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Algebraic structure used in analysis

"Lie bracket" redirects here. For the operation on vector fields, seeLie bracket of vector fields.

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Lie groups andLie algebras

Semisimple Lie algebra

Representation theory

Lie groups inphysics

Scientists

Algebraic structure → Ring theory Ring theory
Basic concepts Rings •Subrings •Ideal •Quotient ring •Fractional ideal •Total ring of fractions •Product of rings • Free product of associative algebras •Tensor product of algebras Ring homomorphisms •Kernel •Inner automorphism •Frobenius endomorphism Algebraic structures •Module •Associative algebra •Graded ring •Involutive ring •Category of rings •Initial ring $\mathbb {Z}$ •Terminal ring $0=\mathbb {Z} /1\mathbb {Z}$ Related structures •Field •Finite field •Non-associative ring •Lie ring •Jordan ring •Semiring •Semifield
Commutative algebra Commutative rings •Integral domain •Integrally closed domain •GCD domain •Unique factorization domain •Principal ideal domain •Euclidean domain •Field •Finite field •Polynomial ring •Formal power series ring Algebraic number theory •Algebraic number field •Integers modulon •Ring of integers •p-adic integers $\mathbb {Z} _{p}$ •p-adic numbers $\mathbb {Q} _{p}$ •Prüferp-ring $\mathbb {Z} (p^{\infty })$
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Inmathematics, aLie algebra (pronounced/liː/LEE) is avector space ${\mathfrak {g}}$ together with an operation called theLie bracket, analternating bilinear map ${\mathfrak {g}}\times {\mathfrak {g}}\rightarrow {\mathfrak {g}}$ , that satisfies theJacobi identity. In other words, a Lie algebra is analgebra 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 $x {\displaystyle x}$ and $y {\displaystyle y}$ is denoted $[x,y]$ . A Lie algebra is typically anon-associative algebra. However, everyassociative algebra gives rise to a Lie algebra, consisting of the same vector space with thecommutator Lie bracket, $[x,y]=xy-yx$ .

Lie algebras are closely related toLie groups, which aregroups that are alsosmooth manifolds: every Lie group gives rise to a Lie algebra, which is thetangent space at the identity. (In this case, the Lie bracket measures the failure ofcommutativity for the Lie group.) Conversely, to any finite-dimensional Lie algebra over thereal orcomplex numbers, there is a correspondingconnected Lie group, unique up tocovering spaces (Lie's third theorem). Thiscorrespondence allows one to study the structure andclassification 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 groupG is (to first order) approximately a real vector space, namely the tangent space ${\mathfrak {g}}$ toG at the identity. To second order, the group operation may be non-commutative, and the second-order terms describing the non-commutativity ofG near the identity give ${\mathfrak {g}}$ the structure of a Lie algebra. It is a remarkable fact that these second-order terms (the Lie algebra) completely determine the group structure ofG near the identity. They even determineG 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 inquantum mechanics and particle physics.

An elementary example (not directly coming from an associative algebra) is the 3-dimensional space ${\mathfrak {g}}=\mathbb {R} ^{3}$ with Lie bracket defined by thecross product $[x,y]=x\times y.$ This is skew-symmetric since $x\times y=-y\times x$ , and instead of associativity it satisfies the Jacobi identity:

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

This is the Lie algebra of the Lie group ofrotations of space, and each vector $v\in \mathbb {R} ^{3}$ may be pictured as an infinitesimal rotation around the axis $v {\displaystyle v}$ , with angular speed equal to the magnitudeof $v {\displaystyle v}$ . The Lie bracket is a measure of the non-commutativity between two rotations. Since a rotation commutes with itself, one has the alternating property $[x,x]=x\times x=0$ .

A fundamental example of a Lie algebra is the space of alllinear maps from a vector space to itself, as discussed below. When the vector space has dimensionn, this Lie algebra is called thegeneral linear Lie algebra, ${\mathfrak {gl}}(n)$ . Equivalently, this is the space of all $n\times n$ matrices. The Lie bracket is defined to be the commutator of matrices (or linear maps), $[X,Y]=XY-YX$ .

History

[edit]

Lie algebras were introduced to study the concept ofinfinitesimal transformations bySophus Lie in the 1870s,^[1] and independently discovered byWilhelm Killing^[2] in the 1880s. The nameLie algebra was given byHermann Weyl in the 1930s; in older texts, the terminfinitesimal group was used.

Definition of a Lie algebra

[edit]

A Lie algebra is a vector space $\,{\mathfrak {g}}$ over afield $F {\displaystyle F}$ together with abinary operation $[\,\cdot \,,\cdot \,]:{\mathfrak {g}}\times {\mathfrak {g}}\to {\mathfrak {g}}$ called the Lie bracket, satisfying the following axioms:^[a]

Bilinearity,

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

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

for all scalars

a, b {\displaystyle a,b}

F {\displaystyle F}

and all elements

x, y, z {\displaystyle x,y,z}

{\mathfrak {g}}

TheAlternating property,

[x,x]=0\

for all

x {\displaystyle x}

{\mathfrak {g}}

TheJacobi identity,

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

for all

x, y, z {\displaystyle x,y,z}

{\mathfrak {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 $x, y {\displaystyle x,y}$ in ${\mathfrak {g}}$ . Thus bilinearity and the alternating property together imply

Anticommutativity,

[x,y]=-[y,x],\

for all

x, y {\displaystyle x,y}

{\mathfrak {g}}

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

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

^[3]

Derivation property, the anti commutativity of the Lie bracket allows to rewrite the Jacobi identity as a "Leibnitz rule" for $\mathrm {ad} _{x}=[x,-]$ :

[x,[y,z]]=[[x,y],z]+[y,[x,z]],\

for all

x, y, z {\displaystyle x,y,z}

{\mathfrak {g}}

It is customary to denote a Lie algebra by a lower-casefraktur letter such as ${\mathfrak {g,h,b,n}}$ . 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 ofSU(n) is ${\mathfrak {su}}(n)$ .

Generators and dimension

[edit]

Thedimension of a Lie algebra over a field means itsdimension as a vector space. In physics, a vector spacebasis of the Lie algebra of a Lie groupG may be called a set ofgenerators forG. (They are "infinitesimal generators" forG, so to speak.) In mathematics, a setS ofgenerators for a Lie algebra ${\mathfrak {g}}$ means a subset of ${\mathfrak {g}}$ such that any Lie subalgebra (as defined below) that containsS must be all of ${\mathfrak {g}}$ . Equivalently, ${\mathfrak {g}}$ is spanned (as a vector space) by all iterated brackets of elements ofS.

Basic examples

[edit]

Abelian Lie algebras

[edit]

A Lie algebra is calledabelian if its Lie bracket is identically zero. Any vector space $V {\displaystyle V}$ 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

[edit]

On an associative algebra $A {\displaystyle A}$ over a field $F {\displaystyle F}$ with multiplication written as $x y {\displaystyle xy}$ , a Lie bracket may be defined by the commutator $[x,y]=xy-yx$ . With this bracket, $A {\displaystyle A}$ is a Lie algebra. (The Jacobi identity follows from the associativity of the multiplication on $A {\displaystyle A}$ .)^[4]
Theendomorphism ring of an $F {\displaystyle F}$ -vector space $V {\displaystyle V}$ with the above Lie bracket is denoted ${\mathfrak {gl}}(V)$ .
For a fieldF and a positive integern, the space ofn ×n matrices overF, denoted ${\mathfrak {gl}}(n,F)$ or ${\mathfrak {gl}}_{n}(F)$ , is a Lie algebra with bracket given by the commutator of matrices: $[X,Y]=XY-YX$ .^[5] This is a special case of the previous example; it is a key example of a Lie algebra. It is called thegeneral linear Lie algebra.

WhenF is the real numbers,

{\mathfrak {gl}}(n,\mathbb {R} )

is the Lie algebra of thegeneral linear group

\mathrm {GL} (n,\mathbb {R} )

, the group ofinvertiblen xn real matrices (or equivalently, matrices with nonzerodeterminant), where the group operation is matrix multiplication. Likewise,

{\mathfrak {gl}}(n,\mathbb {C} )

is the Lie algebra of the complex Lie group

\mathrm {GL} (n,\mathbb {C} )

. The Lie bracket on

{\mathfrak {gl}}(n,\mathbb {R} )

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

{\mathfrak {gl}}(n,F)

can be viewed as the Lie algebra of thealgebraic group

\mathrm {GL} (n)

overF.

Definitions

[edit]

Subalgebras, ideals and homomorphisms

[edit]

The Lie bracket is not required to beassociative, meaning that $[[x,y],z]$ need not be equal to $[x,[y,z]]$ . Nonetheless, much of the terminology for associativerings and algebras (and also for groups) has analogs for Lie algebras. ALie subalgebra is a linear subspace ${\mathfrak {h}}\subseteq {\mathfrak {g}}$ which is closed under the Lie bracket. Anideal ${\mathfrak {i}}\subseteq {\mathfrak {g}}$ is a linear subspace that satisfies the stronger condition:^[6]

[{\mathfrak {g}},{\mathfrak {i}}]\subseteq {\mathfrak {i}}.

In the correspondence between Lie groups and Lie algebras, subgroups correspond to Lie subalgebras, andnormal subgroups correspond to ideals.

A Lie algebrahomomorphism is a linear map compatible with the respective Lie brackets:

\phi \colon {\mathfrak {g}}\to {\mathfrak {h}},\quad \phi ([x,y])=[\phi (x),\phi (y)]\ {\text{for all}}\ x,y\in {\mathfrak {g}}.

Anisomorphism of Lie algebras is abijective homomorphism.

As with normal subgroups in groups, ideals in Lie algebras are precisely thekernels of homomorphisms. Given a Lie algebra ${\mathfrak {g}}$ and an ideal ${\mathfrak {i}}$ in it, thequotient Lie algebra ${\mathfrak {g}}/{\mathfrak {i}}$ is defined, with a surjective homomorphism ${\mathfrak {g}}\to {\mathfrak {g}}/{\mathfrak {i}}$ of Lie algebras. Thefirst isomorphism theorem holds for Lie algebras: for any homomorphism $\phi \colon {\mathfrak {g}}\to {\mathfrak {h}}$ of Lie algebras, the image of $\phi$ is a Lie subalgebra of ${\mathfrak {h}}$ that is isomorphic to ${\mathfrak {g}}/{\text{ker}}(\phi )$ .

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 $x,y\in {\mathfrak {g}}$ are said tocommute if their bracket vanishes: $[x,y]=0$ .

Thecentralizer subalgebra of a subset $S\subset {\mathfrak {g}}$ is the set of elements commuting with $S {\displaystyle S}$ : that is, ${\mathfrak {z}}_{\mathfrak {g}}(S)=\{x\in {\mathfrak {g}}:[x,s]=0\ {\text{ for all }}s\in S\}$ . The centralizer of ${\mathfrak {g}}$ itself is thecenter ${\mathfrak {z}}({\mathfrak {g}})$ . Similarly, for a subspaceS, thenormalizer subalgebra of $S {\displaystyle S}$ is ${\mathfrak {n}}_{\mathfrak {g}}(S)=\{x\in {\mathfrak {g}}:[x,s]\in S\ {\text{ for all}}\ s\in S\}$ .^[7] If $S {\displaystyle S}$ is a Lie subalgebra, ${\mathfrak {n}}_{\mathfrak {g}}(S)$ is the largest subalgebra such that $S {\displaystyle S}$ is an ideal of ${\mathfrak {n}}_{\mathfrak {g}}(S)$ .

Example

[edit]

The subspace ${\mathfrak {t}}_{n}$ of diagonal matrices in ${\mathfrak {gl}}(n,F)$ is an abelian Lie subalgebra. (It is aCartan subalgebra of ${\mathfrak {gl}}(n)$ , analogous to amaximal torus in the theory ofcompact Lie groups.) Here ${\mathfrak {t}}_{n}$ is not an ideal in ${\mathfrak {gl}}(n)$ for $n\geq 2$ . For example, when $n=2$ , this follows from the calculation:

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

(which is not always in ${\mathfrak {t}}_{2}$ ).

Every one-dimensional linear subspace of a Lie algebra ${\mathfrak {g}}$ is an abelian Lie subalgebra, but it need not be an ideal.

Product and semidirect product

[edit]

For two Lie algebras ${\mathfrak {g}}$ and ${\mathfrak {g'}}$ , theproduct Lie algebra is the vector space ${\mathfrak {g}}\times {\mathfrak {g'}}$ consisting of all ordered pairs $(x,x'),\,x\in {\mathfrak {g}},\ x'\in {\mathfrak {g'}}$ , with Lie bracket^[8]

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

This is the product in thecategory of Lie algebras. Note that the copies of ${\mathfrak {g}}$ and ${\mathfrak {g}}'$ in ${\mathfrak {g}}\times {\mathfrak {g'}}$ commute with each other: $[(x,0),(0,x')]=0.$

Let ${\mathfrak {g}}$ be a Lie algebra and ${\mathfrak {i}}$ an ideal of ${\mathfrak {g}}$ . If the canonical map ${\mathfrak {g}}\to {\mathfrak {g}}/{\mathfrak {i}}$ splits (i.e., admits a section ${\mathfrak {g}}/{\mathfrak {i}}\to {\mathfrak {g}}$ , as a homomorphism of Lie algebras), then ${\mathfrak {g}}$ is said to be asemidirect product of ${\mathfrak {i}}$ and ${\mathfrak {g}}/{\mathfrak {i}}$ , ${\mathfrak {g}}={\mathfrak {g}}/{\mathfrak {i}}\ltimes {\mathfrak {i}}$ . See alsosemidirect sum of Lie algebras.

Derivations

[edit]

For analgebraA over a fieldF, aderivation ofA overF is a linear map $D\colon A\to A$ that satisfies theLeibniz rule

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

for all $x,y\in A$ . (The definition makes sense for a possiblynon-associative algebra.) Given two derivations $D_{1}$ and $D_{2}$ , their commutator $[D_{1},D_{2}]:=D_{1}D_{2}-D_{2}D_{1}$ is again a derivation. This operation makes the space ${\text{Der}}_{F}(A)$ of all derivations ofA overF into a Lie algebra.^[9]

Informally speaking, the space of derivations ofA is the Lie algebra of theautomorphism group ofA. (This is literally true when the automorphism group is a Lie group, for example whenF is the real numbers andA 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" ofA. Indeed, writing out the condition that

(1+\epsilon D)(xy)\equiv (1+\epsilon D)(x)\cdot (1+\epsilon D)(y){\pmod {\epsilon ^{2}}}

(where 1 denotes the identity map onA) gives exactly the definition ofD being a derivation.

Example: the Lie algebra of vector fields. LetA be the ring $C^{\infty }(X)$ ofsmooth functions on a smooth manifoldX. Then a derivation ofA over $\mathbb {R}$ is equivalent to avector field onX. (A vector fieldv gives a derivation of the space of smooth functions by differentiating functions in the direction ofv.) This makes the space ${\text{Vect}}(X)$ of vector fields into a Lie algebra (seeLie bracket of vector fields).^[10] Informally speaking, ${\text{Vect}}(X)$ is the Lie algebra of thediffeomorphism group ofX. So the Lie bracket of vector fields describes the non-commutativity of the diffeomorphism group. Anaction of a Lie groupG on a manifoldX determines a homomorphism of Lie algebras ${\mathfrak {g}}\to {\text{Vect}}(X)$ . (An example is illustrated below.)

A Lie algebra can be viewed as a non-associative algebra, and so each Lie algebra ${\mathfrak {g}}$ over a fieldF determines its Lie algebra of derivations, ${\text{Der}}_{F}({\mathfrak {g}})$ . That is, a derivation of ${\mathfrak {g}}$ is a linear map $D\colon {\mathfrak {g}}\to {\mathfrak {g}}$ such that

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

Theinner derivation associated to any $x\in {\mathfrak {g}}$ is the adjoint mapping $\mathrm {ad} _{x}$ defined by $\mathrm {ad} _{x}(y):=[x,y]$ . (This is a derivation as a consequence of the Jacobi identity.) That gives a homomorphism of Lie algebras, $\operatorname {ad} \colon {\mathfrak {g}}\to {\text{Der}}_{F}({\mathfrak {g}})$ . The image ${\text{Inn}}_{F}({\mathfrak {g}})$ is an ideal in ${\text{Der}}_{F}({\mathfrak {g}})$ , and the Lie algebra ofouter derivations is defined as the quotient Lie algebra, ${\text{Out}}_{F}({\mathfrak {g}})={\text{Der}}_{F}({\mathfrak {g}})/{\text{Inn}}_{F}({\mathfrak {g}})$ . (This is exactly analogous to theouter automorphism group of a group.) For asemisimple Lie algebra (defined below) over a field of characteristic zero, every derivation is inner.^[11] This is related to the theorem that the outer automorphism group of a semisimple Lie group is finite.^[12]

In contrast, an abelian Lie algebra has many outer derivations. Namely, for a vector space $V {\displaystyle V}$ with Lie bracket zero, the Lie algebra ${\text{Out}}_{F}(V)$ can be identified with ${\mathfrak {gl}}(V)$ .

Examples

[edit]

Matrix Lie algebras

[edit]

Amatrix group is a Lie group consisting of invertible matrices, $G\subset \mathrm {GL} (n,\mathbb {R} )$ , where the group operation ofG is matrix multiplication. The corresponding Lie algebra ${\mathfrak {g}}$ is the space of matrices which are tangent vectors toG inside the linear space $M_{n}(\mathbb {R} )$ : this consists of derivatives of smooth curves inG at theidentity matrix $I {\displaystyle I}$ :

{\mathfrak {g}}=\{X=c'(0)\in M_{n}(\mathbb {R} ):{\text{ smooth }}c:\mathbb {R} \to G,\ c(0)=I\}.

The Lie bracket of ${\mathfrak {g}}$ is given by the commutator of matrices, $[X,Y]=XY-YX$ . Given a Lie algebra ${\mathfrak {g}}\subset {\mathfrak {gl}}(n,\mathbb {R} )$ , one can recover the Lie group as the subgroup generated by thematrix exponential of elements of ${\mathfrak {g}}$ .^[13] (To be precise, this gives theidentity component ofG, ifG is not connected.) Here the exponential mapping $\exp :M_{n}(\mathbb {R} )\to M_{n}(\mathbb {R} )$ is defined by $\exp(X)=I+X+{\tfrac {1}{2!}}X^{2}+{\tfrac {1}{3!}}X^{3}+\cdots$ , which converges for every matrix $X {\displaystyle X}$ .

The same comments apply to complex Lie subgroups of $GL(n,\mathbb {C} )$ and the complex matrix exponential, $\exp :M_{n}(\mathbb {C} )\to M_{n}(\mathbb {C} )$ (defined by the same formula).

Here are some matrix Lie groups and their Lie algebras.^[14]

For a positive integern, thespecial linear group $\mathrm {SL} (n,\mathbb {R} )$ consists of all realn × n matrices with determinant 1. This is the group of linear maps from $\mathbb {R} ^{n}$ to itself that preserve volume andorientation. More abstractly, $\mathrm {SL} (n,\mathbb {R} )$ is thecommutator subgroup of the general linear group $\mathrm {GL} (n,\mathbb {R} )$ . Its Lie algebra ${\mathfrak {sl}}(n,\mathbb {R} )$ consists of all realn × n matrices withtrace 0. Similarly, one can define the analogous complex Lie group ${\rm {SL}}(n,\mathbb {C} )$ and its Lie algebra ${\mathfrak {sl}}(n,\mathbb {C} )$ .
Theorthogonal group $\mathrm {O} (n)$ plays a basic role in geometry: it is the group of linear maps from $\mathbb {R} ^{n}$ to itself that preserve the length of vectors. For example, rotations and reflections belong to $\mathrm {O} (n)$ . Equivalently, this is the group ofn xn orthogonal matrices, meaning that $A^{\mathrm {T} }=A^{-1}$ , where $A^{\mathrm {T} }$ denotes thetranspose of a matrix. The orthogonal group has two connected components; the identity component is called thespecial orthogonal group $\mathrm {SO} (n)$ , consisting of the orthogonal matrices with determinant 1. Both groups have the same Lie algebra ${\mathfrak {so}}(n)$ , the subspace of skew-symmetric matrices in ${\mathfrak {gl}}(n,\mathbb {R} )$ ( $X^{\rm {T}}=-X$ ). See alsoinfinitesimal rotations with skew-symmetric matrices.

The complex orthogonal group

\mathrm {O} (n,\mathbb {C} )

, its identity component

\mathrm {SO} (n,\mathbb {C} )

, and the Lie algebra

{\mathfrak {so}}(n,\mathbb {C} )

are given by the same formulas applied ton xn complex matrices. Equivalently,

\mathrm {O} (n,\mathbb {C} )

is the subgroup of

\mathrm {GL} (n,\mathbb {C} )

that preserves the standardsymmetric bilinear form on

\mathbb {C} ^{n}

Theunitary group $\mathrm {U} (n)$ is the subgroup of $\mathrm {GL} (n,\mathbb {C} )$ that preserves the length of vectors in $\mathbb {C} ^{n}$ (with respect to the standardHermitian inner product). Equivalently, this is the group ofn × n unitary matrices (satisfying $A^{*}=A^{-1}$ , where $A^{*}$ denotes theconjugate transpose of a matrix). Its Lie algebra ${\mathfrak {u}}(n)$ consists of the skew-hermitian matrices in ${\mathfrak {gl}}(n,\mathbb {C} )$ ( $X^{*}=-X$ ). This is a Lie algebra over $\mathbb {R}$ , not over $\mathbb {C}$ . (Indeed,i times a skew-hermitian matrix is hermitian, rather than skew-hermitian.) Likewise, the unitary group $\mathrm {U} (n)$ is a real Lie subgroup of the complex Lie group $\mathrm {GL} (n,\mathbb {C} )$ . For example, $\mathrm {U} (1)$ is thecircle group, and its Lie algebra (from this point of view) is $i\mathbb {R} \subset \mathbb {C} ={\mathfrak {gl}}(1,\mathbb {C} )$ .
Thespecial unitary group $\mathrm {SU} (n)$ is the subgroup of matrices with determinant 1 in $\mathrm {U} (n)$ . Its Lie algebra ${\mathfrak {su}}(n)$ consists of the skew-hermitian matrices with trace zero.
Thesymplectic group $\mathrm {Sp} (2n,\mathbb {R} )$ is the subgroup of $\mathrm {GL} (2n,\mathbb {R} )$ that preserves the standardalternating bilinear form on $\mathbb {R} ^{2n}$ . Its Lie algebra is thesymplectic Lie algebra ${\mathfrak {sp}}(2n,\mathbb {R} )$ .
Theclassical Lie algebras are those listed above, along with variants over any field.

Two dimensions

[edit]

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

There is a unique nonabelian Lie algebra ${\mathfrak {g}}$ of dimension 2 over any fieldF, up to isomorphism.^[15] Here ${\mathfrak {g}}$ has a basis $X, Y {\displaystyle X,Y}$ 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, ${\mathfrak {g}}$ can be viewed as the Lie algebra of the Lie group $G=\mathrm {Aff} (1,\mathbb {R} )$ ofaffine transformations of the real line, $x\mapsto ax+b$ .

The affine groupG 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\in \mathbb {R}

a\neq 0

. Its Lie algebra is the Lie subalgebra

{\mathfrak {g}}

{\mathfrak {gl}}(2,\mathbb {R} )

consisting of all matrices

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

In these terms, the basis above for

{\mathfrak {g}}

is given by the matrices

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

For any field

F {\displaystyle F}

, the 1-dimensional subspace

F\cdot Y

is an ideal in the 2-dimensional Lie algebra

{\mathfrak {g}}

, by the formula

[X,Y]=Y\in F\cdot Y

. Both of the Lie algebras

F\cdot Y

and

{\mathfrak {g}}/(F\cdot Y)

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

{\mathfrak {g}}

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

Three dimensions

[edit]

TheHeisenberg algebra ${\mathfrak {h}}_{3}(F)$ over a fieldF is the three-dimensional Lie algebra with a basis $X, Y, Z {\displaystyle X,Y,Z}$ such that^[16]

[X,Y]=Z,\quad [X,Z]=0,\quad [Y,Z]=0

It can be viewed as the Lie algebra of 3×3 strictlyupper-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),\quad Y=\left({\begin{array}{ccc}0&0&0\\0&0&1\\0&0&0\end{array}}\right),\quad Z=\left({\begin{array}{ccc}0&0&1\\0&0&0\\0&0&0\end{array}}\right)~.\quad

Over the real numbers,

{\mathfrak {h}}_{3}(\mathbb {R} )

is the Lie algebra of theHeisenberg group

\mathrm {H} _{3}(\mathbb {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 fieldF, the center of

{\mathfrak {h}}_{3}(F)

is the 1-dimensional ideal

F\cdot Z

, and the quotient

{\mathfrak {h}}_{3}(F)/(F\cdot Z)

is abelian, isomorphic to

F^{2}

. In the terminology below, it follows that

{\mathfrak {h}}_{3}(F)

is nilpotent (though not abelian).

The Lie algebra ${\mathfrak {so}}(3)$ of therotation group SO(3) is the space of skew-symmetric 3 x 3 matrices over $\mathbb {R}$ . A basis is given by the three matrices^[17]

F_{1}=\left({\begin{array}{ccc}0&0&0\\0&0&-1\\0&1&0\end{array}}\right),\quad F_{2}=\left({\begin{array}{ccc}0&0&1\\0&0&0\\-1&0&0\end{array}}\right),\quad F_{3}=\left({\begin{array}{ccc}0&-1&0\\1&0&0\\0&0&0\end{array}}\right)~.\quad

The commutation relations among these generators are

[F_{1},F_{2}]=F_{3},

[F_{2},F_{3}]=F_{1},

[F_{3},F_{1}]=F_{2}.

The cross product of vectors in

\mathbb {R} ^{3}

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

{\mathfrak {so}}(3)

. Also,

{\mathfrak {so}}(3)

is equivalent to theSpin (physics) angular-momentum component operators for spin-1 particles inquantum mechanics.^[18]

The Lie algebra

{\mathfrak {so}}(3)

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

{\mathfrak {so}}(3)

Another simple Lie algebra of dimension 3, in this case over $\mathbb {C}$ , is the space ${\mathfrak {sl}}(2,\mathbb {C} )$ 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 action of

{\mathfrak {sl}}(2,\mathbb {C} )

on theRiemann sphere

\mathbb {CP} ^{1}

. In particular, the Lie brackets of the vector fields shown are:

[H,E]=2E

[H,F]=-2F

[E,F]=H

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

{\mathfrak {sl}}(2,\mathbb {C} )

is simple, and classify its finite-dimensional representations (defined below).^[19] In the terminology of quantum mechanics, one can think ofE andF asraising and lowering operators. Indeed, for any representation of

{\mathfrak {sl}}(2,\mathbb {C} )

, the relations above imply thatE maps thec-eigenspace ofH (for a complex numberc) into the

(c+2)

-eigenspace, whileF maps thec-eigenspace into the

(c-2)

-eigenspace.

The Lie algebra

{\mathfrak {sl}}(2,\mathbb {C} )

is isomorphic to thecomplexification of

{\mathfrak {so}}(3)

, meaning thetensor product

{\mathfrak {so}}(3)\otimes _{\mathbb {R} }\mathbb {C}

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

{\mathfrak {sl}}(2,\mathbb {C} )

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

\mathrm {SO} (3)

by relating them to representations of the Lie algebra

{\mathfrak {sl}}(2,\mathbb {C} )

Infinite dimensions

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The Lie algebra of vector fields on a smooth manifold of positive dimension is an infinite-dimensional Lie algebra over $\mathbb {R}$ .
TheKac–Moody algebras are a large class of infinite-dimensional Lie algebras, say over $\mathbb {C}$ , with structure much like that of the finite-dimensional simple Lie algebras (such as ${\mathfrak {sl}}(n,\mathbb {C} )$ ).
TheMoyal algebra is an infinite-dimensional Lie algebra that contains all theclassical Lie algebras as subalgebras.
TheVirasoro algebra is important instring theory.
The functor that takes a Lie algebra over a fieldF to the underlying vector space has aleft adjoint $V\mapsto L(V)$ , called thefree Lie algebra on a vector spaceV. It is spanned by all iterated Lie brackets of elements ofV, modulo only the relations coming from the definition of a Lie algebra. The free Lie algebra $L(V)$ is infinite-dimensional forV of dimension at least 2.^[20]

Representations

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Main article:Lie algebra representation

Definitions

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Given a vector spaceV, let ${\mathfrak {gl}}(V)$ denote the Lie algebra consisting of all linear maps fromV to itself, with bracket given by $[X,Y]=XY-YX$ . Arepresentation of a Lie algebra ${\mathfrak {g}}$ onV is a Lie algebra homomorphism

\pi \colon {\mathfrak {g}}\to {\mathfrak {gl}}(V).

That is, $\pi$ sends each element of ${\mathfrak {g}}$ to a linear map fromV to itself, in such a way that the Lie bracket on ${\mathfrak {g}}$ corresponds to the commutator of linear maps.

A representation is said to befaithful 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.^[21] Equivalently, every finite-dimensional Lie algebra over a fieldF is isomorphic to a Lie subalgebra of ${\mathfrak {gl}}(n,F)$ for some positive integern.

Adjoint representation

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For any Lie algebra ${\mathfrak {g}}$ , theadjoint representation is the representation

\operatorname {ad} \colon {\mathfrak {g}}\to {\mathfrak {gl}}({\mathfrak {g}})

given by $\operatorname {ad} (x)(y)=[x,y]$ . (This is a representation of ${\mathfrak {g}}$ by the Jacobi identity.)

Goals of representation theory

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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 ${\mathfrak {g}}$ . Indeed, in the semisimple case, the adjoint representation is already faithful. Rather, the goal is to understand all possible representations of ${\mathfrak {g}}$ . For a semisimple Lie algebra over a field of characteristic zero,Weyl's theorem^[22] 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 therepresentation theory of semisimple Lie algebras and theWeyl character formula.

Universal enveloping algebra

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Main article:Universal enveloping algebra

The functor that takes an associative algebraA over a fieldF toA as a Lie algebra (by $[X,Y]:=XY-YX$ ) has aleft adjoint ${\mathfrak {g}}\mapsto U({\mathfrak {g}})$ , called theuniversal enveloping algebra. To construct this: given a Lie algebra ${\mathfrak {g}}$ overF, let

T({\mathfrak {g}})=F\oplus {\mathfrak {g}}\oplus ({\mathfrak {g}}\otimes {\mathfrak {g}})\oplus ({\mathfrak {g}}\otimes {\mathfrak {g}}\otimes {\mathfrak {g}})\oplus \cdots

be thetensor algebra on ${\mathfrak {g}}$ , also called the free associative algebra on the vector space ${\mathfrak {g}}$ . Here $\otimes$ denotes thetensor product ofF-vector spaces. LetI be the two-sidedideal in $T({\mathfrak {g}})$ generated by the elements $XY-YX-[X,Y]$ for $X,Y\in {\mathfrak {g}}$ ; then the universal enveloping algebra is the quotient ring $U({\mathfrak {g}})=T({\mathfrak {g}})/I$ . It satisfies thePoincaré–Birkhoff–Witt theorem: if $e_{1},\ldots ,e_{n}$ is a basis for ${\mathfrak {g}}$ as anF-vector space, then a basis for $U({\mathfrak {g}})$ is given by all ordered products $e_{1}^{i_{1}}\cdots e_{n}^{i_{n}}$ with $i_{1},\ldots ,i_{n}$ natural numbers. In particular, the map ${\mathfrak {g}}\to U({\mathfrak {g}})$ isinjective.^[23]

Representations of ${\mathfrak {g}}$ are equivalent tomodules over the universal enveloping algebra. The fact that ${\mathfrak {g}}\to U({\mathfrak {g}})$ is injective implies that every Lie algebra (possibly of infinite dimension) has a faithful representation (of infinite dimension), namely its representation on $U({\mathfrak {g}})$ . This also shows that every Lie algebra is contained in the Lie algebra associated to some associative algebra.

Representation theory in physics

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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 theangular momentum operators, whose commutation relations are those of the Lie algebra ${\mathfrak {so}}(3)$ of the rotation group $\mathrm {SO} (3)$ . 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 thehydrogen atom, for example, quantum mechanics textbooks classify (more or less explicitly) the finite-dimensional irreducible representations of the Lie algebra ${\mathfrak {so}}(3)$ .^[18]

Structure theory and classification

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

Abelian, nilpotent, and solvable

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

A Lie algebra ${\mathfrak {g}}$ isabelian if the Lie bracket vanishes; that is, [x,y] = 0 for allx andy in ${\mathfrak {g}}$ . In particular, the Lie algebra of an abelian Lie group (such as the group $\mathbb {R} ^{n}$ under addition or thetorus group $\mathbb {T} ^{n}$ ) is abelian. Every finite-dimensional abelian Lie algebra over a field $F {\displaystyle F}$ is isomorphic to $F^{n}$ for some $n\geq 0$ , meaning ann-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, thecommutator subalgebra (orderived subalgebra) of a Lie algebra ${\mathfrak {g}}$ is $[{\mathfrak {g}},{\mathfrak {g}}]$ , meaning the linear subspace spanned by all brackets $[x,y]$ with $x,y\in {\mathfrak {g}}$ . The commutator subalgebra is an ideal in ${\mathfrak {g}}$ , in fact the smallest ideal such that the quotient Lie algebra is abelian. It is analogous to thecommutator subgroup of a group.

A Lie algebra ${\mathfrak {g}}$ isnilpotent if thelower central series

{\mathfrak {g}}\supseteq [{\mathfrak {g}},{\mathfrak {g}}]\supseteq [[{\mathfrak {g}},{\mathfrak {g}}],{\mathfrak {g}}]\supseteq [[[{\mathfrak {g}},{\mathfrak {g}}],{\mathfrak {g}}],{\mathfrak {g}}]\supseteq \cdots

becomes zero after finitely many steps. Equivalently, ${\mathfrak {g}}$ is nilpotent if there is a finite sequence of ideals in ${\mathfrak {g}}$ ,

0={\mathfrak {a}}_{0}\subseteq {\mathfrak {a}}_{1}\subseteq \cdots \subseteq {\mathfrak {a}}_{r}={\mathfrak {g}},

such that ${\mathfrak {a}}_{j}/{\mathfrak {a}}_{j-1}$ is central in ${\mathfrak {g}}/{\mathfrak {a}}_{j-1}$ for eachj. ByEngel's theorem, a Lie algebra over any field is nilpotent if and only if for everyu in ${\mathfrak {g}}$ the adjoint endomorphism

\operatorname {ad} (u):{\mathfrak {g}}\to {\mathfrak {g}},\quad \operatorname {ad} (u)v=[u,v]

isnilpotent.^[24]

More generally, a Lie algebra ${\mathfrak {g}}$ is said to besolvable if thederived series:

{\mathfrak {g}}\supseteq [{\mathfrak {g}},{\mathfrak {g}}]\supseteq [[{\mathfrak {g}},{\mathfrak {g}}],[{\mathfrak {g}},{\mathfrak {g}}]]\supseteq [[[{\mathfrak {g}},{\mathfrak {g}}],[{\mathfrak {g}},{\mathfrak {g}}]],[[{\mathfrak {g}},{\mathfrak {g}}],[{\mathfrak {g}},{\mathfrak {g}}]]]\supseteq \cdots

becomes zero after finitely many steps. Equivalently, ${\mathfrak {g}}$ is solvable if there is a finite sequence of Lie subalgebras,

0={\mathfrak {m}}_{0}\subseteq {\mathfrak {m}}_{1}\subseteq \cdots \subseteq {\mathfrak {m}}_{r}={\mathfrak {g}},

such that ${\mathfrak {m}}_{j-1}$ is an ideal in ${\mathfrak {m}}_{j}$ with ${\mathfrak {m}}_{j}/{\mathfrak {m}}_{j-1}$ abelian for eachj.^[25]

Every finite-dimensional Lie algebra over a field has a unique maximal solvable ideal, called itsradical.^[26] Under theLie correspondence, nilpotent (respectively, solvable) Lie groups correspond to nilpotent (respectively, solvable) Lie algebras over $\mathbb {R}$ .

For example, for a positive integern and a fieldF of characteristic zero, the radical of ${\mathfrak {gl}}(n,F)$ is its center, the 1-dimensional subspace spanned by the identity matrix. An example of a solvable Lie algebra is the space ${\mathfrak {b}}_{n}$ of upper-triangular matrices in ${\mathfrak {gl}}(n)$ ; this is not nilpotent when $n\geq 2$ . An example of a nilpotent Lie algebra is the space ${\mathfrak {u}}_{n}$ of strictly upper-triangular matrices in ${\mathfrak {gl}}(n)$ ; this is not abelian when $n\geq 3$ .

Simple and semisimple

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Main article:Semisimple Lie algebra

A Lie algebra ${\mathfrak {g}}$ is calledsimple if it is not abelian and the only ideals in ${\mathfrak {g}}$ are 0 and ${\mathfrak {g}}$ . (In particular, a one-dimensional—necessarily abelian—Lie algebra ${\mathfrak {g}}$ is by definition not simple, even though its only ideals are 0 and ${\mathfrak {g}}$ .) A finite-dimensional Lie algebra ${\mathfrak {g}}$ is calledsemisimple if the only solvable ideal in ${\mathfrak {g}}$ is 0. In characteristic zero, a Lie algebra ${\mathfrak {g}}$ is semisimple if and only if it is isomorphic to a product of simple Lie algebras, ${\mathfrak {g}}\cong {\mathfrak {g}}_{1}\times \cdots \times {\mathfrak {g}}_{r}$ .^[27]

For example, the Lie algebra ${\mathfrak {sl}}(n,F)$ is simple for every $n\geq 2$ and every fieldF of characteristic zero (or just of characteristic not dividingn). The Lie algebra ${\mathfrak {su}}(n)$ over $\mathbb {R}$ is simple for every $n\geq 2$ . The Lie algebra ${\mathfrak {so}}(n)$ over $\mathbb {R}$ is simple if $n=3$ or $n\geq 5$ .^[28] (There are "exceptional isomorphisms" ${\mathfrak {so}}(3)\cong {\mathfrak {su}}(2)$ and ${\mathfrak {so}}(4)\cong {\mathfrak {su}}(2)\times {\mathfrak {su}}(2)$ .)

The concept of semisimplicity for Lie algebras is closely related with the complete reducibility (semisimplicity) of their representations. When the ground fieldF has characteristic zero, every finite-dimensional representation of a semisimple Lie algebra issemisimple (that is, a direct sum of irreducible representations).^[22]

A finite-dimensional Lie algebra over a field of characteristic zero is calledreductive 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.^[29]

For example, ${\mathfrak {gl}}(n,F)$ is reductive forF of characteristic zero: for $n\geq 2$ , it is isomorphic to the product

{\mathfrak {gl}}(n,F)\cong F\times {\mathfrak {sl}}(n,F),

whereF denotes the center of ${\mathfrak {gl}}(n,F)$ , the 1-dimensional subspace spanned by the identity matrix. Since the special linear Lie algebra ${\mathfrak {sl}}(n,F)$ is simple, ${\mathfrak {gl}}(n,F)$ contains few ideals: only 0, the centerF, ${\mathfrak {sl}}(n,F)$ , and all of ${\mathfrak {gl}}(n,F)$ .

Cartan's criterion

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Cartan's criterion (byÉlie Cartan) gives conditions for a finite-dimensional Lie algebra of characteristic zero to be solvable or semisimple. It is expressed in terms of theKilling form, the symmetric bilinear form on ${\mathfrak {g}}$ 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 ${\mathfrak {g}}$ is semisimple if and only if the Killing form isnondegenerate. A Lie algebra ${\mathfrak {g}}$ is solvable if and only if $K({\mathfrak {g}},[{\mathfrak {g}},{\mathfrak {g}}])=0.$ ^[30]

Classification

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TheLevi 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.^[31] 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 analgebraically closed fieldF of characteristic zero were classified by Killing and Cartan in the 1880s and 1890s, usingroot systems. Namely, every simple Lie algebra is of type A_n, B_n, C_n, D_n, E₆, E₇, E₈, F₄, or G₂.^[32] Here the simple Lie algebra of type A_n is ${\mathfrak {sl}}(n+1,F)$ , B_n is ${\mathfrak {so}}(2n+1,F)$ , C_n is ${\mathfrak {sp}}(2n,F)$ , and D_n is ${\mathfrak {so}}(2n,F)$ . The other five are known as theexceptional Lie algebras.

The classification of finite-dimensional simple Lie algebras over $\mathbb {R}$ is more complicated, but it was also solved by Cartan (seesimple Lie group for an equivalent classification). One can analyze a Lie algebra ${\mathfrak {g}}$ over $\mathbb {R}$ by considering its complexification ${\mathfrak {g}}\otimes _{\mathbb {R} }\mathbb {C}$ .

In the years leading up to 2004, the finite-dimensional simple Lie algebras over an algebraically closed field of characteristic $p>3$ were classified byRichard Earl Block, Robert Lee Wilson, Alexander Premet, and Helmut Strade. (Seerestricted 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

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Main article:Lie group–Lie algebra correspondence

The tangent space of asphere at a point $x {\displaystyle x}$ . If $x {\displaystyle x}$ were the identity element of a Lie group, the tangent space would be a Lie algebra.

The tangent space of asphere at a point $x {\displaystyle x}$ . If $x {\displaystyle x}$ were the identity element of a Lie group, the tangent space would be a Lie algebra.

Although Lie algebras can be studied in their own right, historically they arose as a means to studyLie groups.

The relationship between Lie groups and Lie algebras can be summarized as follows. Each Lie group determines a Lie algebra over $\mathbb {R}$ (concretely, the tangent space at the identity). Conversely, for every finite-dimensional Lie algebra ${\mathfrak {g}}$ , there is a connected Lie group $G {\displaystyle G}$ with Lie algebra ${\mathfrak {g}}$ . This isLie's third theorem; see theBaker–Campbell–Hausdorff formula. This Lie group is not determined uniquely; however, any two Lie groups with the same Lie algebra arelocally isomorphic, and more strongly, they have the sameuniversal cover. For instance, the special orthogonal groupSO(3) and the special unitary groupSU(2) have isomorphic Lie algebras, but SU(2) is asimply connected double cover of SO(3).

Forsimply connected Lie groups, there is a complete correspondence: taking the Lie algebra gives anequivalence of categories from simply connected Lie groups to Lie algebras of finite dimension over $\mathbb {R}$ .^[33]

The correspondence between Lie algebras and Lie groups is used in several ways, including in theclassification of Lie groups and therepresentation 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 adiscrete central subgroup.^[34] So classifying Lie groups becomes simply a matter of counting the discrete subgroups of thecenter, 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 localhomeomorphism (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.^[35]

Lie theory also does not work so neatly for infinite-dimensional representations of a finite-dimensional group. Even for the additive group $G=\mathbb {R}$ , an infinite-dimensional representation of $G {\displaystyle G}$ can usually not be differentiated to produce a representation of its Lie algebra on the same space, or vice versa.^[36] The theory ofHarish-Chandra modules is a more subtle relation between infinite-dimensional representations for groups and Lie algebras.

Real form and complexification

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Given acomplex Lie algebra ${\mathfrak {g}}$ , a real Lie algebra ${\mathfrak {g}}_{0}$ is said to be areal form of ${\mathfrak {g}}$ if the complexification ${\mathfrak {g}}_{0}\otimes _{\mathbb {R} }\mathbb {C}$ is isomorphic to ${\mathfrak {g}}$ . A real form need not be unique; for example, ${\mathfrak {sl}}(2,\mathbb {C} )$ has two real forms up to isomorphism, ${\mathfrak {sl}}(2,\mathbb {R} )$ and ${\mathfrak {su}}(2)$ .^[37]

Given a semisimple complex Lie algebra ${\mathfrak {g}}$ , asplit 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). Acompact 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.^[37]

Lie algebra with additional structures

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

For example, thehomotopy groups of a simply connectedtopological space form a graded Lie algebra, using theWhitehead product. In a related construction,Daniel Quillen used differential graded Lie algebras over therational numbers $\mathbb {Q}$ to describerational homotopy theory in algebraic terms.^[38]

Lie ring

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The definition of a Lie algebra over a field extends to define a Lie algebra over anycommutative ringR. Namely, a Lie algebra ${\mathfrak {g}}$ overR is anR-module with an alternatingR-bilinear map $[\ ,\ ]\colon {\mathfrak {g}}\times {\mathfrak {g}}\to {\mathfrak {g}}$ that satisfies the Jacobi identity. A Lie algebra over the ring $\mathbb {Z}$ ofintegers is sometimes called aLie ring. (This is not directly related to the notion of a Lie group.)

Lie rings are used in the study of finitep-groups (for a prime numberp) through theLazard correspondence.^[39] The lower central factors of a finitep-group are finite abelianp-groups. The direct sum of the lower central factors is given the structure of a Lie ring by defining the bracket to be thecommutator of two coset representatives; see the example below.

p-adic Lie groups are related to Lie algebras over the field $\mathbb {Q} _{p}$ ofp-adic numbers as well as over the ring $\mathbb {Z} _{p}$ ofp-adic integers.^[40] Part ofClaude Chevalley's construction of the finitegroups 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) agroup scheme over the integers.^[41]

Examples

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Here is a construction of Lie rings arising from the study of abstract groups. For elements $x, y {\displaystyle x,y}$ of a group, define the commutator $[x,y]=x^{-1}y^{-1}xy$ . Let $G=G_{1}\supseteq G_{2}\supseteq G_{3}\supseteq \cdots \supseteq G_{n}\supseteq \cdots$ be afiltration of a group $G {\displaystyle G}$ , that is, a chain of subgroups such that $[G_{i},G_{j}]$ is contained in $G_{i+j}$ for all $i, j {\displaystyle i,j}$ . (For the Lazard correspondence, one takes the filtration to be the lower central series ofG.) Then

L=\bigoplus _{i\geq 1}G_{i}/G_{i+1}

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

G_{i}/G_{i+1}

), and with Lie bracket

G_{i}/G_{i+1}\times G_{j}/G_{j+1}\to G_{i+j}/G_{i+j+1}

given by commutators in the group:^[42]

[xG_{i+1},yG_{j+1}]:=[x,y]G_{i+j+1}.

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

\mathbb {Z} /2\mathbb {Z}

Definition using category-theoretic notation

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The definition of a Lie algebra can be reformulated more abstractly in the language ofcategory theory. Namely, one can define a Lie algebra in terms of linear maps—that is,morphisms in thecategory 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, twobraiding isomorphisms are needed. IfA is a vector space, theinterchange isomorphism $\tau :A\otimes A\to A\otimes A$ is defined by

\tau (x\otimes y)=y\otimes x.

Thecyclic-permutation braiding $\sigma :A\otimes A\otimes A\to A\otimes A\otimes A$ is defined as

\sigma =(\mathrm {id} \otimes \tau )\circ (\tau \otimes \mathrm {id} ),

where $\mathrm {id}$ is the identity morphism. Equivalently, $\sigma$ is defined by

\sigma (x\otimes y\otimes z)=y\otimes z\otimes x.

With this notation, a Lie algebra can be defined as an object $A {\displaystyle A}$ in the category of vector spaces together with a morphism

[\cdot ,\cdot ]\colon A\otimes A\rightarrow A

that satisfies the two morphism equalities

[\cdot ,\cdot ]\circ (\mathrm {id} +\tau )=0,

and

[\cdot ,\cdot ]\circ ([\cdot ,\cdot ]\otimes \mathrm {id} )\circ (\mathrm {id} +\sigma +\sigma ^{2})=0.

Generalization

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Main article:Generalization of a Lie algebra

Several generalizations of a Lie algebra have been proposed, many from physics. Among them aregraded Lie algebras,Lie superalgebras,Lie n-algebras,

Remarks

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^More generally, one has the notion of a Lie algebra over anycommutative ringR: anR-module with an alternatingR-bilinear map that satisfies the Jacobi identity (Bourbaki (1989, Section 2)).