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The cross product and the Lie bracket operation [,] both satisfy the Jacobi identity. [3] In analytical mechanics , the Jacobi identity is satisfied by the Poisson brackets . In quantum mechanics , it is satisfied by operator commutators on a Hilbert space and equivalently in the phase space formulation of quantum mechanics by the Moyal bracket .
The cross product can be seen as one of the simplest Lie products, and is thus generalized by Lie algebras, which are axiomatized as binary products satisfying the axioms of multilinearity, skew-symmetry, and the Jacobi identity.
The products in zero and one dimensions are trivial, so non-trivial cross products only exist in three and seven dimensions. [18] [19] The failure of the 7-dimension cross product to satisfy the Jacobi identity is related to the nonassociativity of the octonions. In fact,
Using the cross product as a Lie bracket, the algebra of 3-dimensional real vectors is a Lie algebra isomorphic to the Lie algebras of SU(2) and SO(3). The structure constants are f a b c = ϵ a b c {\displaystyle f^{abc}=\epsilon ^{abc}} , where ϵ a b c {\displaystyle \epsilon ^{abc}} is the antisymmetric Levi-Civita symbol .
In Cartesian coordinates, the divergence of a continuously differentiable vector field = + + is the scalar-valued function: = = (, , ) (, , ) = + +.. As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge.
The cross product also satisfies the Jacobi identity. Lie algebras are algebras satisfying anticommutativity and the Jacobi identity. Algebras of vector fields on a differentiable manifold (if K is R or the complex numbers C) or an algebraic variety (for general K); Jordan algebras are algebras which satisfy the commutative law and the Jordan ...
Thus, by the double-duality isomorphism (more precisely, by the double-duality monomorphism, since the vector space needs not be finite-dimensional), the Jacobi identity is satisfied. In particular, note that this proof demonstrates that the cocycle condition d 2 = 0 {\displaystyle d^{2}=0} is in a sense dual to the Jacobi identity.
The following are important identities in vector algebra.Identities that only involve the magnitude of a vector ‖ ‖ and the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension, while identities that use the cross product (vector product) A×B only apply in three dimensions, since the cross product is only defined there.