Jacobi identity

After German mathematician Carl Gustav Jakob Jacobi (1804-1851).

Jacobi identity (countable and uncountable, plural Jacobi identities)

1. (mathematics) Given a binary operation × defined on a set S which also has additive operation + and additive identity 0, the property that a × (b×c) + b × (c×a) + c × (a×b) = 0 for all a, b, c in S.
   • 1995, Stephen L. Adler, Quaternionic Quantum Mechanics and Quantum Fields, Oxford University Press, page 535:

     We give here two proofs of the Jacobi identity for the generalized Poisson bracket defined in Eq. (13.69a).

   • 2004, ames Lepowsky, Haisheng Li, Introduction to Vertex Operator Algebras and Their Representations, Springer (Birkhäuser), page 12,

     As we have already mentioned, the Jacobi identity is actually the generating function of an infinite list of generally highly nontrivial identities, and one needs many of these individual componenent identities in working with the theory.

   • 2005, Martin Kröger, Models for Polymeric and Anisotropic Liquids, Springer, page 113:

     It is sufficient to test the Jacobi identity against three linear functions [354] (this reference also provides a code for evaluating Jacobi identities).

Often stipulated as an axiom. Notably applicable to the cross product in a vector space and to the Lie bracket operation in a Lie algebra.

• super Jacobi identity

• commutator on Wikipedia.Wikipedia
• Hall-Witt identity on Wikipedia.Wikipedia