Lie bracket

A bilinear operation on a vector space that is alternating and satisfies the Jacobi identity. It abstracts the commutator of operators and defines the algebraic structure of a Lie algebra.

Vector space: A set with addition and scalar multiplication satisfying the axioms of linear algebra. A Lie bracket takes two vectors from this underlying space and returns another.
Commutator: The difference between the two possible orders of multiplying operators or elements: XY minus YX. The operator commutator is the standard model of a Lie bracket.
Lie group: A group that is also a smooth manifold, with smooth multiplication and inversion. Its tangent space at the identity carries a Lie bracket encoding infinitesimal group structure.
Associative algebra: A vector space with a bilinear multiplication whose order of grouping does not affect products. Its multiplication is associative, while a Lie bracket generally is not.
Bilinear map: A function of two vector arguments that is linear in each argument separately. Bilinearity makes the bracket linear in each of its two inputs.
Cross product: A three-dimensional vector operation whose result is perpendicular to both input vectors and changes sign when they are swapped. In three dimensions, the cross product is an example of a Lie bracket.
Hamiltonian mechanics: A formulation of classical mechanics in which motion is generated by a Hamiltonian function on phase space. Poisson brackets express the time evolution of observables in this formulation.
Jordan algebra: An algebra with a commutative product satisfying the Jordan identity. It captures a symmetrized product, contrasting with the antisymmetric Lie bracket.
Alternating bilinear map: A bilinear map that vanishes whenever its two inputs are equal. The defining alternating condition forces every vector to bracket with itself to zero.
Poisson bracket: A bilinear operation on functions on a phase space that encodes Hamiltonian dynamics. It is a Lie bracket on observables, with a direct role in classical mechanics.

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