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Lie's theory of continuous symmetries was originally for differential equations, but turns out to be very useful for physics because symmetries are manifest in many physical systems. This is the start of a series on Lie groups, Lie algebras, and Lie brackets.

One of the key ideas in the theory of Lie groups is to replace the global object, the group, with its local or linearized version, which Lie himself called its "infinitesimal group" and which has since become known as its Lie algebra. Lie groups play an enormous role in modern geometry, on several different levels.

A Lie group is first of all a group. Secondly it is a smooth manifold which is a specific kind of geometric object. The circle and the sphere are examples of smooth manifolds. Finally the algebraic structure and the geometric structure must be compatible in a precise way.

In physics, Lie groups appear as symmetry groups of physical systems, and their Lie algebras (tangent vectors near the identity) may be thought of as infinitesimal symmetry motions. Thus Lie algebras and their representations are used extensively in physics, notably in quantum mechanics and particle physics.

A Lie algebra is a vector space g over a field F with an operation [·, ·] : g × g ? g which we call a Lie bracket, such that the following axioms are satisfied: ? It is bilinear. ? It is skew symmetric: [x, x] = 0 which implies [x, y] = ?[y, x] for all x, y ? g.

In mathematics, the mathematician Sophus Lie (/li?/ LEE) initiated lines of study involving integration of differential equations, transformation groups, and contact of spheres that have come to be called Lie theory. For instance, the latter subject is Lie sphere geometry.

The Clifford algebra Cl(V) is an associative algebra of dimension 2n when dimV=n. Then it acquires a Lie algebra structure where the Lie bracket is the (scaled) commutator. This does not change the dimension.