Lie algebra

In mathematics, a Lie algebra (pronounced /liː/ LEE) is a vector space \(\mathfrak g\) together with an operation called the Lie bracket, an alternating bilinear map \(\mathfrak g \times \mathfrak g \rightarrow \mathfrak g\), that satisfies the Jacobi identity. In other words, a Lie algebra is an algebra over a field for which the multiplication operation (called the Lie bracket) is alternating and satisfies the Jacobi identity. The Lie bracket of two vectors \(x\) and \(y\) is denoted \([x,y]\). A Lie algebra is typically a non-associative algebra. However, every associative algebra gives rise to a Lie algebra, consisting of the same vector space with the commutator Lie bracket, \([x,y] = xy - yx\).

Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: every Lie group gives rise to a Lie algebra, which is the tangent space at the identity. (In this case, the Lie bracket measures the failure of commutativity for the Lie group.) Conversely, to any finite-dimensional Lie algebra over the real or complex numbers, there is a corresponding connected Lie group, unique up to covering spaces (Lie's third theorem). This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras, which are simpler objects of linear algebra.

In more detail: for any Lie group, the multiplication operation near the identity element 1 is commutative to first order. In other words, every Lie group G is (to first order) approximately a real vector space, namely the tangent space \(\mathfrak{g}\) to G at the identity. To second order, the group operation may be non-commutative, and the second-order terms describing the non-commutativity of G near the identity give \(\mathfrak{g}\) the structure of a Lie algebra. It is a remarkable fact that these second-order terms (the Lie algebra) completely determine the group structure of G near the identity. They even determine G globally, up to covering spaces.

In physics, Lie groups appear as symmetry groups of physical systems, and their Lie algebras (tangent vectors near the identity) may be thought of as infinitesimal symmetry motions. Thus Lie algebras and their representations are used extensively in physics, notably in quantum mechanics and particle physics.

An elementary example (not directly coming from an associative algebra) is the 3-dimensional space \(\mathfrak{g}=\mathbb{R}^3\) with Lie bracket defined by the cross 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 of rotations of space, and each vector \(v\in\R^3\) 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 \([x,x]=x\times x = 0\).

Condensed: the full section is in Wikipedia.

Lie algebras were introduced to study the concept of infinitesimal transformations by Sophus Lie in the 1870s, and independently discovered by Wilhelm Killing in the 1880s. The name Lie algebra was given by Hermann Weyl in the 1930s; in older texts, the term infinitesimal group was used.

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

• Bilinearity:

\[\begin{align} \left[ax+by,z\right]=a\left[x,z\right]+b\left[y,z\right] & , \\ \left[z,ax+by\right]=a\left[z,x\right]+b\left[z,y\right] \end{align}\]

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

• The alternating property:

\[[x,x]=0\]

for all \(x\) in \(\mathfrak{g}\).

• The Jacobi identity:

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

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 have characteristic 2, then anticommutativity implies the alternating property, since it implies \([x,x]=-[x,x]\).

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

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

Condensed: the full section is in Wikipedia.

The dimension of a Lie algebra over a field means its dimension as a vector space. In physics, a vector space basis of the Lie algebra of a Lie group \(G\) may be called a set of generators for \(G\). (They are "infinitesimal generators" for \(G\), so to speak.) In mathematics, a set \(S\) of generators for a Lie algebra \(\mathfrak{g}\) means a subset of \(\mathfrak{g}\) such that any Lie subalgebra (as defined below) that contains \(S\) must be all of \(\mathfrak{g}\). Equivalently, \(\mathfrak{g}\) is spanned (as a vector space) by all iterated brackets of elements of \(S\).

A Lie algebra is called abelian if its Lie bracket is identically zero. Any vector space \(V\) 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.

• 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 \(A\).)
• The endomorphism ring of an \(F\)-vector space \(V\) with the above Lie bracket is denoted \(\mathfrak{gl}(V)\).
• For a field \(F\) and a positive integer \(n\), the space of \(n\times n\) matrices over \(F\), 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 the general linear Lie algebra.

When \(F\) is the real numbers, \(\mathfrak{gl}(n,\mathbb{R})\) is the Lie algebra of the general linear group \(\mathrm{GL}(n,\mathbb{R})\), the group of invertible \(n\times n\) real matrices (or equivalently, matrices with nonzero determinant), 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 \(\mathfrak{gl}(n,\R)\) describes the failure of commutativity for matrix multiplication, or equivalently for the composition of linear maps. For any field \(F\), \(\mathfrak{gl}(n,F)\) can be viewed as the Lie algebra of the algebraic group \(\mathrm{GL}(n)\) over \(F\).

The Lie bracket is not required to be associative, meaning that \([[x,y],z]\) need not be equal to \([x,[y,z]]\). Nonetheless, much of the terminology for associative rings and algebras (and also for groups) has analogs for Lie algebras. A Lie subalgebra is a linear subspace \(\mathfrak{h} \subseteq \mathfrak{g}\) which is closed under the Lie bracket. An ideal \(\mathfrak i\subseteq\mathfrak{g}\) 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, and normal subgroups correspond to ideals.

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

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

An isomorphism of Lie algebras is a bijective homomorphism.

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

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 to commute if their bracket vanishes: \([x,y]=0\).

The centralizer subalgebra 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 the center \(\mathfrak{z}(\mathfrak{g})\). Similarly, for a subspace S, the normalizer subalgebra 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 \(\mathfrak{n}_{\mathfrak g}(S)\).

For two Lie algebras \(\mathfrak{g}\) and \(\mathfrak{g'}\), the product 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

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

This is the product in the category 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 a semidirect product of \(\mathfrak{i}\) and \(\mathfrak{g}/\mathfrak{i}\), written \(\mathfrak{g}=\mathfrak{g}/\mathfrak{i}\ltimes\mathfrak{i}\). See also semidirect sum of Lie algebras.

For an algebra A over a field F, a derivation of A over F is a linear map \(D\colon A\to A\) that satisfies the Leibniz rule

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

for all \(x,y\in A\). (The definition makes sense for a possibly non-associative algebra.) Given two derivations \(D_1\) and \(D_2\), their commutator \([D_1,D_2]:=D_1D_2-D_2D_1\) is again a derivation. This operation makes the space \(\text{Der}_F(A)\) of all derivations of A over F into a Lie algebra.

Informally speaking, the space of derivations of A is the Lie algebra of the automorphism group of A. (This is literally true when the automorphism group is a Lie group, for example when F is the real numbers and A has finite dimension as a vector space.) For this reason, spaces of derivations are a natural way to construct Lie algebras: they are the "infinitesimal automorphisms" of A. Indeed, writing out the condition that

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

(where 1 denotes the identity map on A) gives exactly the definition of D being a derivation.

Example: the Lie algebra of vector fields. Let A be the ring \(C^\infty(X)\) of smooth functions on a smooth manifold X. Then a derivation of A over \(\mathbb{R}\) is equivalent to a vector field on X. (A vector field v gives a derivation of the space of smooth functions by differentiating functions in the direction of v.) This makes the space \(\operatorname{Vect}(X)\) of vector fields into a Lie algebra (see Lie bracket of vector fields). Informally speaking, \(\text{Vect}(X)\) is the Lie algebra of the diffeomorphism group of X. So the Lie bracket of vector fields describes the non-commutativity of the diffeomorphism group. An action of a Lie group G on a manifold X determines a homomorphism of Lie algebras \(\mathfrak{g}\to \text{Vect}(X)\). (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 field F 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

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

The inner 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 of outer 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 the outer automorphism group of a group.) For a semisimple Lie algebra (defined below) over a field of characteristic zero, every derivation is inner. This is related to the theorem that the outer automorphism group of a semisimple Lie group is finite.

Condensed: the full section is in Wikipedia.

A matrix group is a Lie group consisting of invertible matrices, \(G\subset \mathrm{GL}(n,\mathbb{R})\), where the group operation of G is matrix multiplication. The corresponding Lie algebra \(\mathfrak g\) is the space of matrices which are tangent vectors to G inside the linear space \(M_n(\mathbb{R})\): this consists of derivatives of smooth curves in G at the identity 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 the matrix exponential of elements of \(\mathfrak{g}\). (To be precise, this gives the identity component of G, if G 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 \(X\).

The same comments apply to complex Lie subgroups of \(\operatorname{GL}(n,\mathbb{C})\) 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 integer n, the special linear group \(\mathrm{SL}(n,\mathbb{R})\) consists of all real n × n matrices with determinant 1. This is the group of linear maps from \(\mathbb{R}^n\) to itself that preserve volume and orientation. More abstractly, \(\mathrm{SL}(n,\mathbb{R})\) is the commutator subgroup of the general linear group \(\mathrm{GL}(n,\R)\). Its Lie algebra \(\mathfrak{sl}(n,\mathbb{R})\) consists of all real n × n matrices with trace 0. Similarly, one can define the analogous complex Lie group \({\rm SL}(n,\mathbb{C})\) and its Lie algebra \(\mathfrak{sl}(n,\mathbb{C})\).
• The orthogonal 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 of n x n orthogonal matrices, meaning that \(A^{\mathrm{T}}=A^{-1}\), where \(A^{\mathrm{T}}\) denotes the transpose of a matrix. The orthogonal group has two connected components; the identity component is called the special 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 also infinitesimal 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 to n x n complex matrices. Equivalently, \(\mathrm{O}(n,\mathbb{C})\) is the subgroup of \(\mathrm{GL}(n,\mathbb{C})\) that preserves the standard symmetric bilinear form on \(\mathbb{C}^n\).

Condensed: the full section is in Wikipedia.

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

• There is a unique nonabelian Lie algebra \(\mathfrak{g}\) of dimension 2 over any field F, 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 \(G=\mathrm{Aff}(1,\mathbb{R})\) of affine transformations of the real line, \(x\mapsto ax+b\).

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

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

under matrix multiplication, with \(a,b \in \mathbb{R}\), \(a \neq 0\). 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.

• The Heisenberg algebra \(\mathfrak{h}_3(F)\) over a field F is 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 strictly upper-triangular matrices, with the commutator Lie bracket and the basis

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

Over the real numbers, \(\mathfrak{h}_3(\mathbb{R})\) is the Lie algebra of the Heisenberg 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 field F, 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 the rotation 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 the Spin (physics) angular-momentum component operators for spin-1 particles in quantum mechanics.

The Lie algebra \(\mathfrak{so}(3)\) cannot be broken into pieces in the way that the previous examples can: it is simple, meaning that it is not abelian and its only ideals are 0 and all of \(\mathfrak{so}(3)\).

• 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 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 of E and F as raising and lowering operators. Indeed, for any representation of \(\mathfrak{sl}(2,\mathbb{C})\), the relations above imply that E maps the c-eigenspace of H (for a complex number c) into the \((c+2)\)-eigenspace, while F maps the c-eigenspace into the \((c-2)\)-eigenspace.

Condensed: the full section is in Wikipedia.

• The Lie algebra of vector fields on a smooth manifold of positive dimension is an infinite-dimensional Lie algebra over \(\mathbb{R}\).
• The Kac-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 \(\mathfrak{sl}(n,\C)\)).
• The Moyal algebra is an infinite-dimensional Lie algebra that contains all the classical Lie algebras as subalgebras.
• The Virasoro algebra is important in string theory.
• The functor that takes a Lie algebra over a field F to the underlying vector space has a left adjoint \(V\mapsto L(V)\), called the free Lie algebra on a vector space V. It is spanned by all iterated Lie brackets of elements of V, modulo only the relations coming from the definition of a Lie algebra. The free Lie algebra \(L(V)\) is infinite-dimensional for V of dimension at least 2.

Given a vector space V, let \(\mathfrak{gl}(V)\) denote the Lie algebra consisting of all linear maps from V to itself, with bracket given by \([X,Y]=XY-YX\). A representation of a Lie algebra \(\mathfrak{g}\) on V 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 from V 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 be faithful if its kernel is zero. Ado's theorem states that every finite-dimensional Lie algebra over a field of characteristic zero has a faithful representation on a finite-dimensional vector space. Kenkichi Iwasawa extended this result to finite-dimensional Lie algebras over a field of any characteristic. Equivalently, every finite-dimensional Lie algebra over a field F is isomorphic to a Lie subalgebra of \(\mathfrak{gl}(n,F)\) for some positive integer n.