a Lie algebra
/ lee AL-juh-bruh /
If a Lie group is the space of all symmetries, its Lie algebra is the space of infinitesimal symmetries — the velocities of motions through the identity. Imagine starting at 'do nothing' and beginning to move: the instantaneous direction of every possible motion lives in one vector space. The surprise is that this linear space remembers how the group's symmetries fail to commute, packaged as an antisymmetric product called the bracket. Linearizing the group at the identity loses size but keeps the essential algebra.
Abstractly, a Lie algebra over a field is a vector space g with a bilinear map [.,.]: g x g -> g (the bracket) that is antisymmetric, [X, Y] = -[Y, X], and satisfies the Jacobi identity [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0. The Jacobi identity is the infinitesimal shadow of associativity in the group. The canonical example is the space of all n-by-n matrices with bracket the commutator [A, B] = AB - BA; one checks antisymmetry and Jacobi by direct computation. For a Lie group G the associated Lie algebra g is the tangent space T_e G at the identity, with the bracket inherited from left-invariant vector fields (equivalently, for matrix groups, the matrix commutator). Examples: the Lie algebra of SO(3) is the space of antisymmetric 3-by-3 matrices, with bracket the commutator — equivalently R^3 with the cross product; the Lie algebra of U(n) is the skew-Hermitian matrices.
Lie algebras matter because they are linear and finite-dimensional, hence tractable, yet they capture nearly all the local structure of the group: connected, simply connected groups are determined up to isomorphism by their Lie algebra (Lie's theorems). Classification of semisimple Lie algebras via root systems is one of the triumphs of 20th-century mathematics. The honest caveat: the Lie algebra is a local object. It cannot see global topology — SO(3) and its double cover SU(2) have isomorphic Lie algebras (so(3) ~ su(2)) but are different groups, and disconnected groups like O(n) share a Lie algebra with their identity component SO(n).
so(3) ~ R^3 with the cross product: identify an antisymmetric matrix with the vector it represents (as in v x w). Then the matrix commutator of two such matrices corresponds exactly to the cross product of the vectors. The Jacobi identity for the bracket is the familiar identity a x (b x c) + b x (c x a) + c x (a x b) = 0.
The Lie algebra so(3) is just R^3 under the cross product — Jacobi is the vector-triple-product identity.
A Lie algebra is local: it cannot distinguish groups with the same identity component or differing only by covers. so(3) ~ su(2), yet SO(3) and SU(2) are not isomorphic groups.