Lie algebra

If a Lie group is a curved surface, its Lie algebra is the FLAT tangent plane touching that surface at the identity. It is the linearized version of the group: the directions you can move in starting from 'do nothing'. Because it is a vector space, you can finally use all your linear-algebra tools on the group's local structure.

Concretely, the Lie algebra g of a matrix Lie group G is the set of matrices X such that exp(tX) stays in G for all real t. It is a vector space (closed under addition and scalar multiples) equipped with one extra operation, the Lie bracket [X, Y] = XY - YX, which records how the group fails to commute. A Lie algebra is exactly: a vector space plus a bilinear, antisymmetric bracket satisfying the Jacobi identity.

The standard examples drop out by differentiating the group's defining equation at the identity. Differentiating A^T A = I gives X^T + X = 0, so so(n) = { skew-symmetric matrices }. Differentiating A^* A = I gives X^* + X = 0, so u(n) = { anti-Hermitian matrices }, and su(n) adds trace 0. Differentiating det A = 1 gives trace X = 0, so sl(n) = { traceless matrices }.

Why it matters: the bracket on g determines the group multiplication near the identity (via Baker-Campbell-Hausdorff), so an enormous amount of hard nonlinear group theory becomes linear algebra plus one bracket. A caveat: g is a real vector space for real groups even when the matrices have complex entries (as for su(n)), and the bracket of skew matrices is again skew — closure under [.,.] is the nontrivial part.