Lie algebras capture a whole symmetry group from its tiniest motions
Turn a book a quarter-turn forward and then a quarter-turn sideways, and it ends up differently than if you reverse the order. Rotations do not commute. A Lie algebra measures exactly that mismatch for infinitely small motions, and remarkably, this small-scale information pins down the entire group of symmetries it came from.
Formally, a Lie algebra, pronounced lee, is a vector space equipped with an operation called the Lie bracket. The bracket takes two vectors and returns a third, is linear in each input, gives zero when a vector is paired with itself, and obeys a rule named the Jacobi identity in place of the familiar associative law. Any associative algebra produces one automatically: define the bracket of two elements as their commutator, the product one way minus the product the other way. Square matrices under this bracket form the general linear Lie algebra.
The friendliest example is ordinary three-dimensional space with the cross product from school physics. Each vector there can be read as an infinitely small rotation about its own direction, spinning at a rate equal to its length, and the cross product of two such vectors records how far the corresponding rotations fail to commute. A rotation always commutes with itself, which matches the bracket of any vector with itself being zero.
The link to Lie groups, groups that are also smooth curved spaces, is the heart of the subject. Near its identity element, any such group looks, to first approximation, like a flat vector space, its tangent space. The second-order corrections describing non-commutativity supply the Lie bracket, and these corrections determine the group near the identity and even globally, up to so-called covering spaces. That lets hard questions about curved groups be turned into linear algebra.
The Norwegian mathematician Sophus Lie introduced these objects in the 1870s to study infinitesimal transformations, and Wilhelm Killing found them independently in the following decade. Older books called them infinitesimal groups; Hermann Weyl coined the modern name in the 1930s. Because physical symmetries form Lie groups, the algebras now run through quantum mechanics and particle physics.
Source: Lie algebra
