Jacobi identity

The Jacobi identity requires a bracket operation to satisfy a cyclic sum of nested brackets equal to zero. It is a defining property of Lie algebras.

Lie algebra: A vector space with a bilinear, alternating bracket that satisfies the Jacobi identity. The identity is one of the defining axioms of this algebraic structure.
Cyclic permutation: A rearrangement that moves each item to the next position while preserving cyclic order. The identity sums nested brackets over the three cyclic arrangements of its inputs.
Lie group: A group that is also a smooth manifold, with smooth multiplication and inversion. The tangent space at its identity forms a Lie algebra whose bracket satisfies Jacobi.
Associativity: A binary operation is associative when regrouping three inputs does not change their result. Associative multiplication differs from the bracket law constrained by Jacobi.
Lie bracket: A bilinear operation on a Lie algebra that is alternating and satisfies the Jacobi identity. The identity specifies how this operation must combine nested brackets.
Derivation (mathematics): A linear map satisfying the product rule with respect to a specified algebraic operation. The Jacobi identity says bracketing with a fixed element acts as a derivation of the bracket.
Hamiltonian mechanics: A formulation of classical mechanics in terms of a phase space and a Hamiltonian function. The Poisson bracket's Jacobi identity supports consistent evolution of observables.
Leibniz rule: A product rule specifying how an operation acts on a product of inputs. The Leibniz rule constrains products, while Jacobi constrains nested brackets.
Bilinear map: A function of two vector inputs that is linear in each input separately. Bilinearity is a separate condition commonly imposed on Lie brackets.
Adjoint representation: The representation of a Lie algebra on itself in which each element acts by bracketing with that element. The identity is equivalent to the adjoint action preserving brackets.

Linked from 7 pages