Jacobi identity

English Etymology After German mathematician Carl Gustav Jakob Jacobi (1804-1851). Noun ~ en 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. en 1995 Stephen L. Adler Quaternionic Quantum Mechanics and Quantum Fields https://books.google.com.au/books?id=irt7nOFaR3sC&pg=PA535&dq=%22Jacobi+identity%22%7C%22Jacobi+identities%22&hl=en&sa=X&ved=0ahUKEwjhnPe2mODWAhUFU7wKHQ0JBewQ6AEIYzAK#v=onepage&q=%22Jacobi%20identity%22%7C%22Jacobi%20identities%22&f=false 535 w:Oxford University Press We give here two proofs of the Jacobi identity for the generalized Poisson bracket defined in Eq. (13.69a). 2004, James 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. en 2005 Martin Kröger Models for Polymeric and Anisotropic Liquids https://books.google.com.au/books?id=WAVzyygV5NQC&pg=PA113&dq=%22Jacobi+identity%22%7C%22Jacobi+identities%22&hl=en&sa=X&ved=0ahUKEwjhnPe2mODWAhUFU7wKHQ0JBewQ6AEIUjAH#v=onepage&q=%22Jacobi%20identity%22%7C%22Jacobi%20identities%22&f=false 113 Springer It is sufficient to test the Jacobi identity against three linear functions [354] (this reference also provides a code for evaluating Jacobi identities). Usage notes 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. Derived terms en super Jacobi identity Translations property of a binary operation Italian: it identità di Jacobi f Polish: pl tożsamość Jacobiego f Further reading commutator Hall-Witt identity