In differential geometry, the Schouten–Nijenhuis bracket, also known as the Schouten bracket, is a type of graded Lie bracket defined on multivector fields on a smooth manifold extending the Lie bracket of vector fields. There are two different versions, both rather confusingly called by the same name.

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  • In differential geometry, the Schouten–Nijenhuis bracket, also known as the Schouten bracket, is a type of graded Lie bracket defined on multivector fields on a smooth manifold extending the Lie bracket of vector fields. There are two different versions, both rather confusingly called by the same name. The most common version is defined on alternating multivector fields and makes them into a Gerstenhaber algebra, but there is also another version defined on symmetric multivector fields, which is more or less the same as the Poisson bracket on the cotangent bundle. It was discovered by Jan Arnoldus Schouten (1940, 1953) and its properties were investigated by his student Albert Nijenhuis (1955). It is related to but not the same as the Nijenhuis–Richardson bracket and the Frölicher–Nijenhuis bracket.
  • 在微分几何中，Schouten–Nijenhuis括号（Schouten–Nijenhuis bracket，国际音标：[ˈsχʌutən]-[ˈnɛiənhœys]），也称为斯豪滕括号，是定义在光滑流形上的多重向量场上的一种分次李括号，推广了向量场的李括号。有两种不同的版本，让人相当不解地是有相同的名字。最通常的版本是定义在交错多重向量场上，使得其成为一个格尔斯滕哈伯代数；但另一个版本定义在对称多重向量场上，这或多或少与余切丛上的泊松括号相同。它由扬·阿诺尔德斯·斯豪滕（Jan Arnoldus Schouten）在1940年与1953年发现，其性质为他的学生 Albert Nijenhuis（Albert Nijenhuis）在1955年研究。它与 Nijenhuis–Richardson括号及 Frölicher–Nijenhuis括号有联系但不相同。
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rdfs:comment
  • In differential geometry, the Schouten–Nijenhuis bracket, also known as the Schouten bracket, is a type of graded Lie bracket defined on multivector fields on a smooth manifold extending the Lie bracket of vector fields. There are two different versions, both rather confusingly called by the same name.
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  • Schouten–Nijenhuis bracket
  • Schouten–Nijenhuis括号
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