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Square brackets around two vector fields or Lie algebra elements.
Also known as: Lie bracket, Bracket of vector fields, Jacobi bracket
The Lie bracket measures the failure of two flows to commute — walk along one field then the other, then reverse both, and the bracket is where you end up short. It obeys the Jacobi identity in place of associativity, which is the axiom that defines a Lie algebra.
Denotes the Lie bracket: an antisymmetric bilinear product satisfying the Jacobi identity.
Contexts: Mathematics
Measures the non-commutativity of two flows geometrically: it is the leading term of the failure to return to the starting point.
Contexts: Mathematics
Replaces associativity with the Jacobi identity, which is what distinguishes a Lie algebra from an associative algebra.
Contexts: Mathematics
A pair of square brackets enclosing two field or element names separated by a comma.