Lie group (matrix)
A Lie group is a group whose elements form a SMOOTH SHAPE — a curved surface (manifold) — on which multiplication and inversion vary smoothly. So you can do calculus on the group: take derivatives, draw smooth paths, talk about tangent directions. It captures the idea of a CONTINUOUS symmetry, like the unbroken family of all rotations.
For us the concrete examples are matrix groups: GL(n), O(n), SO(n), U(n), SU(n), SL(n), Sp(2n). Each is a subset of matrices cut out by smooth equations (like A^T A = I), making it a smooth submanifold of M_n, and matrix multiplication and inversion are polynomial/rational hence smooth. A matrix Lie group is, by a clean theorem, just a closed subgroup of GL(n) — no extra hypotheses needed.
The payoff is that geometry and algebra fuse. Sitting at the identity element you find a TANGENT SPACE; that tangent space, with the commutator bracket, is the group's Lie algebra, and it encodes nearly all the group's structure in a flat, linear object you already know how to compute with. The matrix exponential is the bridge that climbs from the algebra back up onto the group.
A caveat on the word 'smooth manifold': it means locally the group looks like flat R^d for some fixed d, the group's dimension. SO(3) is a 3-dimensional manifold, U(1) a 1-dimensional circle, GL(n,R) an open n^2-dimensional region. Compactness, connectedness, and dimension are the first invariants you read off — and they differ sharply across the groups above.
The rotation group SO(2) is parametrized by a single angle t, tracing out a smooth circle — the simplest curved Lie group.
Named after Sophus Lie (pronounced 'Lee'), who studied continuous symmetries of differential equations. The deep slogan: a connected Lie group is almost completely determined by its Lie algebra — the curved global object is captured by a flat local one.