Weyl algebra (original) (raw)
En mathématiques, et plus précisément en algèbre générale, l'algèbre de Weyl est un anneau d'opérateurs différentiels dont les coefficients sont des polynômes à une variable. Cette algèbre (et d'autres la généralisant, appelées elles aussi algèbres de Weyl) a été introduite par Hermann Weyl en 1928 comme outil d'étude du principe d'incertitude en mécanique quantique.
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| dbo:abstract | En mathématiques, et plus précisément en algèbre générale, l'algèbre de Weyl est un anneau d'opérateurs différentiels dont les coefficients sont des polynômes à une variable. Cette algèbre (et d'autres la généralisant, appelées elles aussi algèbres de Weyl) a été introduite par Hermann Weyl en 1928 comme outil d'étude du principe d'incertitude en mécanique quantique. (fr) 抽象代数学におけるワイル代数（ワイルだいすう、英語: Weyl algebra）は多項式係数の微分作用素がなす非可換環である。量子力学におけるハイゼンベルクの不確定性原理の研究においてこの環を導入したヘルマン・ワイルにちなみ、この名前が付けられている。ワイル代数はハイゼンベルク群のリー環の普遍包絡環から、リー環の中心の生成元と普遍包絡環の単位元とを同一視して得られる商になっており、このことからハイゼンベルク代数とも呼ばれる。 (ja) In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), namely expressions of the form More precisely, let F be the underlying field, and let F[X] be the ring of polynomials in one variable, X, with coefficients in F. Then each fi lies in F[X]. ∂X is the derivative with respect to X. The algebra is generated by X and ∂X. The Weyl algebra is an example of a simple ring that is not a matrix ring over a division ring. It is also a noncommutative example of a domain, and an example of an Ore extension. The Weyl algebra is isomorphic to the quotient of the free algebra on two generators, X and Y, by the ideal generated by the element The Weyl algebra is the first in an infinite family of algebras, also known as Weyl algebras. The n-th Weyl algebra, An, is the ring of differential operators with polynomial coefficients in n variables. It is generated by Xi and ∂Xi, i = 1, ..., n. Weyl algebras are named after Hermann Weyl, who introduced them to study the Heisenberg uncertainty principle in quantum mechanics. It is a quotient of the universal enveloping algebra of the Heisenberg algebra, the Lie algebra of the Heisenberg group, by setting the central element of the Heisenberg algebra (namely [X,Y]) equal to the unit of the universal enveloping algebra (called 1 above). The Weyl algebra is also referred to as the symplectic Clifford algebra. Weyl algebras represent the same structure for symplectic bilinear forms that Clifford algebras represent for non-degenerate symmetric bilinear forms. (en) In algebra astratta, l'Algebra di Weyl è l'anello formato dagli operatori differenziali con coefficienti polinomiali in una sola variabile. Le algebre di Weyl prendono il nome da Hermann Weyl, che le introdusse in meccanica quantistica nello studio del principio di indeterminazione di Heisenberg. (it) 환론에서 바일 대수(영어: Weyl algebra)는 다항식 계수의 미분 연산자로 구성되는 단위 결합 대수이다. (ko) In de abstracte algebra, meer specifiek de ringtheorie, een deelgebied van de wiskunde, is de Weyl-algebra de ring van differentiaaloperatoren met coëfficiënten, die een polynoom zijn in één variabele. Meer precies: laat een lichaam (Nederlands) of veld (Belgisch) zijn en laat de ring van polynomen in één variabele, , met coëfficiënten in zijn. Dan ligt elke in . De operator is de afgeleide naar . De algebra wordt gegenereerd door en . De Weyl-algebra is een voorbeeld van een enkelvoudige ring, die geen matrixring over een delingsring (Nederlands) of lichaam (Belgisch) is. Het is ook een voorbeeld van een , dat niet commutatief is, en tevens een voorbeeld van een . De Weyl-algebra is een quotiënt van de op twee generatoren, en , door de ideaal, gegenereerd door de enkele relatie De Weyl-algebra is de eerste in een oneindige familie van algebra's, die ook bekendstaat als de Weyl-algebra's. De -de Weyl-algebra, , is de ring van differentiaaloperatoren met coëfficiënten, die een polynoom zijn in variabelen. De Weyl-algebra wordt gegenereerd door en . Weyl-algebra's zijn vernoemd naar Hermann Weyl, die zij als eerste introduceerde om de onzekerheidsrelatie van Heisenberg in de kwantummechanica te bestuderen. Het is een quotiënt van de van de Lie-algebra van de Heisenberg-groep, door het eenheidselement 1 van de Lie-algebra gelijk te zetten aan de het eenheidselement 1 van de universele omhullende algebra. Om deze reden staan Weyl-algebra's ook wel bekend als Heisenberg-algebra's. (nl) Em álgebra abstrata, a álgebra de Weyl é o anel de operadores diferenciais com coeficientes polinomiais (em uma variável), Mais precisamente, seja F um corpo e F[X] o anel de polinômios em uma variável, X, com coeficiêntes em F. Então cada fi está em F[X]. ∂X é a derivada com relação a X. A álgebra é gerada por X e ∂X. (pt) |
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| rdfs:comment | En mathématiques, et plus précisément en algèbre générale, l'algèbre de Weyl est un anneau d'opérateurs différentiels dont les coefficients sont des polynômes à une variable. Cette algèbre (et d'autres la généralisant, appelées elles aussi algèbres de Weyl) a été introduite par Hermann Weyl en 1928 comme outil d'étude du principe d'incertitude en mécanique quantique. (fr) 抽象代数学におけるワイル代数（ワイルだいすう、英語: Weyl algebra）は多項式係数の微分作用素がなす非可換環である。量子力学におけるハイゼンベルクの不確定性原理の研究においてこの環を導入したヘルマン・ワイルにちなみ、この名前が付けられている。ワイル代数はハイゼンベルク群のリー環の普遍包絡環から、リー環の中心の生成元と普遍包絡環の単位元とを同一視して得られる商になっており、このことからハイゼンベルク代数とも呼ばれる。 (ja) In algebra astratta, l'Algebra di Weyl è l'anello formato dagli operatori differenziali con coefficienti polinomiali in una sola variabile. Le algebre di Weyl prendono il nome da Hermann Weyl, che le introdusse in meccanica quantistica nello studio del principio di indeterminazione di Heisenberg. (it) 환론에서 바일 대수(영어: Weyl algebra)는 다항식 계수의 미분 연산자로 구성되는 단위 결합 대수이다. (ko) Em álgebra abstrata, a álgebra de Weyl é o anel de operadores diferenciais com coeficientes polinomiais (em uma variável), Mais precisamente, seja F um corpo e F[X] o anel de polinômios em uma variável, X, com coeficiêntes em F. Então cada fi está em F[X]. ∂X é a derivada com relação a X. A álgebra é gerada por X e ∂X. (pt) In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), namely expressions of the form More precisely, let F be the underlying field, and let F[X] be the ring of polynomials in one variable, X, with coefficients in F. Then each fi lies in F[X]. ∂X is the derivative with respect to X. The algebra is generated by X and ∂X. The Weyl algebra is an example of a simple ring that is not a matrix ring over a division ring. It is also a noncommutative example of a domain, and an example of an Ore extension. (en) In de abstracte algebra, meer specifiek de ringtheorie, een deelgebied van de wiskunde, is de Weyl-algebra de ring van differentiaaloperatoren met coëfficiënten, die een polynoom zijn in één variabele. Meer precies: laat een lichaam (Nederlands) of veld (Belgisch) zijn en laat de ring van polynomen in één variabele, , met coëfficiënten in zijn. Dan ligt elke in . De operator is de afgeleide naar . De algebra wordt gegenereerd door en . De Weyl-algebra is een quotiënt van de op twee generatoren, en , door de ideaal, gegenereerd door de enkele relatie (nl) |
| rdfs:label | Algèbre de Weyl (fr) Algebra di Weyl (it) 바일 대수 (ko) ワイル代数 (ja) Weyl-algebra (nl) Álgebra de Weyl (pt) Weyl algebra (en) |
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