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Lie algebra

A Lie algebra (pronounced as "lee", named in honor of Sophus Lie) is an algebraic structure in mathematics whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds.

Table of contents

Definition

A Lie algebra is a vector space g over some field F (typically the real or complex numbers) together with a binary operation [·, ·] : g × g -> g, called the Lie bracket, which satisfies the following properties:

  • it is bilinear, i.e., [a x + b y, z] = a [x, z] + b [y, z] and [z, a x + b y] = a [z, x] + b [z, y] for all a, b in F and all x, y, z in g.
  • it satisfies the Jacobi identity, i.e., [[x, y], z] + [[z, x], y] + [[y, z], x] = 0 for all x, y, z in g.
  • [x, x] = 0 for all x in g.

Note that the first and third property together imply [x, y] = - [y, x] for all x, y in g ("anti-symmetry"). Note also that the multiplication represented by the Lie bracket is not in general associative, that is, [[x, y], z] need not equal [x, [y, z]].

Examples

Every vector space becomes a (rather uninteresting) Lie algebra if we define the Lie bracket to be identically zero.

Euclidean space R3 becomes a Lie algebra with the Lie bracket given by the cross-product of vectors.

If an associative algebra A with multiplication * is given, it can be turned into a Lie algebra by defining [xy] = x * y - y * x. This expression is called the commutator of x and y. Conversely, it can be shown that every Lie algebra can be embedded into one that arises from an associative algebra in this fashion.

Other important examples of Lie algebras come from differential topology: the vector fields on a differentiable manifold form an infinite dimensional Lie algebra; for two vector fields X and Y, the Lie bracked [X, Y] is defined by

[X, Y] f = (XY - YX) f for every function f on the manifold
(here we view vector fields as operators that turn functions on a manifold into other functions).

The vector space of left-invariant vector fields on a Lie group is closed under this operation and is therefore a finite dimensional Lie algebra. One may alternatively think of the underlying vector space of the Lie algebra belonging to a Lie group as the tangent space at the group's identity element. The multiplication is the differential of the group commutator, (a,b) |-> aba-1b-1, at the identity element.

As a concrete example, consider the Lie group SL(n,R) of all n-by-n matrices with real entries and determinant 1. The tangent space at the identity matrix may be identified with the space of all real n-by-n matrices with trace 0, and the Lie algebra structure coming from the Lie group coincides with the one arising from commutators of matrix multiplication.

For more examples of Lie groups and their associated Lie algebras, see the Lie group article.

Homomorphisms, Subalgebras and Ideals

A homomorphism φ : g -> h between Lie algebras g and h over the same base field F is an F-linear map such that [φ(x), φ(y)] = [xy] for all x and y in g. The composition of such homomorphisms is again a homomorphisms, and the Lie algebras over the field F, together with these morphisms, form a category. If such a homomorphism is bijective, it is called an isomorphism, and the two Lie algebras g and h are called isomorphic. For all practical purposes, isomorphical Lie algebraas are identical.

A subalgebra of the Lie algebra g is a subspace[?] h of g such that [xy] ∈ h for all x, yh. The subalgebra is then itself a Lie algebra.

An ideal of the Lie algebra g is a subspace h of g such that [ay] ∈ h for all ag and yh. All ideals are subalgebras. If h is an ideal of g, then the quotient space g/h becomes a Lie algebra by defining [x + h, y + h] = [x, y] for all x, yg. The ideals are precisely the kernels of homomorphisms, and the fundamental theorem on homomorphisms is valid for Lie algebras.

Classification of Lie Algebras

Real and complex Lie algebras can be classified to some extent, and this classification helps in understanding Lie groups, which are the truly interesting objects in geometry, mathematical analysis and physics since they capture symmetries of analytical structures. Lie algebras were originally introduced and studied by Sophus Lie and independently by Wilhelm Killing[?] starting in the 1870s for this reason.

A Lie algebra is abelian if the Lie bracket vanishes, i.e. [x, y] = 0 for all x and y. An abelian subalgebra of a Lie algebra is often called a torus. A maximal torus is also called a Cartan subalgebra.

A Lie algeba g is solvable or nilpotent if the lower central series[?]

g > [g, g] > [[g, g], g] > [[[g, g], g], g] > ...
becomes zero eventually. By Engels theorem, a Lie algebra is solvable iff for every u in g the map ad(u): g -> g defined by
ad(u)(v) = [u,v]
is nilpotent. A maximal solvable subalgebra is called a Borel subalgebra.

A Lie algebra g is called semi simple if one of the following equivalent conditions is satisfied

  • the Lie algebra homomorphism ad : g ->End(g) is injective.
  • the Killing form K(u,v) = tr(ad(u)ad(v)) is non degenerate; here tr denotes the trace operator.
  • every representation[?] is completely reducible, that is for every invariant subspace of the representation there is an invariant complement (theorem XXXX by whom XXXX ???)

A Lie Algebra is simple if it has no non-trivial ideals. In particular since the center Z must be trivial, a simple Lie Algebra is semi simple.

Semi simple Lie algebras are classified through the representations of their Cartan subalgebras, and more particular their Root system[?].

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