◽  
Present Article ◽  
Learning Process ◽  
Theoretical Perspective ◽  
Mathematical Education ◽  
School Teachers ◽  
Mathematical Communication ◽  
Basic Concepts ◽  
Teaching Learning ◽  
Theoretical Results

El objetivo del presente artículo es comunicar el proceso y los resultados teóricos de unainvestigación relacionada con la formación didáctico matemática de docentes, generada a partirde las insuficiencias detectadas en el desempeño de docentes de Matemática en el ejercicio dela profesión. La investigación aborda aspectos relacionados con: la comprensión de los objetosde la Geometría Plana; la gestión didáctica de las demostraciones matemáticas, la promoción deuna valoración positiva hacia la Matemática, la producción de problemas, el dominio deconceptos básicos del Álgebra Lineal y la expresión al comunicarse en matemática, desde unaperspectiva teórica que integra el enfoque histórico cultural del desarrollo humano y losenfoques ontosemiótico, comunicativo y contextualizado de la enseñanza-aprendizaje de laMatemática. Los resultados responden a la labor realizada por un equipo de trabajo constituidopor profesores que imparten docencia en carreras de formación de docentes de Matemática,maestrías y doctorados de la Universidad de Camagüey y la Universidad Autónoma de SantoDomingo.Palabras clave: Formación Didáctico-Matemática, Educación Matemática, Docentes deMatemática, Gestión DidácticaA Formação Didático-Matemática de Pedagogos: implicações teóricasAbstractO objetivo deste artigo presente é comunicar o processo e os resultados teóricos de umainvestigação relacionados com o processo de aprendizagem didático-matemático de pedagogos;este estudo começou a levar em conta as insuficiências descobriu no processo de ensinoaprendizagemde Matemática administrado por professores escolares. Esta investigação chegaaspectos relacionados com: a compreensão de objetos de Geometria Planos; a administraçãodidática de demonstrações de Matemáticas, a promoção de uma estimação positiva paraMatemática, a geração de problemas didáticos, o dominando de conceitos básicos de ÁlgebraLinear e comunicação matemática, de uma perspectiva teórica que integra a aproximaçãohistórica cultural de desenvolvimento humano e o ontosemiotico, comunicativo e contextualfocaliza do ensino-aprendizagem de Matemática. Os resultados apresentados são um resultadodo trabalho levado a cabo por um time constituiu por professores que ensinam Matemática empre-serviço, mestres e cursos de doutorado em Universidade de Camaguey e a UniversidadeAutônoma de Santo Domingo.Palavras chave: Formação Didático-Matemática, Educação Matemática, Educação deMatemática, Administração didática.The Didactic and Mathematical Formation of Teachers: theoric resultsAbstractThe objective of this present article is to communicate the process and the theoretical results of an investigation related with the didactic-mathematical learning process of educators; this study began taking into account the inadequacies detected in the teaching-learning process of Mathematics managed by school teachers. This investigation approaches aspects related with: the understanding of Plane Geometry objects; the didactic management of Maths demonstrations, the promotion of a positive valuation towards Mathematics, the generation of didactic problems, the mastering of basic concepts of Lineal Algebra and mathematical communication, from a theoretical perspective that integrates the cultural historical approach of human development and the ontosemiotic, communicative and contextual focuses of the teaching-learning of Mathematics. The results presented is a result of the work carried out by a team constituted by professors that teach Mathematics in pre-service, masters and doctorates courses at Camaguey University and the Autonomous University of Santo Domingo.Key words: Didactic-Mathematical Formation, Mathematical Education, Education of Mathematics, Didactic Management.

scholarly journals Remarks on the Representation Theory of the Moyal Plane

2011 ◽  
Vol 2011 ◽  
pp. 1-9
Author(s):  
J. M. Isidro ◽  
P. Fernández de Córdoba ◽  
J. M. Rivera-Rebolledo ◽  
J. L. G. Santander
Keyword(s):  
Representation Theory ◽  
Unitary Representation ◽  
Explicit Construction ◽  
Commutator Algebra ◽  
Moyal Plane

We present an explicit construction of a unitary representation of the commutator algebra satisfied by position and momentum operators on the Moyal plane.

scholarly journals Simple Finite-Dimensional Modules and Monomial Bases from the Gelfand-Testlin Patterns

2021 ◽  
pp. 60-65
Author(s):  
Amadou Keita
Keyword(s):  
Representation Theory ◽  
Lie Algebras ◽  
Simple Module ◽  
Explicit Construction ◽  
Research Area ◽  
The Past ◽  
Finite Dimensional ◽  
Dimensional Module

One of the most important classes of Lie algebras is sl_n, which are the n×n matrices with trace 0. The representation theory for sl_n has been an interesting research area for the past hundred years and in it, the simple finite-dimensional modules have become very important. They were classified and Gelfand and Tsetlin actually gave an explicit construction of a basis for every simple finite-dimensional module. This paper extends their work by providing theorems and proofs and constructs monomial bases of the simple module.

scholarly journals LA FORMACIÓN DIDÁCTICO-MATEMÁTICA DE DOCENTES: RESULTADOS TEÓRICOS

PARADIGMA ◽  
2020 ◽  
pp. 271-288
Author(s):  
Nancy Montes de Oca Recio
Keyword(s):  
Human Development ◽  
Present Article ◽  
Learning Process ◽  
Theoretical Perspective ◽  
Mathematical Education ◽  
School Teachers ◽  
Mathematical Communication ◽  
Basic Concepts ◽  
Teaching Learning ◽  
Theoretical Results

El objetivo del presente artículo es comunicar el proceso y los resultados teóricos de unainvestigación relacionada con la formación didáctico matemática de docentes, generada a partirde las insuficiencias detectadas en el desempeño de docentes de Matemática en el ejercicio dela profesión. La investigación aborda aspectos relacionados con: la comprensión de los objetosde la Geometría Plana; la gestión didáctica de las demostraciones matemáticas, la promoción deuna valoración positiva hacia la Matemática, la producción de problemas, el dominio deconceptos básicos del Álgebra Lineal y la expresión al comunicarse en matemática, desde unaperspectiva teórica que integra el enfoque histórico cultural del desarrollo humano y losenfoques ontosemiótico, comunicativo y contextualizado de la enseñanza-aprendizaje de laMatemática. Los resultados responden a la labor realizada por un equipo de trabajo constituidopor profesores que imparten docencia en carreras de formación de docentes de Matemática,maestrías y doctorados de la Universidad de Camagüey y la Universidad Autónoma de SantoDomingo.Palabras clave: Formación Didáctico-Matemática, Educación Matemática, Docentes deMatemática, Gestión DidácticaA Formação Didático-Matemática de Pedagogos: implicações teóricasAbstractO objetivo deste artigo presente é comunicar o processo e os resultados teóricos de umainvestigação relacionados com o processo de aprendizagem didático-matemático de pedagogos;este estudo começou a levar em conta as insuficiências descobriu no processo de ensinoaprendizagemde Matemática administrado por professores escolares. Esta investigação chegaaspectos relacionados com: a compreensão de objetos de Geometria Planos; a administraçãodidática de demonstrações de Matemáticas, a promoção de uma estimação positiva paraMatemática, a geração de problemas didáticos, o dominando de conceitos básicos de ÁlgebraLinear e comunicação matemática, de uma perspectiva teórica que integra a aproximaçãohistórica cultural de desenvolvimento humano e o ontosemiotico, comunicativo e contextualfocaliza do ensino-aprendizagem de Matemática. Os resultados apresentados são um resultadodo trabalho levado a cabo por um time constituiu por professores que ensinam Matemática empre-serviço, mestres e cursos de doutorado em Universidade de Camaguey e a UniversidadeAutônoma de Santo Domingo.Palavras chave: Formação Didático-Matemática, Educação Matemática, Educação deMatemática, Administração didática.The Didactic and Mathematical Formation of Teachers: theoric resultsAbstractThe objective of this present article is to communicate the process and the theoretical results of an investigation related with the didactic-mathematical learning process of educators; this study began taking into account the inadequacies detected in the teaching-learning process of Mathematics managed by school teachers. This investigation approaches aspects related with: the understanding of Plane Geometry objects; the didactic management of Maths demonstrations, the promotion of a positive valuation towards Mathematics, the generation of didactic problems, the mastering of basic concepts of Lineal Algebra and mathematical communication, from a theoretical perspective that integrates the cultural historical approach of human development and the ontosemiotic, communicative and contextual focuses of the teaching-learning of Mathematics. The results presented is a result of the work carried out by a team constituted by professors that teach Mathematics in pre-service, masters and doctorates courses at Camaguey University and the Autonomous University of Santo Domingo.Key words: Didactic-Mathematical Formation, Mathematical Education, Education of Mathematics, Didactic Management.

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