Lie algebra explained

akg

together with an operation called the Lie bracket, an alternating bilinear map

akg x akg → akg

, 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

x

and

y

is denoted

[x,y]

. 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,

[x,y]=xy-yx

.

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

ak{g}

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

ak{g}

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

ak{g}=R3

with Lie bracket defined by the cross product

[x,y]=x x y.

This is skew-symmetric since

x x y=-y x x

, and instead of associativity it satisfies the Jacobi identity:

x x (y x z)+ y x (z x x)+ z x (x x y) = 0.

This is the Lie algebra of the Lie group of rotations of space, and each vector

v\in\R3

may be pictured as an infinitesimal rotation around the axis

v

, with angular speed equal to the magnitudeof

v

. The Lie bracket is a measure of the non-commutativity between two rotations. Since a rotation commutes with itself, one has the alternating property

[x,x]=x x x=0

.

History

Lie algebras were introduced to study the concept of infinitesimal transformations by Sophus Lie in the 1870s,[1] and independently discovered by Wilhelm Killing[2] 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

ak{g}

over a field

F

together with a binary operation

[ ⋅ , ⋅ ]:ak{g} x ak{g}\toak{g}

called the Lie bracket, satisfying the following axioms:

[ax+by,z]=a[x,z]+b[y,z],

[z,ax+by]=a[z,x]+b[z,y]

for all scalars

a,b

in

F

and all elements

x,y,z

in

ak{g}

.

[x,x]=0 

for all

x

in

ak{g}

.

[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0 

for all

x,y,z

in

ak{g}

.

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

[x+y,x+y]

and using the alternating property shows that

[x,y]+[y,x]=0

for all

x,y

in

ak{g}

. Thus bilinearity and the alternating property together imply

[x,y]=-[y,x], 

for all

x,y

in

ak{g}

. If the field does not have characteristic 2, then anticommutativity implies the alternating property, since it implies

[x,x]=-[x,x].

It is customary to denote a Lie algebra by a lower-case fraktur letter such as

ak{g,h,b,n}

. 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

ak{su}(n)

.

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. (They are "infinitesimal generators" for G, so to speak.) In mathematics, a set S of generators for a Lie algebra

ak{g}

means a subset of

ak{g}

such that any Lie subalgebra (as defined below) that contains S must be all of

ak{g}

. Equivalently,

ak{g}

is spanned (as a vector space) by all iterated brackets of elements of S.

Basic examples

Abelian Lie algebras

Any vector space

V

endowed with the identically zero Lie bracket becomes a Lie algebra. Such a Lie algebra is called abelian. Every one-dimensional Lie algebra is abelian, by the alternating property of the Lie bracket.

The Lie algebra of matrices

A

over a field

F

with multiplication written as

xy

, a Lie bracket may be defined by the commutator

[x,y]=xy-yx

. With this bracket,

A

is a Lie algebra. (The Jacobi identity follows from the associativity of the multiplication on

A

.)

F

-vector space

V

with the above Lie bracket is denoted

ak{gl}(V)

.

ak{gl}(n,F)

or

ak{gl}n(F)

, is a Lie algebra with bracket given by the commutator of matrices:

[X,Y]=XY-YX

. 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 F is the real numbers,

ak{gl}(n,R)

is the Lie algebra of the general linear group

GL(n,R)

, the group of invertible n x n real matrices (or equivalently, matrices with nonzero determinant), where the group operation is matrix multiplication. Likewise,

ak{gl}(n,C)

is the Lie algebra of the complex Lie group

GL(n,C)

. The Lie bracket on

ak{gl}(n,\R)

describes the failure of commutativity for matrix multiplication, or equivalently for the composition of linear maps. For any field F,

ak{gl}(n,F)

can be viewed as the Lie algebra of the algebraic group

GL(n)

over F.

Definitions

Subalgebras, ideals and homomorphisms

The Lie bracket is not required to be associative, meaning that

[[x,y],z]

need not be equal to

[x,[y,z]]

. 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

ak{h}\subseteqak{g}

which is closed under the Lie bracket. An ideal

aki\subseteqak{g}

is a linear subspace that satisfies the stronger condition:[3]

[ak{g},aki]\subseteqaki.

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:

\phi\colonak{g}\toak{h},   \phi([x,y])=[\phi(x),\phi(y)] forall x,y\inakg.

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

ak{g}

and an ideal

aki

in it, the quotient Lie algebra

ak{g}/aki

is defined, with a surjective homomorphism

ak{g}\toak{g}/ak{i}

of Lie algebras. The first isomorphism theorem holds for Lie algebras: for any homomorphism

\phi\colonak{g}\toak{h}

of Lie algebras, the image of

\phi

is a Lie subalgebra of

ak{h}

that is isomorphic to

ak{g}/ker(\phi)

.

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

x,y\inakg

are said to commute if their bracket vanishes:

[x,y]=0

.

The centralizer subalgebra of a subset

S\subsetak{g}

is the set of elements commuting with

S

: that is,

ak{z}akg(S)=\{x\inakg:[x,s]=0 foralls\inS\}

. The centralizer of

ak{g}

itself is the center

ak{z}(ak{g})

. Similarly, for a subspace S, the normalizer subalgebra of

S

 is

ak{n}akg(S)=\{x\inakg:[x,s]\inS forall s\inS\}

. If

S

is a Lie subalgebra,

ak{n}akg(S)

is the largest subalgebra such that

S

is an ideal of

ak{n}akg(S)

.

Example

The subspace

ak{t}n

of diagonal matrices in

ak{gl}(n,F)

is an abelian Lie subalgebra. (It is a Cartan subalgebra of

ak{gl}(n)

, analogous to a maximal torus in the theory of compact Lie groups.) Here

ak{t}n

is not an ideal in

ak{gl}(n)

for

n\geq2

. For example, when

n=2

, this follows from the calculation:

\begin{align} \left[ \begin{bmatrix} a&b\\ c&d \end{bmatrix}, \begin{bmatrix} x&0\\ 0&y \end{bmatrix} \right]&=\begin{bmatrix} ax&by\\ cx&dy\\ \end{bmatrix}-\begin{bmatrix} ax&bx\\ cy&dy\\ \end{bmatrix}\\ &=\begin{bmatrix} 0&b(y-x)\\ c(x-y)&0 \end{bmatrix} \end{align}

(which is not always in

ak{t}2

).

Every one-dimensional linear subspace of a Lie algebra

ak{g}

is an abelian Lie subalgebra, but it need not be an ideal.

Product and semidirect product

For two Lie algebras

ak{g}

and

ak{g'}

, the product Lie algebra is the vector space

ak{g} x ak{g'}

consisting of all ordered pairs

(x,x'),x\inak{g}, x'\inak{g'}

, with Lie bracket

[(x,x'),(y,y')]=([x,y],[x',y']).

This is the product in the category of Lie algebras. Note that the copies of

akg

and

akg'

in

ak{g} x ak{g'}

commute with each other:

[(x,0),(0,x')]=0.

Let

ak{g}

be a Lie algebra and

ak{i}

an ideal of

ak{g}

. If the canonical map

ak{g}\toak{g}/ak{i}

splits (i.e., admits a section

ak{g}/ak{i}\toak{g}

, as a homomorphism of Lie algebras), then

ak{g}

is said to be a semidirect product of

ak{i}

and

ak{g}/ak{i}

,

ak{g}=ak{g}/ak{i}\ltimesak{i}

. 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

D\colonA\toA

that satisfies the Leibniz rule

D(xy)=D(x)y+xD(y)

for all

x,y\inA

. (The definition makes sense for a possibly non-associative algebra.) Given two derivations

D1

and

D2

, their commutator

[D1,D2]:=D1D2-D2D1

is again a derivation. This operation makes the space

Derk(A)

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

(1+\epsilonD)(xy)\equiv(1+\epsilonD)(x) ⋅ (1+\epsilonD)(y)\pmod{\epsilon2}

(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

Cinfty(X)

of smooth functions on a smooth manifold X. Then a derivation of A over

R

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

Vect(X)

of vector fields into a Lie algebra (see Lie bracket of vector fields). Informally speaking,

Vect(X)

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

ak{g}\toVect(X)

. (An example is illustrated below.)

A Lie algebra can be viewed as a non-associative algebra, and so each Lie algebra

ak{g}

over a field F determines its Lie algebra of derivations,

DerF(ak{g})

. That is, a derivation of

ak{g}

is a linear map

D\colonak{g}\toak{g}

such that

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

.The inner derivation associated to any

x\inakg

is the adjoint mapping

adx

defined by

adx(y):=[x,y]

. (This is a derivation as a consequence of the Jacobi identity.) That gives a homomorphism of Lie algebras,

\operatorname{ad}\colonak{g}\toDerF(ak{g})

. The image

InnF(ak{g})

is an ideal in

DerF(ak{g})

, and the Lie algebra of outer derivations is defined as the quotient Lie algebra,

OutF(ak{g})=DerF(ak{g})/InnF(ak{g})

. (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

V

with Lie bracket zero, the Lie algebra

OutF(V)

can be identified with

ak{gl}(V)

.

Examples

Matrix Lie algebras

A matrix group is a Lie group consisting of invertible matrices,

G\subsetGL(n,R)

, where the group operation of G is matrix multiplication. The corresponding Lie algebra

akg

is the space of matrices which are tangent vectors to G inside the linear space

Mn(R)

: this consists of derivatives of smooth curves in G at the identity matrix

I

:

ak{g}=\{X=c'(0)\inMn(R):smoothc:R\toG, c(0)=I\}.

The Lie bracket of

ak{g}

is given by the commutator of matrices,

[X,Y]=XY-YX

. Given a Lie algebra

ak{g}\subsetak{gl}(n,R)

, one can recover the Lie group as the subgroup generated by the matrix exponential of elements of

ak{g}

. (To be precise, this gives the identity component of G, if G is not connected.) Here the exponential mapping

\exp:Mn(R)\toMn(R)

is defined by

\exp(X)=I+X+\tfrac{1}{2!}X2+\tfrac{1}{3!}X3+ …

, which converges for every matrix

X

.

The same comments apply to complex Lie subgroups of

GL(n,C)

and the complex matrix exponential,

\exp:Mn(C)\toMn(C)

(defined by the same formula).

Here are some matrix Lie groups and their Lie algebras.

SL(n,R)

consists of all real matrices with determinant 1. This is the group of linear maps from

Rn

to itself that preserve volume and orientation. More abstractly,

SL(n,R)

is the commutator subgroup of the general linear group

GL(n,\R)

. Its Lie algebra

ak{sl}(n,R)

consists of all real matrices with trace 0. Similarly, one can define the analogous complex Lie group

{\rmSL}(n,C)

and its Lie algebra

ak{sl}(n,C)

.

O(n)

plays a basic role in geometry: it is the group of linear maps from

Rn

to itself that preserve the length of vectors. For example, rotations and reflections belong to

O(n)

. Equivalently, this is the group of n x n orthogonal matrices, meaning that

AT=A-1

, where

AT

denotes the transpose of a matrix. The orthogonal group has two connected components; the identity component is called the special orthogonal group

SO(n)

, consisting of the orthogonal matrices with determinant 1. Both groups have the same Lie algebra

ak{so}(n)

, the subspace of skew-symmetric matrices in

ak{gl}(n,R)

(

X\rm=-X

). See also infinitesimal rotations with skew-symmetric matrices.

The complex orthogonal group

O(n,C)

, its identity component

SO(n,C)

, and the Lie algebra

ak{so}(n,C)

are given by the same formulas applied to n x n complex matrices. Equivalently,

O(n,C)

is the subgroup of

GL(n,C)

that preserves the standard symmetric bilinear form on

Cn

.

U(n)

is the subgroup of

GL(n,C)

that preserves the length of vectors in

Cn

(with respect to the standard Hermitian inner product). Equivalently, this is the group of n × n unitary matrices (satisfying

A*=A-1

, where

A*

denotes the conjugate transpose of a matrix). Its Lie algebra

ak{u}(n)

consists of the skew-hermitian matrices in

ak{gl}(n,C)

(

X*=-X

). This is a Lie algebra over

R

, not over

C

. (Indeed, i times a skew-hermitian matrix is hermitian, rather than skew-hermitian.) Likewise, the unitary group

U(n)

is a real Lie subgroup of the complex Lie group

GL(n,C)

. For example,

U(1)

is the circle group, and its Lie algebra (from this point of view) is

iR\subsetC=ak{gl}(1,C)

.

SU(n)

is the subgroup of matrices with determinant 1 in

U(n)

. Its Lie algebra

su(n)

consists of the skew-hermitian matrices with trace zero.

Sp(2n,\R)

is the subgroup of

GL(2n,R)

that preserves the standard alternating bilinear form on

R2n

. Its Lie algebra is the symplectic Lie algebra

ak{sp}(2n,R)

.

Two dimensions

Some Lie algebras of low dimension are described here. See the classification of low-dimensional real Lie algebras for further examples.

ak{g}

of dimension 2 over any field F, up to isomorphism. Here

ak{g}

has a basis

X,Y

for which the bracket is given by

\left[X,Y\right]=Y

. (This determines the Lie bracket completely, because the axioms imply that

[X,X]=0

and

[Y,Y]=0

.) Over the real numbers,

ak{g}

can be viewed as the Lie algebra of the Lie group

G=Aff(1,R)

of affine transformations of the real line,

x\mapstoax+b

.

The affine group G can be identified with the group of matrices

\left(\begin{array}{cc}a&b\ 0&1\end{array}\right)

under matrix multiplication, with

a,b\inR

,

a ≠ 0

. Its Lie algebra is the Lie subalgebra

ak{g}

of

ak{gl}(2,R)

consisting of all matrices

\left(\begin{array}{cc}c&d\ 0&0\end{array}\right).

In these terms, the basis above for

ak{g}

is given by the matrices

X=\left(\begin{array}{cc}1&0\ 0&0\end{array}\right),    Y=\left(\begin{array}{cc}0&1\ 0&0\end{array}\right).

For any field

F

, the 1-dimensional subspace

F ⋅ Y

is an ideal in the 2-dimensional Lie algebra

ak{g}

, by the formula

[X,Y]=Y\inF ⋅ Y

. Both of the Lie algebras

F ⋅ Y

and

ak{g}/(F ⋅ Y)

are abelian (because 1-dimensional). In this sense,

ak{g}

can be broken into abelian "pieces", meaning that it is solvable (though not nilpotent), in the terminology below.

Three dimensions

ak{h}3(F)

over a field F is the three-dimensional Lie algebra with a basis

X,Y,Z

such that

[X,Y]=Z,   [X,Z]=0,   [Y,Z]=0

.

It can be viewed as the Lie algebra of 3×3 strictly upper-triangular matrices, with the commutator Lie bracket and the basis

X=\left(\begin{array}{ccc} 0&1&0\\ 0&0&0\\ 0&0&0 \end{array}\right),   Y=\left(\begin{array}{ccc} 0&0&0\\ 0&0&1\\ 0&0&0 \end{array}\right),   Z=\left(\begin{array}{ccc} 0&0&1\\ 0&0&0\\ 0&0&0 \end{array}\right)~.  

Over the real numbers,

ak{h}3(R)

is the Lie algebra of the Heisenberg group

H3(R)

, that is, the group of matrices

\left(\begin{array}{ccc} 1&a&c\\ 0&1&b\\ 0&0&1 \end{array}\right)

under matrix multiplication.

For any field F, the center of

ak{h}3(F)

is the 1-dimensional ideal

F ⋅ Z

, and the quotient

ak{h}3(F)/(F ⋅ Z)

is abelian, isomorphic to

F2

. In the terminology below, it follows that

ak{h}3(F)

is nilpotent (though not abelian).

ak{so}(3)

of the rotation group SO(3) is the space of skew-symmetric 3 x 3 matrices over

R

. A basis is given by the three matrices

F1=\left(\begin{array}{ccc} 0&0&0\\ 0&0&-1\\ 0&1&0 \end{array}\right),   F2=\left(\begin{array}{ccc} 0&0&1\\ 0&0&0\\ -1&0&0 \end{array}\right),   F3=\left(\begin{array}{ccc} 0&-1&0\\ 1&0&0\\ 0&0&0 \end{array}\right)~.  

The commutation relations among these generators are

[F1,F2]=F3,

[F2,F3]=F1,

[F3,F1]=F2.

The cross product of vectors in

R3

is given by the same formula in terms of the standard basis; so that Lie algebra is isomorphic to

ak{so}(3)

. Also,

ak{so}(3)

is equivalent to the Spin (physics) angular-momentum component operators for spin-1 particles in quantum mechanics.

The Lie algebra

ak{so}(3)

cannot be broken into pieces in the way that the previous examples can: it is simple, meaning that it is not abelian and its only ideals are 0 and all of

ak{so}(3)

.

C

, is the space

ak{sl}(2,C)

of 2 x 2 matrices of trace zero. A basis is given by the three matrices

H=\left(\begin{array}{cc}1&0\ 0&-1\end{array}\right), E=\left (\begin{array}{cc}0&1\ 0&0\end{array}\right), F=\left(\begin{array}{cc}0&0\ 1&0\end{array}\right).

The Lie bracket is given by:

[H,E]=2E,

[H,F]=-2F,

[E,F]=H.

Using these formulas, one can show that the Lie algebra

ak{sl}(2,C)

is simple, and classify its finite-dimensional representations (defined below). In the terminology of quantum mechanics, one can think of E and F as raising and lowering operators. Indeed, for any representation of

ak{sl}(2,C)

, the relations above imply that E maps the c-eigenspace of H (for a complex number c) into the

(c+2)

-eigenspace, while F maps the c-eigenspace into the

(c-2)

-eigenspace.

The Lie algebra

ak{sl}(2,C)

is isomorphic to the complexification of

ak{so}(3)

, meaning the tensor product

ak{so}(3) ⊗ RC

. The formulas for the Lie bracket are easier to analyze in the case of

ak{sl}(2,C)

. As a result, it is common to analyze complex representations of the group

SO(3)

by relating them to representations of the Lie algebra

ak{sl}(2,C)

.

Infinite dimensions

R

.

C

, with structure much like that of the finite-dimensional simple Lie algebras (such as

ak{sl}(n,\C)

).

V\mapstoL(V)

, called the free Lie algebra on a vector space V. It is spanned by all iterated Lie brackets of elements of V, modulo only the relations coming from the definition of a Lie algebra. The free Lie algebra

L(V)

is infinite-dimensional for V of dimension at least 2.

Representations

See main article: Lie algebra representation.

Definitions

Given a vector space V, let

ak{gl}(V)

denote the Lie algebra consisting of all linear maps from V to itself, with bracket given by

[X,Y]=XY-YX

. A representation of a Lie algebra

ak{g}

on V is a Lie algebra homomorphism

\pi\colonakg\toak{gl}(V).

That is,

\pi

sends each element of

ak{g}

to a linear map from V to itself, in such a way that the Lie bracket on

ak{g}

corresponds to the commutator of linear maps.

A representation is said to be faithful if its kernel is zero. Ado's theorem states that every finite-dimensional Lie algebra over a field of characteristic zero has a faithful representation on a finite-dimensional vector space. Kenkichi Iwasawa extended this result to finite-dimensional Lie algebras over a field of any characteristic. Equivalently, every finite-dimensional Lie algebra over a field F is isomorphic to a Lie subalgebra of

ak{gl}(n,F)

for some positive integer n.

Adjoint representation

For any Lie algebra

ak{g}

, the adjoint representation is the representation

\operatorname{ad}\colonak{g}\toak{gl}(ak{g})

given by

\operatorname{ad}(x)(y)=[x,y]

. (This is a representation of

ak{g}

by the Jacobi identity.)

Goals of representation theory

One important aspect of the study of Lie algebras (especially semisimple Lie algebras, as defined below) is the study of their representations. Although Ado's theorem is an important result, the primary goal of representation theory is not to find a faithful representation of a given Lie algebra

ak{g}

. Indeed, in the semisimple case, the adjoint representation is already faithful. Rather, the goal is to understand all possible representations of

ak{g}

. For a semisimple Lie algebra over a field of characteristic zero, Weyl's theorem says that every finite-dimensional representation is a direct sum of irreducible representations (those with no nontrivial invariant subspaces). The finite-dimensional irreducible representations are well understood from several points of view; see the representation theory of semisimple Lie algebras and the Weyl character formula.

Universal enveloping algebra

See main article: Universal enveloping algebra. The functor that takes an associative algebra A over a field F to A as a Lie algebra (by

[X,Y]:=XY-YX

) has a left adjoint

ak{g}\mapstoU(ak{g})

, called the universal enveloping algebra. To construct this: given a Lie algebra

ak{g}

over F, let

T(ak{g})=F ⊕ ak{g} ⊕ (ak{g} ⊗ ak{g}) ⊕ (ak{g} ⊗ ak{g} ⊗ ak{g}) ⊕ …

be the tensor algebra on

ak{g}

, also called the free associative algebra on the vector space

ak{g}

. Here

⊗

denotes the tensor product of F-vector spaces. Let I be the two-sided ideal in

T(ak{g})

generated by the elements

XY-YX-[X,Y]

for

X,Y\inak{g}

; then the universal enveloping algebra is the quotient ring

U(ak{g})=T(ak{g})/I

. It satisfies the Poincaré–Birkhoff–Witt theorem: if

e1,\ldots,en

is a basis for

ak{g}

as an F-vector space, then a basis for

U(ak{g})

is given by all ordered products
i1
e
1

…

in
e
n
with

i1,\ldots,in

natural numbers. In particular, the map

ak{g}\toU(ak{g})

is injective.

Representations of

ak{g}

are equivalent to modules over the universal enveloping algebra. The fact that

ak{g}\toU(ak{g})

is injective implies that every Lie algebra (possibly of infinite dimension) has a faithful representation (of infinite dimension), namely its representation on

U(ak{g})

. This also shows that every Lie algebra is contained in the Lie algebra associated to some associative algebra.

Representation theory in physics

The representation theory of Lie algebras plays an important role in various parts of theoretical physics. There, one considers operators on the space of states that satisfy certain natural commutation relations. These commutation relations typically come from a symmetry of the problem—specifically, they are the relations of the Lie algebra of the relevant symmetry group. An example is the angular momentum operators, whose commutation relations are those of the Lie algebra

ak{so}(3)

of the rotation group

SO(3)

. Typically, the space of states is far from being irreducible under the pertinent operators, butone can attempt to decompose it into irreducible pieces. In doing so, one needs to know the irreducible representations of the given Lie algebra. In the study of the hydrogen atom, for example, quantum mechanics textbooks classify (more or less explicitly) the finite-dimensional irreducible representations of the Lie algebra

ak{so}(3)

.

Structure theory and classification

Lie algebras can be classified to some extent. This is a powerful approach to the classification of Lie groups.

Abelian, nilpotent, and solvable

Analogously to abelian, nilpotent, and solvable groups, one can define abelian, nilpotent, and solvable Lie algebras.

A Lie algebra

ak{g}

is abelian if the Lie bracket vanishes; that is, [''x'',''y''] = 0 for all x and y in

ak{g}

. In particular, the Lie algebra of an abelian Lie group (such as the group

Rn

under addition or the torus group

Tn

) is abelian. Every finite-dimensional abelian Lie algebra over a field

F

is isomorphic to

Fn

for some

n\geq0

, meaning an n-dimensional vector space with Lie bracket zero.

A more general class of Lie algebras is defined by the vanishing of all commutators of given length. First, the commutator subalgebra (or derived subalgebra) of a Lie algebra

ak{g}

is

[ak{g},ak{g}]

, meaning the linear subspace spanned by all brackets

[x,y]

with

x,y\inak{g}

. The commutator subalgebra is an ideal in

ak{g}

, in fact the smallest ideal such that the quotient Lie algebra is abelian. It is analogous to the commutator subgroup of a group.

A Lie algebra

ak{g}

is nilpotent if the lower central series

ak{g}\supseteq[ak{g},ak{g}]\supseteq[[ak{g},ak{g}],ak{g}]\supseteq[[[ak{g},ak{g}],ak{g}],ak{g}]\supseteq …

becomes zero after finitely many steps. Equivalently,

ak{g}

is nilpotent if there is a finite sequence of ideals in

ak{g}

,

0=ak{a}0\subseteqak{a}1\subseteq … \subseteqak{a}r=ak{g},

such that

ak{a}j/ak{a}j-1

is central in

ak{g}/ak{a}j-1

for each j. By Engel's theorem, a Lie algebra over any field is nilpotent if and only if for every u in

ak{g}

the adjoint endomorphism

\operatorname{ad}(u):ak{g}\toak{g},   \operatorname{ad}(u)v=[u,v]

is nilpotent.

More generally, a Lie algebra

ak{g}

is said to be solvable if the derived series:

ak{g}\supseteq[ak{g},ak{g}]\supseteq[[ak{g},ak{g}],[ak{g},ak{g}]]\supseteq[[[ak{g},ak{g}],[ak{g},ak{g}]],[[ak{g},ak{g}],[ak{g},ak{g}]]]\supseteq …

becomes zero after finitely many steps. Equivalently,

ak{g}

is solvable if there is a finite sequence of Lie subalgebras,

0=ak{m}0\subseteqak{m}1\subseteq … \subseteqak{m}r=ak{g},

such that

ak{m}j-1

is an ideal in

ak{m}j

with

ak{m}j/ak{m}j-1

abelian for each j.

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

R

.

For example, for a positive integer n and a field F of characteristic zero, the radical of

ak{gl}(n,F)

is its center, the 1-dimensional subspace spanned by the identity matrix. An example of a solvable Lie algebra is the space

ak{b}n

of upper-triangular matrices in

ak{gl}(n)

; this is not nilpotent when

n\geq2

. An example of a nilpotent Lie algebra is the space

ak{u}n

of strictly upper-triangular matrices in

ak{gl}(n)

; this is not abelian when

n\geq3

.

Simple and semisimple

See main article: Semisimple Lie algebra. A Lie algebra

ak{g}

is called simple if it is not abelian and the only ideals in

ak{g}

are 0 and

ak{g}

. (In particular, a one-dimensional—necessarily abelian—Lie algebra

ak{g}

is by definition not simple, even though its only ideals are 0 and

ak{g}

.) A finite-dimensional Lie algebra

ak{g}

is called semisimple if the only solvable ideal in

ak{g}

is 0. In characteristic zero, a Lie algebra

ak{g}

is semisimple if and only if it is isomorphic to a product of simple Lie algebras,

ak{g}\congak{g}1 x … x ak{g}r

.

For example, the Lie algebra

ak{sl}(n,F)

is simple for every

n\geq2

and every field F of characteristic zero (or just of characteristic not dividing n). The Lie algebra

ak{su}(n)

over

R

is simple for every

n\geq2

. The Lie algebra

ak{so}(n)

over

R

is simple if

n=3

or

n\geq5

. (There are "exceptional isomorphisms"

ak{so}(3)\congak{su}(2)

and

ak{so}(4)\congak{su}(2) x ak{su}(2)

.)

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, every finite-dimensional representation of a semisimple Lie algebra is semisimple (that is, a direct sum of irreducible representations).

A finite-dimensional Lie algebra over a field of characteristic zero is called reductive if its adjoint representation is semisimple. Every reductive Lie algebra is isomorphic to the product of an abelian Lie algebra and a semisimple Lie algebra.

For example,

ak{gl}(n,F)

is reductive for F of characteristic zero: for

n\geq2

, it is isomorphic to the product

ak{gl}(n,F)\congF x ak{sl}(n,F),

where F denotes the center of

ak{gl}(n,F)

, the 1-dimensional subspace spanned by the identity matrix. Since the special linear Lie algebra

ak{sl}(n,F)

is simple,

ak{gl}(n,F)

contains few ideals: only 0, the center F,

ak{sl}(n,F)

, and all of

ak{gl}(n,F)

.

Cartan's criterion

Cartan's criterion (by Élie Cartan) gives conditions for a finite-dimensional Lie algebra of characteristic zero to be solvable or semisimple. It is expressed in terms of the Killing form, the symmetric bilinear form on

ak{g}

defined by

K(u,v)=\operatorname{tr}(\operatorname{ad}(u)\operatorname{ad}(v)),

where tr denotes the trace of a linear operator. Namely: a Lie algebra

ak{g}

is semisimple if and only if the Killing form is nondegenerate. A Lie algebra

ak{g}

is solvable if and only if

K(ak{g},[ak{g},ak{g}])=0.

Classification

The Levi decomposition asserts that every finite-dimensional Lie algebra over a field of characteristic zero is a semidirect product of its solvable radical and a semisimple Lie algebra. Moreover, a semisimple Lie algebra in characteristic zero is a product of simple Lie algebras, as mentioned above. This focuses attention on the problem of classifying the simple Lie algebras.

The simple Lie algebras of finite dimension over an algebraically closed field F of characteristic zero were classified by Killing and Cartan in the 1880s and 1890s, using root systems. Namely, every simple Lie algebra is of type An, Bn, Cn, Dn, E6, E7, E8, F4, or G2. Here the simple Lie algebra of type An is

ak{sl}(n+1,F)

, Bn is

ak{so}(2n+1,F)

, Cn is

ak{sp}(2n,F)

, and Dn is

ak{so}(2n,F)

. The other five are known as the exceptional Lie algebras.

The classification of finite-dimensional simple Lie algebras over

R

is more complicated, but it was also solved by Cartan (see simple Lie group for an equivalent classification). One can analyze a Lie algebra

ak{g}

over

R

by considering its complexification

ak{g} ⊗ RC

.

In the years leading up to 2004, the finite-dimensional simple Lie algebras over an algebraically closed field of characteristic

p>3

were classified by Richard Earl Block, Robert Lee Wilson, Alexander Premet, and Helmut Strade. (See restricted Lie algebra#Classification of simple Lie algebras.) It turns out that there are many more simple Lie algebras in positive characteristic than in characteristic zero.

Relation to Lie groups

See main article: Lie group–Lie algebra correspondence. Although Lie algebras can be studied in their own right, historically they arose as a means to study Lie groups.

The relationship between Lie groups and Lie algebras can be summarized as follows. Each Lie group determines a Lie algebra over

R

(concretely, the tangent space at the identity). Conversely, for every finite-dimensional Lie algebra

akg

, there is a connected Lie group

G

with Lie algebra

akg

. This is Lie's third theorem; see the Baker–Campbell–Hausdorff formula. This Lie group is not determined uniquely; however, any two Lie groups with the same Lie algebra are locally isomorphic, and more strongly, they have the same universal cover. For instance, the special orthogonal group SO(3) and the special unitary group SU(2) have isomorphic Lie algebras, but SU(2) is a simply connected double cover of SO(3).

For simply connected Lie groups, there is a complete correspondence: taking the Lie algebra gives an equivalence of categories from simply connected Lie groups to Lie algebras of finite dimension over

R

.

The correspondence between Lie algebras and Lie groups is used in several ways, including in the classification of Lie groups and the representation theory of Lie groups. For finite-dimensional representations, there is an equivalence of categories between representations of a real Lie algebra and representations of the corresponding simply connected Lie group. This simplifies the representation theory of Lie groups: it is often easier to classify the representations of a Lie algebra, using linear algebra.

Every connected Lie group is isomorphic to its universal cover modulo a discrete central subgroup. So classifying Lie groups becomes simply a matter of counting the discrete subgroups of the center, once the Lie algebra is known. For example, the real semisimple Lie algebras were classified by Cartan, and so the classification of semisimple Lie groups is well understood.

For infinite-dimensional Lie algebras, Lie theory works less well. The exponential map need not be a local homeomorphism (for example, in the diffeomorphism group of the circle, there are diffeomorphisms arbitrarily close to the identity that are not in the image of the exponential map). Moreover, in terms of the existing notions of infinite-dimensional Lie groups, some infinite-dimensional Lie algebras do not come from any group.

Lie theory also does not work so neatly for infinite-dimensional representations of a finite-dimensional group. Even for the additive group

G=R

, an infinite-dimensional representation of

G

can usually not be differentiated to produce a representation of its Lie algebra on the same space, or vice versa. The theory of Harish-Chandra modules is a more subtle relation between infinite-dimensional representations for groups and Lie algebras.

Real form and complexification

akg

, a real Lie algebra

ak{g}0

is said to be a real form of

akg

if the complexification

ak{g}0 ⊗ RC

is isomorphic to

ak{g}

. A real form need not be unique; for example,

ak{sl}(2,C)

has two real forms up to isomorphism,

ak{sl}(2,R)

and

ak{su}(2)

.

Given a semisimple complex Lie algebra

akg

, a split form of it is a real form that splits; i.e., it has a Cartan subalgebra which acts via an adjoint representation with real eigenvalues. A split form exists and is unique (up to isomorphism). A compact form is a real form that is the Lie algebra of a compact Lie group. A compact form exists and is also unique up to isomorphism.

Lie algebra with additional structures

A Lie algebra may be equipped with additional structures that are compatible with the Lie bracket. For example, a graded Lie algebra is a Lie algebra (or more generally a Lie superalgebra) with a compatible grading. A differential graded Lie algebra also comes with a differential, making the underlying vector space a chain complex.

For example, the homotopy groups of a simply connected topological space form a graded Lie algebra, using the Whitehead product. In a related construction, Daniel Quillen used differential graded Lie algebras over the rational numbers

Q

to describe rational homotopy theory in algebraic terms.

Lie ring

The definition of a Lie algebra over a field extends to define a Lie algebra over any commutative ring R. Namely, a Lie algebra

ak{g}

over R is an R-module with an alternating R-bilinear map

[ , ]\colonak{g} x ak{g}\toak{g}

that satisfies the Jacobi identity. A Lie algebra over the ring

Z

of integers is sometimes called a Lie ring. (This is not directly related to the notion of a Lie group.)

Lie rings are used in the study of finite p-groups (for a prime number p) through the Lazard correspondence. The lower central factors of a finite p-group are finite abelian p-groups. 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; see the example below.

p-adic Lie groups are related to Lie algebras over the field

Qp

of p-adic numbers as well as over the ring

Zp

of p-adic integers. Part of Claude Chevalley's construction of the finite groups of Lie type involves showing that a simple Lie algebra over the complex numbers comes from a Lie algebra over the integers, and then (with more care) a group scheme over the integers.

Examples

x,y

of a group, define the commutator

[x,y]=x-1y-1xy

. Let

G=G1\supseteqG2\supseteqG3\supseteq … \supseteqGn\supseteq …

be a filtration of a group

G

, that is, a chain of subgroups such that

[Gi,Gj]

is contained in

Gi+j

for all

i,j

. (For the Lazard correspondence, one takes the filtration to be the lower central series of G.) Then

L=oplusi\geqGi/Gi+1

is a Lie ring, with addition given by the group multiplication (which is abelian on each quotient group

Gi/Gi+1

), and with Lie bracket

Gi/Gi+1 x Gj/Gj+1\toGi+j/Gi+j+1

given by commutators in the group:

[xGi+1,yGj+1]:=[x,y]Gi+j+1.

For example, the Lie ring associated to the lower central series on the dihedral group of order 8 is the Heisenberg Lie algebra of dimension 3 over the field

Z/2Z

.

Definition using category-theoretic notation

The definition of a Lie algebra can be reformulated more abstractly in the language of category theory. Namely, one can define a Lie algebra in terms of linear maps—that is, morphisms in the category of vector spaces—without considering individual elements. (In this section, the field over which the algebra is defined is assumed to be of characteristic different from 2.)

For the category-theoretic definition of Lie algebras, two braiding isomorphisms are needed. If is a vector space, the interchange isomorphism

\tau:A ⊗ A\toA ⊗ A

is defined by

\tau(x ⊗ y)=y ⊗ x.

The cyclic-permutation braiding

\sigma:A ⊗ A ⊗ A\toA ⊗ A ⊗ A

is defined as

\sigma=(id ⊗ \tau)\circ(\tau ⊗ id),

where

id

is the identity morphism. Equivalently,

\sigma

is defined by

\sigma(x ⊗ y ⊗ z)=y ⊗ z ⊗ x.

With this notation, a Lie algebra can be defined as an object

A

in the category of vector spaces together with a morphism

[ ⋅ , ⋅ ]\colonA ⊗ A → A

that satisfies the two morphism equalities

[ ⋅ , ⋅ ]\circ(id+\tau)=0,

and

[ ⋅ , ⋅ ]\circ([ ⋅ , ⋅ ] ⊗ id)\circ(id+\sigma+\sigma2)=0.

See also

Sources

External links

Notes and References

  1. .
  2. .
  3. By the anticommutativity of the commutator, the notions of a left and right ideal in a Lie algebra coincide.

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Lie algebra".

Except where otherwise indicated, Everything.Explained.Today is © Copyright 2009-2024, A B Cryer, All Rights Reserved. Cookie policy.