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Clifford algebra Алгебра Кліфорда クリフォード代数 Algèbre de Clifford 클리퍼드 대수 Alĝebro de Clifford Álgebra de Clifford Clifford-Algebra Алгебра Клиффорда Algebra di Clifford Clifford-algebra Álgebra de Clifford Cliffordalgebra 克利福德代数
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As álgebras de Clifford são álgebras associativas de importância na matemática, em particular na teoria da forma quadrática e do grupo ortogonal e na física. São nomeadas em homenagem a William Kingdon Clifford. In algebra lineare, un'algebra di Clifford è una struttura algebrica che generalizza la nozione di numero complesso e di quaternione. Lo studio delle algebre di Clifford è strettamente legato alla teoria delle forme quadratiche, e ha importanti applicazioni nella geometria e nella fisica teorica. Il loro nome deriva da quello del matematico William Kingdon Clifford che le introdusse nel 1878, partendo dallo studio di altri due oggetti algebrici analoghi, l'algebra dei quaternioni e le algebre di Grassmann. 数学において、クリフォード代数 (クリフォードだいすう、英: Clifford algebra) は結合多元環の一種である。K-代数として、それらは実数、複素数、四元数、そしていくつかの他の超複素数系を一般化する。クリフォード代数の理論は二次形式と直交変換の理論と親密に関係がある。クリフォード代数は幾何学、理論物理学、デジタル画像処理を含む種々の分野において重要な応用を持つ。それらはイギリス人幾何学者にちなんで名づけられている。 最もよく知られたクリフォード代数、あるいは直交クリフォード代数 (ちょっこうクリフォードだいすう、英: orthogonal Clifford algebra) は、リーマンクリフォード代数 (リーマンクリフォードだいすう、英: Riemannian Clifford algebra) とも呼ばれる。 En mathématiques, l'algèbre de Clifford est un objet d'algèbre multilinéaire associé à une forme quadratique. C'est une algèbre associative sur un corps, permettant un type de calcul étendu, englobant les vecteurs, les scalaires et des « multivecteurs » obtenus par produits de vecteurs, et avec une règle de calcul qui traduit la géométrie de la forme quadratique sous-jacente.Le nom de cette structure est un hommage au mathématicien anglais William Kingdon Clifford. 환론에서 클리퍼드 대수(Clifford代數, 영어: Clifford algebra)는 이차 형식에 의하여 정의되는 결합 대수의 한 종류이다. 복소수체와 사원수환의 일반화이며, 외대수의 양자화로 여길 수 있다. Las álgebras de Clifford son álgebras asociativas de importancia en matemáticas, en particular en teoría de la forma cuadrática y del grupo ortogonal y en la física. Se nombran así por William Kingdon Clifford. Je algebro, la alĝebro de Clifford estas asocieca alĝebro, generita de vektora spaco (aŭ modulo), tia ke la kvadrato de ĉiu unugrada elemento (t.e. elemento de la generinta vektora spaco) egalas la valoron de kvadrata formo. Die Clifford-Algebra ist ein nach William Kingdon Clifford benanntes mathematisches Objekt aus der Algebra, welches die komplexen und hyperkomplexen Zahlensysteme erweitert. Sie findet in der Differentialgeometrie sowie in der Quantenphysik Anwendung. Sie dient der Definition der Spin-Gruppe und ihrer Darstellungen, der Konstruktion von Spinorfeldern / -bündeln, die wiederum zur Beschreibung von Elektronen und anderen Elementarteilchen wichtig sind, sowie zur Bestimmung von Invarianten auf Mannigfaltigkeiten. Алгебра Кліфорда — це унітарна асоціативна алгебра, що містись і утворена за допомогою векторного простору V з квадратичною формою Q. Її можна розглядати як одне з можливих узагальнень комплексних чисел та кватерніонів. Теорія алгебр Кліфорда тісно пов'язана з теорією квадратичних форм і . Алгебра Кліффорда має важливі додатки в різних областях, в тому числі геометрії та теоретичної фізики. Вона названа на честь англійського математика Вільяма Кліфорда. In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing. They are named after the English mathematician William Kingdon Clifford. Алгебра Клиффорда — специального вида ассоциативная алгебра с единицей над некоторым коммутативным кольцом ( — векторное пространство или, более общо, свободный -модуль) с некоторой операцией [«умножения»], совпадающей с заданной на билинейной формой . Cliffordalgebra är en typ av vektoralgebra som kan betraktas som en generalisering av komplexa tal och kvaternioner. Algebran är uppkallad efter William Kingdon Clifford. Geometrisk algebra, Cliffords ursprungliga algebra från vilken begreppet Cliffordalgebra generaliserats, har en mängd tillämpningar inom datorgrafik och fysik. In de abstracte algebra is een clifford-algebra een unitaire (d.w.z. met eenheidselement) associatieve algebra die een uitbreiding vormt van de complexe getallen en de hypercomplexe getalsystemen. Clifford-algebra's zijn genoemd naar William Kingdon Clifford, die ze in 1878 ontdekte. 數學上,克利福德代数(Clifford algebra)是由具有二次型的向量空間生成的單位結合代數。作為域上的代數,其推廣實數系、複數系、四元數系等超複數系,以及外代数。此代數結構得名自英國數學家威廉·金顿·克利福德。 研究克里福代数的理論有時也稱為克里福代數,其與二次型論和正交群理論緊密聯繫。其在几何、理論物理、中有很多应用。其主要贡献者有:威廉·哈密顿(四元数),赫尔曼·格拉斯曼(外代数),威廉·金顿·克利福德,等。 最常見的克里福代數是正交克里福代數,又稱(偽)黎曼克里福代數。另一類是扭對稱克里福代數。
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Clifford algebra
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In de abstracte algebra is een clifford-algebra een unitaire (d.w.z. met eenheidselement) associatieve algebra die een uitbreiding vormt van de complexe getallen en de hypercomplexe getalsystemen. Clifford-algebra's zijn genoemd naar William Kingdon Clifford, die ze in 1878 ontdekte. De theorie van clifford-algebra's is nauw verbonden met de theorie van kwadratische vormen en orthogonale transformaties. Clifford-algebra's vinden brede toepassing in onder meer de meetkunde, de kwantumfysica, bij de definitie en voorstelling van spingroepen en spinoren, de bepaling van invarianten op varieteiten en de digitale beeldbewerking. En mathématiques, l'algèbre de Clifford est un objet d'algèbre multilinéaire associé à une forme quadratique. C'est une algèbre associative sur un corps, permettant un type de calcul étendu, englobant les vecteurs, les scalaires et des « multivecteurs » obtenus par produits de vecteurs, et avec une règle de calcul qui traduit la géométrie de la forme quadratique sous-jacente.Le nom de cette structure est un hommage au mathématicien anglais William Kingdon Clifford. Les algèbres de Clifford constituent l'une des généralisations possibles des nombres complexes et des quaternions. En mathématiques, elles offrent un cadre unificateur pour étudier des problèmes de géométrie tels que la théorie des formes quadratiques, et les groupes orthogonaux et introduire les spineurs et la (en). Mais elles fournissent aussi un cadre de calcul pertinent à de nombreux domaines physiques, des plus théoriques (relativité, mécanique quantique) aux plus appliqués (vision par ordinateur, robotique). Pour ces applications, une approche simplifiée est parfois pratiquée, avec une introduction différente, limitée aux corps des réels et complexes, ce qui conduit à la structure très proche d'algèbre géométrique. Une certaine familiarité avec les bases de l'algèbre multilinéaire sera très utile à la lecture de cet article.De nombreux résultats supposent que la caractéristique du corps de base K n'est pas 2 (c'est-à-dire que la division par 2 est possible) ; on fait cette hypothèse dans toute la suite, sauf dans une section dédiée au cas particulier de la caractéristique 2. In algebra lineare, un'algebra di Clifford è una struttura algebrica che generalizza la nozione di numero complesso e di quaternione. Lo studio delle algebre di Clifford è strettamente legato alla teoria delle forme quadratiche, e ha importanti applicazioni nella geometria e nella fisica teorica. Il loro nome deriva da quello del matematico William Kingdon Clifford che le introdusse nel 1878, partendo dallo studio di altri due oggetti algebrici analoghi, l'algebra dei quaternioni e le algebre di Grassmann. Die Clifford-Algebra ist ein nach William Kingdon Clifford benanntes mathematisches Objekt aus der Algebra, welches die komplexen und hyperkomplexen Zahlensysteme erweitert. Sie findet in der Differentialgeometrie sowie in der Quantenphysik Anwendung. Sie dient der Definition der Spin-Gruppe und ihrer Darstellungen, der Konstruktion von Spinorfeldern / -bündeln, die wiederum zur Beschreibung von Elektronen und anderen Elementarteilchen wichtig sind, sowie zur Bestimmung von Invarianten auf Mannigfaltigkeiten. Cliffordalgebra är en typ av vektoralgebra som kan betraktas som en generalisering av komplexa tal och kvaternioner. Algebran är uppkallad efter William Kingdon Clifford. Geometrisk algebra, Cliffords ursprungliga algebra från vilken begreppet Cliffordalgebra generaliserats, har en mängd tillämpningar inom datorgrafik och fysik. Je algebro, la alĝebro de Clifford estas asocieca alĝebro, generita de vektora spaco (aŭ modulo), tia ke la kvadrato de ĉiu unugrada elemento (t.e. elemento de la generinta vektora spaco) egalas la valoron de kvadrata formo. Las álgebras de Clifford son álgebras asociativas de importancia en matemáticas, en particular en teoría de la forma cuadrática y del grupo ortogonal y en la física. Se nombran así por William Kingdon Clifford. 數學上,克利福德代数(Clifford algebra)是由具有二次型的向量空間生成的單位結合代數。作為域上的代數,其推廣實數系、複數系、四元數系等超複數系,以及外代数。此代數結構得名自英國數學家威廉·金顿·克利福德。 研究克里福代数的理論有時也稱為克里福代數,其與二次型論和正交群理論緊密聯繫。其在几何、理論物理、中有很多应用。其主要贡献者有:威廉·哈密顿(四元数),赫尔曼·格拉斯曼(外代数),威廉·金顿·克利福德,等。 最常見的克里福代數是正交克里福代數,又稱(偽)黎曼克里福代數。另一類是扭對稱克里福代數。 환론에서 클리퍼드 대수(Clifford代數, 영어: Clifford algebra)는 이차 형식에 의하여 정의되는 결합 대수의 한 종류이다. 복소수체와 사원수환의 일반화이며, 외대수의 양자화로 여길 수 있다. Алгебра Клиффорда — специального вида ассоциативная алгебра с единицей над некоторым коммутативным кольцом ( — векторное пространство или, более общо, свободный -модуль) с некоторой операцией [«умножения»], совпадающей с заданной на билинейной формой . Смысл конструкции состоит в ассоциативном расширении пространства E⊕K и операции умножения на нём так, чтобы квадрат последней совпал с заданной квадратичной формой Q.Впервые рассмотрена Клиффордом.Алгебры Клиффорда обобщают комплексные числа, паракомплексные числа и дуальные числа, также , кватернионы и т.п.: их семейство исчерпывающе охватывает все ассоциативные гиперкомплексные числа. As álgebras de Clifford são álgebras associativas de importância na matemática, em particular na teoria da forma quadrática e do grupo ortogonal e na física. São nomeadas em homenagem a William Kingdon Clifford. 数学において、クリフォード代数 (クリフォードだいすう、英: Clifford algebra) は結合多元環の一種である。K-代数として、それらは実数、複素数、四元数、そしていくつかの他の超複素数系を一般化する。クリフォード代数の理論は二次形式と直交変換の理論と親密に関係がある。クリフォード代数は幾何学、理論物理学、デジタル画像処理を含む種々の分野において重要な応用を持つ。それらはイギリス人幾何学者にちなんで名づけられている。 最もよく知られたクリフォード代数、あるいは直交クリフォード代数 (ちょっこうクリフォードだいすう、英: orthogonal Clifford algebra) は、リーマンクリフォード代数 (リーマンクリフォードだいすう、英: Riemannian Clifford algebra) とも呼ばれる。 In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing. They are named after the English mathematician William Kingdon Clifford. The most familiar Clifford algebras, the orthogonal Clifford algebras, are also referred to as (pseudo-)Riemannian Clifford algebras, as distinct from symplectic Clifford algebras. Алгебра Кліфорда — це унітарна асоціативна алгебра, що містись і утворена за допомогою векторного простору V з квадратичною формою Q. Її можна розглядати як одне з можливих узагальнень комплексних чисел та кватерніонів. Теорія алгебр Кліфорда тісно пов'язана з теорією квадратичних форм і . Алгебра Кліффорда має важливі додатки в різних областях, в тому числі геометрії та теоретичної фізики. Вона названа на честь англійського математика Вільяма Кліфорда.
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