In mathematics, a Chevalley basis for a simple complex Lie algebra is a basis constructed by Claude Chevalley with the property that all structure constants are integers. Chevalley used these bases to construct analogues of Lie groups over finite fields, called Chevalley groups. The Chevalley basis is the Cartan-Weyl basis, but with a different normalization. The generators of a Lie group are split into the generators H and E indexed by simple roots and their negatives . The Cartan-Weyl basis may be written as Defining the dual root or coroot of as One may perform a change of basis to define
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| - Chevalley basis (en)
- 슈발레 기저 (ko)
- Base Chevalley (pt)
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| - 리 군론에서 슈발레 기저(Chevalley基底, 영어: Chevalley basis)는 모든 구조 상수가 정수인, 반단순 리 대수의 특별한 기저이다. 이를 통해, 정수환 또는 임의의 가환환을 계수로 하는 반단순 리 대수의 형태를 정의할 수 있다. (ko)
- Em matemática, uma base Chevalley para uma simples álgebra de Lie complexa é uma base construída por Claude Chevalley com a propriedade de que todas as estruturas constantes são inteiras. Chevalley usou essas bases para a construção de análogos de grupos de Lie sobre corpos finitos, chamados . Os geradores de um grupo de Lie são divididos em geradores H e E tal que: onde p = m se β + γ é uma raiz e m é o maior inteiro positivo tal que γ − mβ é uma raiz. (pt)
- In mathematics, a Chevalley basis for a simple complex Lie algebra is a basis constructed by Claude Chevalley with the property that all structure constants are integers. Chevalley used these bases to construct analogues of Lie groups over finite fields, called Chevalley groups. The Chevalley basis is the Cartan-Weyl basis, but with a different normalization. The generators of a Lie group are split into the generators H and E indexed by simple roots and their negatives . The Cartan-Weyl basis may be written as Defining the dual root or coroot of as One may perform a change of basis to define (en)
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| - In mathematics, a Chevalley basis for a simple complex Lie algebra is a basis constructed by Claude Chevalley with the property that all structure constants are integers. Chevalley used these bases to construct analogues of Lie groups over finite fields, called Chevalley groups. The Chevalley basis is the Cartan-Weyl basis, but with a different normalization. The generators of a Lie group are split into the generators H and E indexed by simple roots and their negatives . The Cartan-Weyl basis may be written as Defining the dual root or coroot of as One may perform a change of basis to define The Cartan integers are The resulting relations among the generators are the following: where in the last relation is the greatest positive integer such that is a root and we consider if is not a root. For determining the sign in the last relation one fixes an ordering of roots which respects addition, i.e., if then provided that all four are roots. We then call an extraspecial pair of roots if they are both positive and is minimal among all that occur in pairs of positive roots satisfying . The sign in the last relation can be chosen arbitrarily whenever is an extraspecial pair of roots. This then determines the signs for all remaining pairs of roots. (en)
- 리 군론에서 슈발레 기저(Chevalley基底, 영어: Chevalley basis)는 모든 구조 상수가 정수인, 반단순 리 대수의 특별한 기저이다. 이를 통해, 정수환 또는 임의의 가환환을 계수로 하는 반단순 리 대수의 형태를 정의할 수 있다. (ko)
- Em matemática, uma base Chevalley para uma simples álgebra de Lie complexa é uma base construída por Claude Chevalley com a propriedade de que todas as estruturas constantes são inteiras. Chevalley usou essas bases para a construção de análogos de grupos de Lie sobre corpos finitos, chamados . Os geradores de um grupo de Lie são divididos em geradores H e E tal que: onde p = m se β + γ é uma raiz e m é o maior inteiro positivo tal que γ − mβ é uma raiz. (pt)
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