Lie algebra

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

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 fieldfor 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 anon-associative algebra.However, everyassociative algebragives rise to a Lie algebra, consisting of the same vector space with thecommutatorLie bracket,$[x,y]=xy-yx$.

Lie algebras are closely related toLie groups,which aregroupsthat are alsosmooth manifolds:every Lie group gives rise to a Lie algebra, which is thetangent spaceat the identity. (In this case, the Lie bracket measures the failure ofcommutativityfor the Lie group.) Conversely, to any finite-dimensional Lie algebra over therealorcomplex numbers,there is a correspondingconnectedLie group, unique up tocovering spaces(Lie's third theorem). Thiscorrespondenceallows one to study the structure andclassificationof 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 groupGis (to first order) approximately a real vector space, namely the tangent space${\mathfrak {g}}$toGat the identity. To second order, the group operation may be non-commutative, and the second-order terms describing the non-commutativity ofGnear 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 ofGnear the identity. They even determineGglobally, 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 mechanicsand 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$v$,with angular speed equal to the magnitude of$v$.The Lie bracket is a measure of the non-commutativity between two rotations. Since a rotation commutes with itself, one has the alternating property$[x,x]=x\times x=0$.

History

Lie algebras were introduced to study the concept ofinfinitesimal transformationsbySophus Liein the 1870s,and independently discovered byWilhelm Killingin the 1880s. The nameLie algebrawas given byHermann Weylin the 1930s; in older texts, the terminfinitesimal groupwas used.

Definition of a Lie algebra

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

• Bilinearity,

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

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

for all scalars$a,b$in$F$and all elements$x,y,z$in${\mathfrak {g}}$.

• TheAlternatingproperty,

$[x,x]=0\$

for all$x$in${\mathfrak {g}}$.

• TheJacobi identity,

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

for all$x,y,z$in${\mathfrak {g}}$.

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

• Anticommutativity,

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

for all$x,y$in${\mathfrak {g}}$.If the field does not havecharacteristic2, then anticommutativity implies the alternating property, since it implies$[x,x]=-[x,x].$

It is customary to denote a Lie algebra by a lower-casefrakturletter 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

Thedimensionof a Lie algebra over a field means itsdimension as a vector space.In physics, a vector spacebasisof the Lie algebra of a Lie groupGmay be called a set ofgeneratorsforG.(They are "infinitesimal generators" forG,so to speak.) In mathematics, a setSofgeneratorsfor a Lie algebra${\mathfrak {g}}$means a subset of${\mathfrak {g}}$such that any Lie subalgebra (as defined below) that containsSmust be all of${\mathfrak {g}}$.Equivalently,${\mathfrak {g}}$is spanned (as a vector space) by all iterated brackets of elements ofS.

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 calledabelian.Every one-dimensional Lie algebra is abelian, by the alternating property of the Lie bracket.

The Lie algebra of matrices

• On an associative algebra$A$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$A$.)
• Theendomorphism ringof an$F$-vector space$V$with the above Lie bracket is denoted${\mathfrak {gl}}(V)$.
• For a fieldFand a positive integern,the space ofn×nmatricesoverF,denoted${\mathfrak {gl}}(n,F)$or${\mathfrak {gl}}_{n}(F)$,is a Lie algebra with bracket given by the commutator of matrices:$[X,Y]=XY-YX$.This is a special case of the previous example; it is a key example of a Lie algebra. It is called thegeneral linearLie algebra.

WhenFis the real numbers,${\mathfrak {gl}}(n,\mathbb {R} )$is the Lie algebra of thegeneral linear group$\mathrm {GL} (n,\mathbb {R} )$,the group ofinvertiblenxnreal 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 oflinear maps.For any fieldF,${\mathfrak {gl}}(n,F)$can be viewed as the Lie algebra of thealgebraic group$\mathrm {GL} (n)$overF.

Definitions

Subalgebras, ideals and homomorphisms

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 associativeringsand algebras (and also for groups) has analogs for Lie algebras. ALie subalgebrais 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:

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

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

A Lie algebrahomomorphismis a linear map compatible with the respective Lie brackets:

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

Anisomorphismof Lie algebras is abijectivehomomorphism.

As with normal subgroups in groups, ideals in Lie algebras are precisely thekernelsof 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 theoremholds 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 tocommuteif their bracket vanishes:$[x,y]=0$.

Thecentralizersubalgebra of a subset$S\subset {\mathfrak {g}}$is the set of elements commuting with$S$: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,thenormalizersubalgebra of$S$is${\mathfrak {n}}_{\mathfrak {g}}(S)=\{x\in {\mathfrak {g}}:[x,s]\in S\ {\text{ for all}}\ s\in S\}$.If$S$is a Lie subalgebra,${\mathfrak {n}}_{\mathfrak {g}}(S)$is the largest subalgebra such that$S$is an ideal of${\mathfrak {n}}_{\mathfrak {g}}(S)$.

Example

The subspace${\mathfrak {t}}_{n}$of diagonal matrices in${\mathfrak {gl}}(n,F)$is an abelian Lie subalgebra. (It is aCartan subalgebraof${\mathfrak {gl}}(n)$,analogous to amaximal torusin 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

For two Lie algebras${\mathfrak {g}}$and${\mathfrak {g'}}$,theproductLie 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

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

This is the product in thecategoryof 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 productof${\mathfrak {i}}$and${\mathfrak {g}}/{\mathfrak {i}}$,${\mathfrak {g}}={\mathfrak {g}}/{\mathfrak {i}}\ltimes {\mathfrak {i}}$.See alsosemidirect sum of Lie algebras.

Derivations

For analgebraAover a fieldF,aderivationofAoverFis 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${\text{Der}}_{k}(A)$of all derivations ofAoverFinto a Lie algebra.

Informally speaking, the space of derivations ofAis the Lie algebra of theautomorphism groupofA.(This is literally true when the automorphism group is a Lie group, for example whenFis the real numbers andAhas 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 ofDbeing a derivation.

Example: the Lie algebra of vector fields.LetAbe the ring$C^{\infty }(X)$ofsmooth functionson a smooth manifoldX.Then a derivation ofAover$\mathbb {R}$is equivalent to avector fieldonX.(A vector fieldvgives 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).Informally speaking,${\text{Vect}}(X)$is the Lie algebra of thediffeomorphism groupofX.So the Lie bracket of vector fields describes the non-commutativity of the diffeomorphism group. Anactionof a Lie groupGon a manifoldXdetermines 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 fieldFdetermines 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 derivationassociated 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 derivationsis 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 groupof a group.) For asemisimple 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${\text{Out}}_{F}(V)$can be identified with${\mathfrak {gl}}(V)$.

Examples

Matrix Lie algebras

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

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

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 exponentialof elements of${\mathfrak {g}}$.(To be precise, this gives theidentity componentofG,ifGis 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$X$.

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.

• For a positive integern,thespecial linear group$\mathrm {SL} (n,\mathbb {R} )$consists of all realn×nmatrices 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 subgroupof the general linear group$\mathrm {GL} (n,\mathbb {R} )$.Its Lie algebra${\mathfrak {sl}}(n,\mathbb {R} )$consists of all realn×nmatrices withtrace0. 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 ofnxnorthogonal matrices, meaning that$A^{\mathrm {T} }=A^{-1}$,where$A^{\mathrm {T} }$denotes thetransposeof 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 tonxncomplex matrices. Equivalently,$\mathrm {O} (n,\mathbb {C} )$is the subgroup of$\mathrm {GL} (n,\mathbb {C} )$that preserves the standardsymmetric bilinear formon$\mathbb {C} ^{n}$.

• 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×nunitary matrices (satisfying$A^{*}=A^{-1}$,where$A^{*}$denotes theconjugate transposeof 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,itimes 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$\mathrm {su} (n)$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 formon$\mathbb {R} ^{2n}$.Its Lie algebra is thesymplectic Lie algebra${\mathfrak {sp}}(2n,\mathbb {R} )$.
• Theclassical Lie algebrasare those listed above, along with variants over any field.

Two dimensions

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

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

The affine groupGcan 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${\mathfrak {g}}$of${\mathfrak {gl}}(2,\mathbb {R} )$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$F$,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

• TheHeisenberg algebra${\mathfrak {h}}_{3}(F)$over a fieldFis the three-dimensional Lie algebra with a basis$X,Y,Z$such that

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

It can be viewed as the Lie algebra of 3×3 strictlyupper-triangularmatrices, 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

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

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.

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

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

H

E

F

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).In the terminology of quantum mechanics, one can think ofEandFasraising and lowering operators.Indeed, for any representation of${\mathfrak {sl}}(2,\mathbb {C} )$,the relations above imply thatEmaps thec-eigenspaceofH(for a complex numberc) into the$(c+2)$-eigenspace, whileFmaps thec-eigenspace into the$(c-2)$-eigenspace.

The Lie algebra${\mathfrak {sl}}(2,\mathbb {C} )$is isomorphic to thecomplexificationof${\mathfrak {so}}(3)$,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

• The Lie algebra of vector fields on a smooth manifold of positive dimension is an infinite-dimensional Lie algebra over$\mathbb {R}$.
• TheKac–Moody algebrasare 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 algebrais an infinite-dimensional Lie algebra that contains all theclassical Lie algebrasas subalgebras.
• TheVirasoro algebrais important instring theory.
• The functor that takes a Lie algebra over a fieldFto the underlying vector space has aleft adjoint$V\mapsto L(V)$,called thefree Lie algebraon 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 forVof dimension at least 2.

Representations

Main article:Lie algebra representation

Definitions

Given a vector spaceV,let${\mathfrak {gl}}(V)$denote the Lie algebra consisting of all linear maps fromVto itself, with bracket given by$[X,Y]=XY-YX$.Arepresentationof a Lie algebra${\mathfrak {g}}$onVis 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 fromVto itself, in such a way that the Lie bracket on${\mathfrak {g}}$corresponds to the commutator of linear maps.

A representation is said to befaithfulif its kernel is zero.Ado's theoremstates that every finite-dimensional Lie algebra over a field of characteristic zero has a faithful representation on a finite-dimensional vector space.Kenkichi Iwasawaextended this result to finite-dimensional Lie algebras over a field of any characteristic.Equivalently, every finite-dimensional Lie algebra over a fieldFis isomorphic to a Lie subalgebra of${\mathfrak {gl}}(n,F)$for some positive integern.

Adjoint representation

For any Lie algebra${\mathfrak {g}}$,theadjoint representationis 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

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 theoremsays 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 algebrasand theWeyl character formula.

Universal enveloping algebra

Main article:Universal enveloping algebra

The functor that takes an associative algebraAover a fieldFtoAas 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 algebraon${\mathfrak {g}}$,also called the free associative algebra on the vector space${\mathfrak {g}}$.Here$\otimes$denotes thetensor productofF-vector spaces. LetIbe the two-sidedidealin$T({\mathfrak {g}})$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.

Representations of${\mathfrak {g}}$are equivalent tomodulesover 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

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, but one 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${\mathfrak {so}}(3)$.

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

A Lie algebra${\mathfrak {g}}$isabelianif the Lie bracket vanishes; that is, [x,y] = 0 for allxandyin${\mathfrak {g}}$.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$F$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 subgroupof a group.

A Lie algebra${\mathfrak {g}}$isnilpotentif 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 everyuin${\mathfrak {g}}$the adjoint endomorphism

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

isnilpotent.

More generally, a Lie algebra${\mathfrak {g}}$is said to besolvableif 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.

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

For example, for a positive integernand a fieldFof 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

Main article:Semisimple Lie algebra

A Lie algebra${\mathfrak {g}}$is calledsimpleif 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 calledsemisimpleif 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,${\mathfrak {g}}\cong {\mathfrak {g}}_{1}\times \cdots \times {\mathfrak {g}}_{r}$.

For example, the Lie algebra${\mathfrak {sl}}(n,F)$is simple for every$n\geq 2$and every fieldFof 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$n\geq 5$.(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 fieldFhas characteristic zero, every finite-dimensional representation of a semisimple Lie algebra issemisimple(that is, a direct sum of irreducible representations).

A finite-dimensional Lie algebra over a field of characteristic zero is calledreductiveif 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,${\mathfrak {gl}}(n,F)$is reductive forFof characteristic zero: for$n\geq 2$,it is isomorphic to the product

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

whereFdenotes 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

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$K({\mathfrak {g}},[{\mathfrak {g}},{\mathfrak {g}}])=0.$

Classification

TheLevi decompositionasserts 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 analgebraically closed fieldFof 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.Here the simple Lie algebra of type A_nis${\mathfrak {sl}}(n+1,F)$,B_nis${\mathfrak {so}}(2n+1,F)$,C_nis${\mathfrak {sp}}(2n,F)$,and D_nis${\mathfrak {so}}(2n,F)$.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 groupfor 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

Main article:Lie group–Lie algebra correspondence

The tangent space of asphereat a point$x$.If$x$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$G$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 connecteddouble cover of SO(3).

Forsimply connectedLie groups, there is a complete correspondence: taking the Lie algebra gives anequivalence of categoriesfrom simply connected Lie groups to Lie algebras of finite dimension over$\mathbb {R}$.

The correspondence between Lie algebras and Lie groups is used in several ways, including in theclassification of Lie groupsand therepresentation theoryof 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 adiscretecentral subgroup.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.

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

Real form and complexification

Given acomplex Lie algebra${\mathfrak {g}}$,a real Lie algebra${\mathfrak {g}}_{0}$is said to be areal formof${\mathfrak {g}}$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${\mathfrak {su}}(2)$.

Given a semisimple complex Lie algebra${\mathfrak {g}}$,asplit formof 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 formis 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, agraded Lie algebrais a Lie algebra (or more generally aLie superalgebra) with a compatible grading. Adifferential graded Lie algebraalso comes with a differential, making the underlying vector space achain complex.

For example, thehomotopy groupsof a simply connectedtopological spaceform a graded Lie algebra, using theWhitehead product.In a related construction,Daniel Quillenused differential graded Lie algebras over therational numbers$\mathbb {Q}$to describerational homotopy theoryin algebraic terms.

Lie ring

The definition of a Lie algebra over a field extends to define a Lie algebra over anycommutative ringR.Namely, a Lie algebra${\mathfrak {g}}$overRis anR-modulewith an alternatingR-bilinear map$[\,\ ]\colon {\mathfrak {g}}\times {\mathfrak {g}}\to {\mathfrak {g}}$that satisfies the Jacobi identity. A Lie algebra over the ring$\mathbb {Z}$ofintegersis 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.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 thecommutatorof two coset representatives; see the example below.

p-adic Lie groupsare related to Lie algebras over the field$\mathbb {Q} _{p}$ofp-adic numbersas well as over the ring$\mathbb {Z} _{p}$ofp-adic integers.Part ofClaude Chevalley's construction of the finitegroups of Lie typeinvolves showing that a simple Lie algebra over the complex numbers comes from a Lie algebra over the integers, and then (with more care) agroup schemeover the integers.

Examples

• Here is a construction of Lie rings arising from the study of abstract groups. For elements$x,y$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 afiltrationof a group$G$,that is, a chain of subgroups such that$[G_{i},G_{j}]$is contained in$G_{i+j}$for all$i,j$.(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:

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

For example, the Lie ring associated to the lower central series on thedihedral groupof order 8 is the Heisenberg Lie algebra of dimension 3 over the field$\mathbb {Z} /2\mathbb {Z}$.

Definition using category-theoretic notation

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,morphismsin 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 isomorphismsare needed. IfAis 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$A$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

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

See also

• Affine Lie algebra
• Automorphism of a Lie algebra
• Frobenius integrability theorem(the integrability being the same as being a Lie subalgebra)
• Gelfand–Fuks cohomology
• Hopf algebra
• Index of a Lie algebra
• Leibniz algebra
• Lie algebra cohomology
• Lie algebra extension
• Lie algebra representation
• Lie bialgebra
• Lie coalgebra
• Lie operad
• Particle physics and representation theory
• Lie superalgebra
• Orthogonal symmetric Lie algebra
• Poisson algebra
• Pre-Lie algebra
• Quantum groups
• Moyal algebra
• Quasi-Frobenius Lie algebra
• Quasi-Lie algebra
• Restricted Lie algebra
• Serre relations

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External links

• Kac, Victor G.;et al.Course notes for MIT 18.745: Introduction to Lie Algebras.Archived fromthe originalon 2010-04-20.
• "Lie algebra",Encyclopedia of Mathematics,EMS Press,2001 [1994]
• McKenzie, Douglas (2015)."An Elementary Introduction to Lie Algebras for Physicists".