Lie algebra

Definitions

A Lie algebra is a vector space g over some field F together with a binary operation [ ⋅ , ⋅ ] : g × g → g called the Lie bracket that satisfies the following axioms:

Bilinearity,

for all scalars a, b in F and all elements x, y, z in g .

Alternativity,

for all x in g .

The Jacobi identity,

for all x, y, z in g .

Using bilinearity to expand the Lie bracket [ x + y , x + y ] and using alternativity shows that [ x , y ] + [ y , x ] = 0   for all elements x, y in g , showing that bilinearity and alternativity together imply

Anticommutativity,

for all elements x, y in g . If the field's characteristic is not 2 then anticommutativity implies alternativity.

It is customary to express a Lie algebra in lower-case fraktur, like g . If a Lie algebra is associated with a Lie group, then the spelling of the Lie algebra is the same as that Lie group. For example, the Lie algebra of SU(n) is written as s u ( n ) .

Generators and dimension

Elements of a Lie algebra g are said to be generators of the Lie algebra if the smallest subalgebra of g containing them is g itself. The dimension of a Lie algebra is its dimension as a vector space over F. The cardinality of a minimal generating set of a Lie algebra is always less than or equal to its dimension.

Subalgebras, ideals and homomorphisms

The Lie bracket is not associative in general, meaning that [ [ x , y ] , z ] need not equal [ x , [ y , z ] ] . (However, it is flexible.) Nonetheless, much of the terminology that was developed in the theory of associative rings or associative algebras is commonly applied to Lie algebras. A subspace h ⊆ g that is closed under the Lie bracket is called a Lie subalgebra. If a subspace I ⊆ g satisfies a stronger condition that

[ g , I ] ⊆ I ,

then I is called an ideal in the Lie algebra g . A homomorphism between two Lie algebras (over the same base field) is a linear map that is compatible with the respective Lie brackets:

f : g → g ′ , f ( [ x , y ] ) = [ f ( x ) , f ( y ) ] ,

for all elements x and y in g . As in the theory of associative rings, ideals are precisely the kernels of homomorphisms; given a Lie algebra g and an ideal I in it, one constructs the factor (or quotient) algebra g / I , and the first isomorphism theorem holds for Lie algebras.

Let S be a subset of g . The set of elements x such that [ x , s ] = 0 for all s in S forms a subalgebra called the centralizer of S. The centralizer of g itself is called the center of g . Similar to centralizers, if S is a subspace, then the set of x such that [ x , s ] is in S for all s in S forms a subalgebra called the normalizer of S.

Direct sum and semidirect product

Given two Lie algebras g and g ′ , their direct sum is the Lie algebra consisting of the vector space g ⊕ g ′ , of the pairs ( x , x ′ ) , x ∈ g , x ′ ∈ g ′ , with the operation

[ ( x , x ′ ) , ( y , y ′ ) ] = ( [ x , y ] , [ x ′ , y ′ ] ) , x , y ∈ g , x ′ , y ′ ∈ g ′ .

Let g be a Lie algebra and i an ideal of g . If the canonical map g → g / i splits (i.e., admits a section), then g is said to be a semidirect product of i and g / i , g = g / i ⋉ i .

Levi's theorem says that a finite-dimensional Lie algebra is a semidirect product of its radical and the complementary subalgebra (Levi subalgebra).

Admits an enveloping algebra

For any associative algebra A with multiplication ∗ , one can construct a Lie algebra L(A). As a vector space, L(A) is the same as A. The Lie bracket of two elements of L(A) is defined to be their commutator in A:

[ a , b ] = a ∗ b − b ∗ a .  

The associativity of the multiplication ∗ in A implies the Jacobi identity of the commutator in L(A). For example, the associative algebra of n × n matrices over a field F gives rise to the general linear Lie algebra g l n ( F ) . The associative algebra A is called an enveloping algebra of the Lie algebra L(A). Every Lie algebra can be embedded into one that arises from an associative algebra in this fashion; see universal enveloping algebra.

Representation

Given a vector space V, let g l ( V ) denote the Lie algebra enveloped by the associative algebra of all linear endomorphisms of V. A representation of a Lie algebra g on V is a Lie algebra homomorphism

π : g → g l ( V ) .

A representation is said to be faithful if its kernel is trivial. Every finite-dimensional Lie algebra has a faithful representation on a finite-dimensional vector space (Ado's theorem).

For example,

ad : g → g l ( g )

given by ad ⁡ ( x ) ( y ) = [ x , y ] is a representation of g on the vector space g called the adjoint representation. A derivation on the Lie algebra g (in fact on any non-associative algebra) is a linear map δ : g → g that obeys the Leibniz law, that is,

δ ( [ x , y ] ) = [ δ ( x ) , y ] + [ x , δ ( y ) ]

for all x and y in the algebra. For any x, ad ⁡ ( x ) is a derivation; a consequence of the Jacobi identity. Thus, the image of ad lies in the subalgebra of g l ( g ) consisting of derivations on g . A derivation that happens to be in the image of ad is called an inner derivation. If g is semisimple, every derivation on g is inner.

Vector spaces

Any vector space V endowed with the identically zero Lie bracket becomes a Lie algebra. Such Lie algebras are called abelian, cf. below. Any one-dimensional Lie algebra over a field is abelian, by the antisymmetry of the Lie bracket.

The real vector space of all n × n skew-hermitian matrices is closed under the commutator and forms a real Lie algebra denoted u ( n ) . This is the Lie algebra of the unitary group U(n).

Subspaces

The subspace of the general linear Lie algebra g l n ( F ) consisting of matrices of trace zero is a subalgebra, the special linear Lie algebra, denoted s l n ( F ) .

Real matrix groups

Any Lie group G defines an associated real Lie algebra g =Lie(G). The definition in general is somewhat technical, but in the case of real matrix groups, it can be formulated via the exponential map, or the matrix exponent. The Lie algebra g consists of those matrices X for which exp(tX) ∈ G, ∀ real numbers t.

The Lie bracket of g is given by the commutator of matrices. As a concrete example, consider the special linear group SL(n,R), consisting of all n × n matrices with real entries and determinant 1. This is a matrix Lie group, and its Lie algebra consists of all n × n matrices with real entries and trace 0.

Three dimensions

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

The Heisenberg algebra H₃(R) is a three-dimensional Lie algebra generated by elements x, y and z with Lie brackets

It is explicitly realized as the space of 3×3 strictly upper-triangular matrices, with the Lie bracket given by the matrix commutator, Any element of the Heisenberg group is thus representable as a product of group generators, i.e., matrix exponentials of these Lie algebra generators,

The commutation relations between the x, y, and z components of the angular momentum operator in quantum mechanics are the same as those of s u ( 2 ) and s o ( 3 ) ,

(The physicist convention for Lie algebras is used in the above equations, hence the factor of i.) The Lie algebra formed by these operators have, in fact, representations of all finite dimensions.

Infinite dimensions

An important class of infinite-dimensional real Lie algebras arises in differential topology. The space of smooth vector fields on a differentiable manifold M forms a Lie algebra, where the Lie bracket is defined to be the commutator of vector fields. One way of expressing the Lie bracket is through the formalism of Lie derivatives, which identifies a vector field X with a first order partial differential operator L_X acting on smooth functions by letting L_X(f) be the directional derivative of the function f in the direction of X. The Lie bracket [X,Y] of two vector fields is the vector field defined through its action on functions by the formula:

A Kac–Moody algebra is an example of an infinite-dimensional Lie algebra.

The Moyal algebra is an infinite-dimensional Lie algebra that contains all classical Lie algebras as subalgebras.

Structure theory and classification

Lie algebras can be classified to some extent. In particular, this has an application to the classification of Lie groups.

Abelian, nilpotent, and solvable

Analogously to abelian, nilpotent, and solvable groups, defined in terms of the derived subgroups, one can define abelian, nilpotent, and solvable Lie algebras.

A Lie algebra g is abelian if the Lie bracket vanishes, i.e. [x,y] = 0, for all x and y in g . Abelian Lie algebras correspond to commutative (or abelian) connected Lie groups such as vector spaces K n or tori T n , and are all of the form k n , meaning an n-dimensional vector space with the trivial Lie bracket.

A more general class of Lie algebras is defined by the vanishing of all commutators of given length. A Lie algebra g is nilpotent if the lower central series

g > [ g , g ] > [ [ g , g ] , g ] > [ [ [ g , g ] , g ] , g ] > ⋯

becomes zero eventually. By Engel's theorem, a Lie algebra is nilpotent if and only if for every u in g the adjoint endomorphism

ad ⁡ ( u ) : g → g , ad ⁡ ( u ) v = [ u , v ]

is nilpotent.

More generally still, a Lie algebra g is said to be solvable if the derived series:

g > [ g , g ] > [ [ g , g ] , [ g , g ] ] > [ [ [ g , g ] , [ g , g ] ] , [ [ g , g ] , [ g , g ] ] ] > ⋯

becomes zero eventually.

Every finite-dimensional Lie algebra has a unique maximal solvable ideal, called its radical. Under the Lie correspondence, nilpotent (respectively, solvable) connected Lie groups correspond to nilpotent (respectively, solvable) Lie algebras.

Simple and semisimple

A Lie algebra is "simple" if it has no non-trivial ideals and is not abelian. A Lie algebra g is called semisimple if its radical is zero. Equivalently, g is semisimple if it does not contain any non-zero abelian ideals. In particular, a simple Lie algebra is semisimple. Conversely, it can be proven that any semisimple Lie algebra is the direct sum of its minimal ideals, which are canonically determined simple Lie algebras.

The concept of semisimplicity for Lie algebras is closely related with the complete reducibility (semisimplicity) of their representations. When the ground field F has characteristic zero, any finite-dimensional representation of a semisimple Lie algebra is semisimple (i.e., direct sum of irreducible representations.) In general, a Lie algebra is called reductive if the adjoint representation is semisimple. Thus, a semisimple Lie algebra is reductive.

Cartan's criterion

Cartan's criterion gives conditions for a Lie algebra to be nilpotent, solvable, or semisimple. It is based on the notion of the Killing form, a symmetric bilinear form on g defined by the formula

K ( u , v ) = tr ⁡ ( ad ⁡ ( u ) ad ⁡ ( v ) ) ,

where tr denotes the trace of a linear operator. A Lie algebra g is semisimple if and only if the Killing form is nondegenerate. A Lie algebra g is solvable if and only if K ( g , [ g , g ] ) = 0.

Classification

The Levi decomposition expresses an arbitrary Lie algebra as a semidirect sum of its solvable radical and a semisimple Lie algebra, almost in a canonical way. Furthermore, semisimple Lie algebras over an algebraically closed field have been completely classified through their root systems. However, the classification of solvable Lie algebras is a 'wild' problem, and cannot be accomplished in general.

Relation to Lie groups

Although Lie algebras are often studied in their own right, historically they arose as a means to study Lie groups.

Lie's fundamental theorems describe a relation between Lie groups and Lie algebras. In particular, any Lie group gives rise to a canonically determined Lie algebra (concretely, the tangent space at the identity); and, conversely, for any finite-dimensional Lie algebra there is a corresponding connected Lie group (Lie's third theorem; see the Baker–Campbell–Hausdorff formula). This Lie group is not determined uniquely; however, any two connected Lie groups with the same Lie algebra are locally isomorphic, and in particular, have the same universal cover. For instance, the special orthogonal group SO(3) and the special unitary group SU(2) give rise to the same Lie algebra, which is isomorphic to R³ with the cross-product, while SU(2) is a simply-connected twofold cover of SO(3).

Given a Lie group, a Lie algebra can be associated to it either by endowing the tangent space to the identity with the differential of the adjoint map, or by considering the left-invariant vector fields as mentioned in the examples. In the case of real matrix groups, the Lie algebra g consists of those matrices X for which exp(tX) ∈ G for all real numbers t, where exp is the exponential map.

Some examples of Lie algebras corresponding to Lie groups are the following:

The Lie algebra g l n ( C ) for the group G L n ( C ) is the algebra of complex n×n matrices

The Lie algebra s l n ( C ) for the group S L n ( C ) is the algebra of complex n×n matrices with trace 0

The Lie algebras o n ( R ) for the group O n ( R ) and s o n ( R ) for S O n ( R ) are both the algebra of real anti-symmetric n×n matrices (See Antisymmetric matrix: Infinitesimal rotations for a discussion)

The Lie algebra u n ( C ) for the group U n ( C ) is the algebra of skew-Hermitian complex n×n matrices while the Lie algebra s u n ( C ) for S U n ( C ) is the algebra of skew-Hermitian, traceless complex n×n matrices.

In the above examples, the Lie bracket [ X , Y ] (for X and Y matrices in the Lie algebra) is defined as [ X , Y ] = X Y − Y X .

Given a set of generators T^a, the structure constants f ^abc express the Lie brackets of pairs of generators as linear combinations of generators from the set, i.e., [T^a, T^b] = f ^abc T^c. The structure constants determine the Lie brackets of elements of the Lie algebra, and consequently nearly completely determine the group structure of the Lie group. The structure of the Lie group near the identity element is displayed explicitly by the Baker–Campbell–Hausdorff formula, an expansion in Lie algebra elements X, Y and their Lie brackets, all nested together within a single exponent, exp(tX) exp(tY) = exp(tX+tY+½ t²[X,Y] + O(t³) ).

The mapping from Lie groups to Lie algebras is functorial, which implies that homomorphisms of Lie groups lift to homomorphisms of Lie algebras, and various properties are satisfied by this lifting: it commutes with composition, it maps Lie subgroups, kernels, quotients and cokernels of Lie groups to subalgebras, kernels, quotients and cokernels of Lie algebras, respectively.

The functor L that takes each Lie group to its Lie algebra and each homomorphism to its differential is faithful and exact. It is however not an equivalence of categories: different Lie groups may have isomorphic Lie algebras (for example SO(3) and SU(2) ), and there are (infinite dimensional) Lie algebras that are not associated to any Lie group.

However, when the Lie algebra g is finite-dimensional, one can associate to it a simply connected Lie group having g as its Lie algebra. More precisely, the Lie algebra functor L has a left adjoint functor Γ from finite-dimensional (real) Lie algebras to Lie groups, factoring through the full subcategory of simply connected Lie groups. In other words, there is a natural isomorphism of bifunctors

The adjunction g → L ( Γ ( g ) ) (corresponding to the identity on Γ ( g ) ) is an isomorphism, and the other adjunction Γ ( L ( H ) ) → H is the projection homomorphism from the universal cover group of the identity component of H to H. It follows immediately that if G is simply connected, then the Lie algebra functor establishes a bijective correspondence between Lie group homomorphisms G→H and Lie algebra homomorphisms L(G)→L(H).

The universal cover group above can be constructed as the image of the Lie algebra under the exponential map. More generally, we have that the Lie algebra is homeomorphic to a neighborhood of the identity. But globally, if the Lie group is compact, the exponential will not be injective, and if the Lie group is not connected, simply connected or compact, the exponential map need not be surjective.

If the Lie algebra is infinite-dimensional, the issue is more subtle. In many instances, the exponential map is not even locally a homeomorphism (for example, in Diff(S¹), one may find diffeomorphisms arbitrarily close to the identity that are not in the image of exp). Furthermore, some infinite-dimensional Lie algebras are not the Lie algebra of any group.

The correspondence between Lie algebras and Lie groups is used in several ways, including in the classification of Lie groups and the related matter of the representation theory of Lie groups. Every representation of a Lie algebra lifts uniquely to a representation of the corresponding connected, simply connected Lie group, and conversely every representation of any Lie group induces a representation of the group's Lie algebra; the representations are in one-to-one correspondence. Therefore, knowing the representations of a Lie algebra settles the question of representations of the group.

As for classification, it can be shown that any connected Lie group with a given Lie algebra is isomorphic to the universal cover mod a discrete central subgroup. So classifying Lie groups becomes simply a matter of counting the discrete subgroups of the center, once the classification of Lie algebras is known (solved by Cartan et al. in the semisimple case).

Category theoretic definition

Using the language of category theory, a Lie algebra can be defined as an object A in Vec_k, the category of vector spaces over a field k of characteristic not 2, together with a morphism [.,.]: A ⊗ A → A, where ⊗ refers to the monoidal product of Vec_k, such that

[ ⋅ , ⋅ ] ∘ ( i d + τ A , A ) = 0

[ ⋅ , ⋅ ] ∘ ( [ ⋅ , ⋅ ] ⊗ i d ) ∘ ( i d + σ + σ 2 ) = 0

where τ(a ⊗ b) := b ⊗ a and σ is the cyclic permutation braiding (id ⊗ τ_A,A) ° (τ_A,A ⊗ id). In diagrammatic form:

Lie ring

A Lie ring arises as a generalisation of Lie algebras, or through the study of the lower central series of groups. A Lie ring is defined as a nonassociative ring with multiplication that is anticommutative and satisfies the Jacobi identity. More specifically we can define a Lie ring L to be an abelian group with an operation [ ⋅ , ⋅ ] that has the following properties:

Bilinearity:

for all x, y, z ∈ L.

The Jacobi identity:

for all x, y, z in L.

For all x in L:

Lie rings need not be Lie groups under addition. Any Lie algebra is an example of a Lie ring. Any associative ring can be made into a Lie ring by defining a bracket operator [ x , y ] = x y − y x . Conversely to any Lie algebra there is a corresponding ring, called the universal enveloping algebra.

Lie rings are used in the study of finite p-groups through the Lazard correspondence. The lower central factors of a p-group are finite abelian p-groups, so modules over Z/pZ. The direct sum of the lower central factors is given the structure of a Lie ring by defining the bracket to be the commutator of two coset representatives. The Lie ring structure is enriched with another module homomorphism, the pth power map, making the associated Lie ring a so-called restricted Lie ring.

Lie rings are also useful in the definition of a p-adic analytic groups and their endomorphisms by studying Lie algebras over rings of integers such as the p-adic integers. The definition of finite groups of Lie type due to Chevalley involves restricting from a Lie algebra over the complex numbers to a Lie algebra over the integers, and the reducing modulo p to get a Lie algebra over a finite field.

Examples

Any Lie algebra over a general ring instead of a field is an example of a Lie ring. Lie rings are not Lie groups under addition, despite the name.

Any associative ring can be made into a Lie ring by defining a bracket operator [ x , y ] = x y − y x .

For an example of a Lie ring arising from the study of groups, let G be a group with ( x , y ) = x − 1 y − 1 x y the commutator operation, and let G = G 0 ⊇ G 1 ⊇ G 2 ⊇ ⋯ ⊇ G n ⊇ ⋯ be a central series in G — that is the commutator subgroup ( G i , G j ) is contained in G i + j for any i , j . Then

is a Lie ring with addition supplied by the group operation (which will be commutative in each homogeneous part), and the bracket operation given by extended linearly. Note that the centrality of the series ensures the commutator ( x , y ) gives the bracket operation the appropriate Lie theoretic properties.