Lie algebra


In mathematics, a Lie algebra is a vector space together with an operation called the Lie bracket, an alternating bilinear map, that satisfies the Jacobi identity. In other words, a Lie algebra is an algebra over a field for which the multiplication operation is alternating and satisfies the Jacobi identity. The Lie bracket of two vectors and is denoted. A Lie algebra is typically a non-associative algebra. However, every associative algebra gives rise to a Lie algebra, consisting of the same vector space with the commutator Lie bracket,.
Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: every Lie group gives rise to a Lie algebra, which is the tangent space at the identity. Conversely, to any finite-dimensional Lie algebra over the real or complex numbers, there is a corresponding connected Lie group, unique up to covering spaces. This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras, which are simpler objects of linear algebra.
In more detail: for any Lie group, the multiplication operation near the identity element 1 is commutative to first order. In other words, every Lie group G is approximately a real vector space, namely the tangent space to G at the identity. To second order, the group operation may be non-commutative, and the second-order terms describing the non-commutativity of G near the identity give the structure of a Lie algebra. It is a remarkable fact that these second-order terms completely determine the group structure of G near the identity. They even determine G globally, up to covering spaces.
In physics, Lie groups appear as symmetry groups of physical systems, and their Lie algebras 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.
An elementary example is the 3-dimensional space with Lie bracket defined by the cross product This is skew-symmetric since, and instead of associativity it satisfies the Jacobi identity:
This is the Lie algebra of the Lie group of rotations of space, and each vector may be pictured as an infinitesimal rotation around the axis, with angular speed equal to the magnitude
of. The Lie bracket is a measure of the non-commutativity between two rotations. Since a rotation commutes with itself, one has the alternating property.
A fundamental example of a Lie algebra is the space of all linear maps from a vector space to itself, as discussed below. When the vector space has dimension n, this Lie algebra is called the general linear Lie algebra,. Equivalently, this is the space of all matrices. The Lie bracket is defined to be the commutator of matrices,.

History

Lie algebras were introduced to study the concept of infinitesimal transformations by Sophus Lie in the 1870s, and independently discovered by Wilhelm Killing in the 1880s. The name Lie algebra was given by Hermann Weyl in the 1930s; in older texts, the term infinitesimal group was used.

Definition of a Lie algebra

A Lie algebra is a vector space over a field together with a binary operation called the Lie bracket, satisfying the following axioms:
  • Bilinearity,
  • The Alternating property,
  • The Jacobi identity,
Given a Lie group, the Jacobi identity for its Lie algebra follows from the associativity of the group operation.
Using bilinearity to expand the Lie bracket and using the alternating property shows that for all in. Thus bilinearity and the alternating property together imply
It is customary to denote a Lie algebra by a lower-case fraktur letter such as. If a Lie algebra is associated with a Lie group, then the algebra is denoted by the fraktur version of the group's name: for example, the Lie algebra of SU is.

Generators and dimension

The dimension of a Lie algebra over a field means its dimension as a vector space. In physics, a vector space basis of the Lie algebra of a Lie group G may be called a set of generators for G. In mathematics, a set S of generators for a Lie algebra means a subset of such that any Lie subalgebra that contains S must be all of. Equivalently, is spanned by all iterated brackets of elements of S.

Basic examples

Abelian Lie algebras

A Lie algebra is called abelian if its Lie bracket is identically zero. Any vector space endowed with the identically zero Lie bracket becomes a Lie algebra. Every one-dimensional Lie algebra is abelian, by the alternating property of the Lie bracket.

The Lie algebra of matrices

  • On an associative algebra over a field with multiplication written as, a Lie bracket may be defined by the commutator. With this bracket, is a Lie algebra.
  • The endomorphism ring of an -vector space with the above Lie bracket is denoted.
  • For a field F and a positive integer n, the space of n × n matrices over F, denoted or, is a Lie algebra with bracket given by the commutator of matrices:. This is a special case of the previous example; it is a key example of a Lie algebra. It is called the general linear Lie algebra.

    Definitions

Subalgebras, ideals and homomorphisms

The Lie bracket is not required to be associative, meaning that need not be equal to. Nonetheless, much of the terminology for associative rings and algebras has analogs for Lie algebras. A Lie subalgebra is a linear subspace which is closed under the Lie bracket. An ideal is a linear subspace that satisfies the stronger condition:
In the correspondence between Lie groups and Lie algebras, subgroups correspond to Lie subalgebras, and normal subgroups correspond to ideals.
A Lie algebra homomorphism is a linear map compatible with the respective Lie brackets:
An isomorphism of Lie algebras is a bijective homomorphism.
As with normal subgroups in groups, ideals in Lie algebras are precisely the kernels of homomorphisms. Given a Lie algebra and an ideal in it, the quotient Lie algebra is defined, with a surjective homomorphism of Lie algebras. The first isomorphism theorem holds for Lie algebras: for any homomorphism of Lie algebras, the image of is a Lie subalgebra of that is isomorphic to.
For the Lie algebra of a Lie group, the Lie bracket is a kind of infinitesimal commutator. As a result, for any Lie algebra, two elements are said to commute if their bracket vanishes:.
The centralizer subalgebra of a subset is the set of elements commuting with ': that is,. The centralizer of itself is the center . Similarly, for a subspace S, the normalizer subalgebra of ' is. If is a Lie subalgebra, is the largest subalgebra such that is an ideal of.

Example

The subspace of diagonal matrices in is an abelian Lie subalgebra. Here is not an ideal in for. For example, when, this follows from the calculation:

.
Every one-dimensional linear subspace of a Lie algebra is an abelian Lie subalgebra, but it need not be an ideal.

Product and semidirect product

For two Lie algebras and, the product Lie algebra is the vector space consisting of all ordered pairs, with Lie bracket
This is the product in the category of Lie algebras. Note that the copies of and in commute with each other:
Let be a Lie algebra and an ideal of. If the canonical map splits, then is said to be a semidirect product of and,. See also semidirect sum of Lie algebras.

Derivations

For an algebra A over a field F, a derivation of A over F is a linear map that satisfies the Leibniz rule
for all. Given two derivations and, their commutator is again a derivation. This operation makes the space of all derivations of A over F into a Lie algebra.
Informally speaking, the space of derivations of A is the Lie algebra of the automorphism group of A. For this reason, spaces of derivations are a natural way to construct Lie algebras: they are the "infinitesimal automorphisms" of A. Indeed, writing out the condition that
gives exactly the definition of D being a derivation.
Example: the Lie algebra of vector fields. Let A be the ring of smooth functions on a smooth manifold X. Then a derivation of A over is equivalent to a vector field on X. This makes the space of vector fields into a Lie algebra. Informally speaking, is the Lie algebra of the diffeomorphism group of X. So the Lie bracket of vector fields describes the non-commutativity of the diffeomorphism group. An action of a Lie group G on a manifold X determines a homomorphism of Lie algebras.
A Lie algebra can be viewed as a non-associative algebra, and so each Lie algebra over a field F determines its Lie algebra of derivations,. That is, a derivation of is a linear map such that
The inner derivation associated to any is the adjoint mapping defined by. That gives a homomorphism of Lie algebras,. The image is an ideal in, and the Lie algebra of outer derivations is defined as the quotient Lie algebra,. For a semisimple Lie algebra over a field of characteristic zero, every derivation is inner. This is related to the theorem that the outer automorphism group of a semisimple Lie group is finite.
In contrast, an abelian Lie algebra has many outer derivations. Namely, for a vector space with Lie bracket zero, the Lie algebra can be identified with.

Examples