In mathematics, a Lie algebra (pronounced ) 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 (called the Lie bracket) 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. (In this case, the Lie bracket measures the failure of commutativity for the Lie group.) 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 (Lie's third theorem). 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 (to first order) 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 (the Lie algebra) 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 (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.
An elementary example (not directly coming from an associative algebra) 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 (or linear maps), .
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 A Lie algebra is a vector space over a field together with a binary operation called the Lie bracket, satisfying the following axioms:
Bilinearity:
for all scalars in and all elements in . The alternating property:
for all in . The Jacobi identity:
for all in .
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 Anticommutativity:
for all in . If the field does not have characteristic 2, then anticommutativity implies the alternating property, since it implies . Derivation property, the anti commutativity of the Lie bracket allows to rewrite the Jacobi identity as a "Leibnitz rule" for :
for all in .
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(n) 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 may be called a set of generators for . (They are "infinitesimal generators" for , so to speak.) In mathematics, a set of generators for a Lie algebra means a subset of such that any Lie subalgebra (as defined below) that contains must be all of . Equivalently, is spanned (as a vector space) by all iterated brackets of elements of .
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 Jacobi identity follows from the associativity of the multiplication on .) The endomorphism ring of an -vector space with the above Lie bracket is denoted . For a field and a positive integer , the space of matrices over , 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.
When is the real numbers, is the Lie algebra of the general linear group , the group of invertible real matrices (or equivalently, matrices with nonzero determinant), where the group operation is matrix multiplication. Likewise, is the Lie algebra of the complex Lie group . The Lie bracket on describes the failure of commutativity for matrix multiplication, or equivalently for the composition of linear maps. For any field , can be viewed as the Lie algebra of the algebraic group over .
Basic constructions 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 (and also for groups) 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. (It is a Cartan subalgebra of , analogous to a maximal torus in the theory of compact Lie groups.) Here is not an ideal in for . For example, when , this follows from the calculation:
(which is not always in ).
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 (i.e., admits a section , as a homomorphism of Lie algebras), then is said to be a semidirect product of and , written . 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 . (The definition makes sense for a possibly non-associative algebra.) 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. (This is literally true when the automorphism group is a Lie group, for example when F is the real numbers and A has finite dimension as a vector space.) 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
(where 1 denotes the identity map on A) 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. (A vector field v gives a derivation of the space of smooth functions by differentiating functions in the direction of v.) This makes the space of vector fields into a Lie algebra (see Lie bracket of vector fields). 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 . (An example is illustrated below.)
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 . (This is a derivation as a consequence of the Jacobi identity.) 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, . (This is exactly analogous to the outer automorphism group of a group.) For a semisimple Lie algebra (defined below) 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 Matrix Lie algebras A matrix group is a Lie group consisting of invertible matrices, , where the group operation of G is matrix multiplication. The corresponding Lie algebra is the space of matrices which are tangent vectors to G inside the linear space : this consists of derivatives of smooth curves in G at the identity matrix :