About: Maurer?Cartan form     Goto   Sponge   NotDistinct   Permalink

An Entity of Type : yago:Message106598915, within Data Space : dbpedia.org associated with source document(s)
QRcode icon
http://dbpedia.org/describe/?url=http%3A%2F%2Fdbpedia.org%2Fresource%2FMaurer%E2%80%93Cartan_form

In mathematics, the Maurer–Cartan form for a Lie group G is a distinguished differential one-form on G that carries the basic infinitesimal information about the structure of G. It was much used by Élie Cartan as a basic ingredient of his method of moving frames, and bears his name together with that of Ludwig Maurer.

AttributesValues
rdf:type
rdfs:label
  • Maurer-Cartan-Form
  • Maurer–Cartan form
  • Forme de Maurer-Cartan
  • Forma di Maurer-Cartan
  • 마우러-카르탕 형식
  • Форма Маурера-Картана
  • Форма Маурера — Картана
  • 马尤厄-嘉当形式
rdfs:comment
  • Die Maurer-Cartan-Form ist eine in Differentialgeometrie und Mathematischer Physik häufig verwendete Lie-Algebra-wertige Differentialform auf Lie-Gruppen. Sie ist benannt nach dem deutschen Mathematiker und Hochschullehrer Ludwig Maurer und dem französischen Mathematiker Élie Cartan.
  • En géométrie différentielle, la 1-forme de Maurer-Cartan est une 1-forme différentielle particulière sur un groupe de Lie.
  • In matematica, la forma di Maurer–Cartan associata ad ogni gruppo di Lie è una particolare 1-forma differenziale su che codifica l'informazione a livello infinitesimo circa la struttura del gruppo . Fu usata dal matematico Élie Cartan come ingrediente fondamentale del suo metodo dei riferimenti mobili e porta il suo nome accanto a quello di Ludwig Maurer.
  • 미분기하학에서, 마우러-카르탕 형식(Maurer-Cartan形式, 영어: Maurer–Cartan form)은 리 군 위에 정의된, 리 대수 값의 1차 미분 형식이다. 리 군의 연산 구조를 나타낸다.
  • 数学上,一个李群G的Maurer-Cartan形式是一个特别的微分形式,它包含关于这个李群的结构的基本的无穷小信息。它被埃里·嘉当多次使用,作为他的移动标架法的基本组成。 设是李群在幺元的切空间(它的李代数)。G可以由左平移作用在自身 , 这个诱导出切丛到自身的一个映射 . 一个左移不变向量场是的一个截面,使得 ∀ Maurer-Cartan形式 是在g值(在g中取值)的G上的1形式,根据公式作用在向量上。若X是G上的左移不变向量场,则在G为常数。而且,若X和Y都是左移不变,则 其中左边的括号为向量场的李括号,而右边的括号为李代数g的李括号。(这可以作为g上的李括号的定义。)这些事实可以用来建立李代数的同构 G上的左移不变向量场 . 根据微分的定义,若X和Y为任意向量场,则 . 实用上,若X和Y为左移不变,则 , 所以 但是左边只是一个2-形式(其值只和X,Y在一点的取值有关,所以跟X,Y作为场在周围的变化无关),所以方程不依赖于X和Y是左移不变的条件。所以这个方程对所有向量场X和Y成立。这被称为Maurer-Cartan方程. 如果G嵌入到GL(n,R),则可以把的公式显式的写成 若我们在李群G上引入主丛,并把G上的定义为变换函数,则联络形式是的。实际上 和Maurer-Cartan方程完全一致。
  • Форма Маурера — Картана в теорії груп Лі — диференціальна форма визначена на групі Лі, що приймає значення у відповідній алгебрі Лі. Названа начесть німецького математика Людвіга Маурера і французького математика Елі Картана.
  • In mathematics, the Maurer–Cartan form for a Lie group G is a distinguished differential one-form on G that carries the basic infinitesimal information about the structure of G. It was much used by Élie Cartan as a basic ingredient of his method of moving frames, and bears his name together with that of Ludwig Maurer.
  • Форма Маурера — Картана.Возникла в математике, в теории групп Ли. Представляет собой особую дифференциальную 1-форму на группе Ли G, несущую основную инфинитезимальную информацию о структуре этой группы. Широко использовалась Эли Картаном как основная составляющая его метода подвижных реперов. Помимо имени Картана носит имя Людвига Маурера.
rdfs:seeAlso
foaf:isPrimaryTopicOf
dct:subject
Wikipage page ID
Wikipage revision ID
Link from a Wikipage to another Wikipage
Link from a Wikipage to an external page
sameAs
dbp:wikiPageUsesTemplate
has abstract
  • In mathematics, the Maurer–Cartan form for a Lie group G is a distinguished differential one-form on G that carries the basic infinitesimal information about the structure of G. It was much used by Élie Cartan as a basic ingredient of his method of moving frames, and bears his name together with that of Ludwig Maurer. As a one-form, the Maurer–Cartan form is peculiar in that it takes its values in the Lie algebra associated to the Lie group G. The Lie algebra is identified with the tangent space of G at the identity, denoted TeG. The Maurer–Cartan form ω is thus a one-form defined globally on G which is a linear mapping of the tangent space TgG at each g ∈ G into TeG. It is given as the pushforward of a vector in TgG along the left-translation in the group:
Faceted Search & Find service v1.17_git51 as of Sep 16 2020


Alternative Linked Data Documents: PivotViewer | iSPARQL | ODE     Content Formats:       RDF       ODATA       Microdata      About   
This material is Open Knowledge   W3C Semantic Web Technology [RDF Data] Valid XHTML + RDFa
OpenLink Virtuoso version 08.03.3321 as of Jun 2 2021, on Linux (x86_64-generic-linux-glibc25), Single-Server Edition (61 GB total memory)
Data on this page belongs to its respective rights holders.
Virtuoso Faceted Browser Copyright © 2009-2021 OpenLink Software