LieBracket

The Lie bracket is the fundamental binary operation that endows a Lie algebra with its algebraic structure. Given a vector space g over a field of characteristic zero, a Lie bracket is a bilinear map [·,·]: g × g → g that is antisymmetric and satisfies the Jacobi identity. Antisymmetric means [x,y] = -[y,x] for all x,y in g, and bilinearity means [ax+by,z] = a[x,z] + b[y,z] and [z,ax+by] = a[z,x] + b[z,y] for all scalars a,b.

The Jacobi identity is [x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0 for all x,y,z in g. This condition encodes a compatibility between the bracket and the vector-space structure, ensuring that the adjoint action ad_x(y) = [x,y] is a derivation: [x,[y,z]] = [[x,y],z] + [y,[x,z]].

Examples include: the set of all n-by-n real or complex matrices with the commutator bracket [A,B] = AB − BA, which makes gl(n) into a Lie algebra; and R^3 with the cross product, where identifying R^3 with so(3) yields a Lie algebra structure via [u,v] = u × v.

Lie algebras can have subalgebras (subspaces closed under the bracket) and ideals (subalgebras stable under bracket with all elements of g). Representations of a Lie algebra are linear actions on vector spaces that respect the bracket.

Lie algebras arise as infinitesimal objects attached to Lie groups; the bracket corresponds to the commutator of left-invariant vector fields, and many Lie groups are studied through their associated Lie algebras, via the correspondence between Lie algebras and simply connected Lie groups.