Lie algebra

 ( Lie Groups ) 

In mathematics, a Lie algebra (pronounced ) is a vector space $\backslash mathfrak\; g$ together with an operation called the Lie bracket, an alternating bilinear map $\backslash mathfrak\; g\; \backslash times\; \backslash mathfrak\; g\; \backslash rightarrow\; \backslash mathfrak\; g$, 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 , 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 number or , there is a corresponding connected space Lie group, unique up to (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 $\backslash mathfrak\{g\}$ 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 $\backslash mathfrak\{g\}$ 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 $\backslash mathfrak\{g\}=\backslash mathbb\{R\}^3$ with Lie bracket defined by the cross product $x,y=x\backslash times\; y.$ This is skew-symmetric since $x\backslash times\; y\; =\; -y\backslash times\; x$, and instead of associativity it satisfies the Jacobi identity:

$x\backslash times(y\backslash times\; z)+\backslash \; y\backslash times(z\backslash times\; x)+\backslash \; z\backslash times(x\backslash times\; y)\backslash \; =\backslash \; 0.$

This is the Lie algebra of the Lie group of rotations of space, and each vector $v\backslash in\backslash R^3$ 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 $x,x=x\backslash times\; x\; =\; 0$.

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

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History 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.

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Definition of a Lie algebra A Lie algebra is a vector space $\backslash ,\backslash mathfrak\{g\}$ over a field $F$ together with a binary operation $\backslash ,\backslash cdot\backslash ,,\backslash cdot\backslash ,:\; \backslash mathfrak\{g\}\backslash times\backslash mathfrak\{g\}\backslash to\backslash mathfrak\{g\}$ called the Lie bracket, satisfying the following axioms:

• Bilinearity,

:$a\; =\; a\; x,\; +\; b\; y,,$

:$z,\; =\; az,\; +\; b\; z,$

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

• The Alternating property,

:$x,x=0\backslash$

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

• The Jacobi identity,

: $x,[y,z]\; +\; z,[x,y]\; +\; y,[z,x]\; =\; 0\; \backslash$

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

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 $\backslash mathfrak\{g\}$. Thus bilinearity and the alternating property together imply

• Anticommutativity,

: $x,y\; =\; -y,x,\backslash$

for all $x,y$ in $\backslash mathfrak\{g\}$. 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 $\backslash mathrm\{ad\}\_x\; =\; x,-$:

: $x,[y,]\; =\; [x,y,z]\; +\; y,],\backslash$

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

It is customary to denote a Lie algebra by a lower-case fraktur letter such as $\backslash mathfrak\{g,\; h,\; b,\; n\}$. 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 of SU( n) is $\backslash mathfrak\{su\}(n)$.

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

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Basic examples

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Abelian Lie algebras 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.

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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 $A$.)
• The endomorphism ring of an $F$-vector space $V$ with the above Lie bracket is denoted $\backslash mathfrak\{gl\}(V)$.
• For a field F and a positive integer n, the space of n × n matrices over F, denoted $\backslash mathfrak\{gl\}(n,\; F)$ or $\backslash mathfrak\{gl\}\_n(F)$, 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, $\backslash mathfrak\{gl\}(n,\backslash mathbb\{R\})$ is the Lie algebra of the general linear group $\backslash mathrm\{GL\}(n,\backslash mathbb\{R\})$, the group of invertible n x n real matrices (or equivalently, matrices with nonzero determinant), where the group operation is matrix multiplication. Likewise, $\backslash mathfrak\{gl\}(n,\backslash mathbb\{C\})$ is the Lie algebra of the complex Lie group $\backslash mathrm\{GL\}(n,\backslash mathbb\{C\})$. The Lie bracket on $\backslash mathfrak\{gl\}(n,\backslash R)$ describes the failure of commutativity for matrix multiplication, or equivalently for the composition of linear maps. For any field F, $\backslash mathfrak\{gl\}(n,F)$ can be viewed as the Lie algebra of the algebraic group $\backslash mathrm\{GL\}(n)$ over F.

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Definitions

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Subalgebras, ideals and homomorphisms The Lie bracket is not required to be associative, meaning that $[x,y,z]$ need not be equal to $x,[y,z]$. Nonetheless, much of the terminology for associative rings and algebras (and also for groups) has analogs for Lie algebras. A Lie subalgebra is a linear subspace $\backslash mathfrak\{h\}\; \backslash subseteq\; \backslash mathfrak\{g\}$ which is closed under the Lie bracket. An ideal $\backslash mathfrak\; i\backslash subseteq\backslash mathfrak\{g\}$ is a linear subspace that satisfies the stronger condition:By the anticommutativity of the commutator, the notions of a left and right ideal in a Lie algebra coincide.

$\backslash mathfrak\{g\},\backslash mathfrak\backslash subseteq\; \backslash mathfrak\; i.$

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

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

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

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 $\backslash mathfrak\{g\}$ and an ideal $\backslash mathfrak\; i$ in it, the quotient Lie algebra $\backslash mathfrak\{g\}/\backslash mathfrak\{i\}$ is defined, with a surjective homomorphism $\backslash mathfrak\{g\}\backslash to\backslash mathfrak\{g\}/\backslash mathfrak\{i\}$ of Lie algebras. The first isomorphism theorem holds for Lie algebras: for any homomorphism $\backslash phi\backslash colon\backslash mathfrak\{g\}\backslash to\backslash mathfrak\{h\}$ of Lie algebras, the image of $\backslash phi$ is a Lie subalgebra of $\backslash mathfrak\{h\}$ that is isomorphic to $\backslash mathfrak\{g\}/\backslash text\{ker\}(\backslash phi)$.

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 $x,y\backslash in\backslash mathfrak\; g$ are said to commute if their bracket vanishes: $x,y=0$.

The centralizer subalgebra of a subset $S\backslash subset\; \backslash mathfrak\{g\}$ is the set of elements commuting with $S$: that is, $\backslash mathfrak\{z\}\_\{\backslash mathfrak\; g\}(S)\; =\; \backslash \{x\backslash in\backslash mathfrak\; g\; :\; x,\; =\; 0\; \backslash \; \backslash text\{\; for\; all\; \}\; s\backslash in\; S\backslash \}$. The centralizer of $\backslash mathfrak\{g\}$ itself is the center $\backslash mathfrak\{z\}(\backslash mathfrak\{g\})$. Similarly, for a subspace S, the normalizer subalgebra of $S$ is $\backslash mathfrak\{n\}\_\{\backslash mathfrak\; g\}(S)\; =\; \backslash \{x\backslash in\backslash mathfrak\; g\; :\; x,s\backslash in\; S\; \backslash \; \backslash text\{\; for\; all\}\backslash \; s\backslash in\; S\backslash \}$. If $S$ is a Lie subalgebra, $\backslash mathfrak\{n\}\_\{\backslash mathfrak\; g\}(S)$ is the largest subalgebra such that $S$ is an ideal of $\backslash mathfrak\{n\}\_\{\backslash mathfrak\; g\}(S)$.

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Example The subspace $\backslash mathfrak\{t\}\_n$ of diagonal matrices in $\backslash mathfrak\{gl\}(n,F)$ is an abelian Lie subalgebra. (It is a Cartan subalgebra of $\backslash mathfrak\{gl\}(n)$, analogous to a maximal torus in the theory of compact Lie groups.) Here $\backslash mathfrak\{t\}\_n$ is not an ideal in $\backslash mathfrak\{gl\}(n)$ for $n\backslash geq\; 2$. For example, when $n=2$, this follows from the calculation:

(which is not always in $\backslash mathfrak\{t\}\_2$).

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

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Product and semidirect product For two Lie algebras $\backslash mathfrak\{g\}$ and $\backslash mathfrak\{g\text{'}\}$, the direct product Lie algebra is the vector space $\backslash mathfrak\{g\}\backslash times\; \backslash mathfrak\{g\text{'}\}$ consisting of all ordered pairs $(x,x\text{'}),\; \backslash ,x\backslash in\backslash mathfrak\{g\},\; \backslash \; x\text{'}\backslash in\backslash mathfrak\{g\text{'}\}$, with Lie bracket

$(x,x\text{'}),(y,y\text{'})=(x,y,x\text{'},y\text{'}).$

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

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

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Derivations For an algebra A over a field F, a derivation of A over F is a linear map $D\backslash colon\; A\backslash to\; A$ that satisfies the product rule

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

for all $x,y\backslash in\; A$. (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 $\backslash text\{Der\}\_k(A)$ 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

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

(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 $C^\{\backslash infty\}(X)$ of on a smooth manifold X. Then a derivation of A over $\backslash mathbb\{R\}$ 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 $\backslash text\{Vect\}(X)$ of vector fields into a Lie algebra (see Lie bracket of vector fields). Informally speaking, $\backslash text\{Vect\}(X)$ 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 group action of a Lie group G on a manifold X determines a homomorphism of Lie algebras $\backslash mathfrak\{g\}\backslash to\; \backslash text\{Vect\}(X)$. (An example is illustrated below.)

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

$D(x,y)=D(x),y+x,D(y)$.

The inner derivation associated to any $x\backslash in\backslash mathfrak\; g$ is the adjoint mapping $\backslash mathrm\{ad\}\_x$ defined by $\backslash mathrm\{ad\}\_x(y):=x,y$. (This is a derivation as a consequence of the Jacobi identity.) That gives a homomorphism of Lie algebras, $\backslash operatorname\{ad\}\backslash colon\backslash mathfrak\{g\}\backslash to\; \backslash text\{Der\}\_F(\backslash mathfrak\{g\})$. The image $\backslash text\{Inn\}\_F(\backslash mathfrak\{g\})$ is an ideal in $\backslash text\{Der\}\_F(\backslash mathfrak\{g\})$, and the Lie algebra of outer derivations is defined as the quotient Lie algebra, $\backslash text\{Out\}\_F(\backslash mathfrak\{g\})=\backslash text\{Der\}\_F(\backslash mathfrak\{g\})/\backslash text\{Inn\}\_F(\backslash mathfrak\{g\})$. (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.

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

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Examples

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Matrix Lie algebras A Linear group is a Lie group consisting of invertible matrices, $G\backslash subset\; \backslash mathrm\{GL\}(n,\backslash mathbb\{R\})$, where the group operation of G is matrix multiplication. The corresponding Lie algebra $\backslash mathfrak\; g$ is the space of matrices which are tangent vectors to G inside the linear space $M\_n(\backslash mathbb\{R\})$: this consists of derivatives of smooth curves in G at the identity matrix $I$:

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

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

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

Here are some matrix Lie groups and their Lie algebras.

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

The complex orthogonal group $\backslash mathrm\{O\}(n,\backslash mathbb\{C\})$, its identity component $\backslash mathrm\{SO\}(n,\backslash mathbb\{C\})$, and the Lie algebra $\backslash mathfrak\{so\}(n,\backslash mathbb\{C\})$ are given by the same formulas applied to n x n complex matrices. Equivalently, $\backslash mathrm\{O\}(n,\backslash mathbb\{C\})$ is the subgroup of $\backslash mathrm\{GL\}(n,\backslash mathbb\{C\})$ that preserves the standard symmetric bilinear form on $\backslash mathbb\{C\}^n$.

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

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Two dimensions 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 $\backslash mathfrak\{g\}$ of dimension 2 over any field F, up to isomorphism. Here $\backslash mathfrak\{g\}$ has a basis $X,Y$ for which the bracket is given by $\backslash left\; X,\; =\; Y$. (This determines the Lie bracket completely, because the axioms imply that $X,X=0$ and $Y,Y=0$.) Over the real numbers, $\backslash mathfrak\{g\}$ can be viewed as the Lie algebra of the Lie group $G=\backslash mathrm\{Aff\}(1,\backslash mathbb\{R\})$ of Affine group of the real line, $x\backslash mapsto\; ax+b$.

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

:

under matrix multiplication, with $a,b\; \backslash in\; \backslash mathbb\{R\}$, $a\; \backslash neq\; 0$. Its Lie algebra is the Lie subalgebra $\backslash mathfrak\{g\}$ of $\backslash mathfrak\{gl\}(2,\backslash mathbb\{R\})$ consisting of all matrices

:

In these terms, the basis above for $\backslash mathfrak\{g\}$ is given by the matrices

:

For any field $F$, the 1-dimensional subspace $F\backslash cdot\; Y$ is an ideal in the 2-dimensional Lie algebra $\backslash mathfrak\{g\}$, by the formula $X,Y=Y\backslash in\; F\backslash cdot\; Y$. Both of the Lie algebras $F\backslash cdot\; Y$ and $\backslash mathfrak\{g\}/(F\backslash cdot\; Y)$ are abelian (because 1-dimensional). In this sense, $\backslash mathfrak\{g\}$ can be broken into abelian "pieces", meaning that it is solvable (though not nilpotent), in the terminology below.

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Three dimensions

• The Heisenberg algebra $\backslash mathfrak\{h\}\_3(F)$ over a field F is the three-dimensional Lie algebra with a basis $X,Y,Z$ such that

:$X,Y\; =\; Z,\backslash quad\; X,Z\; =\; 0,\; \backslash quad\; Y,Z\; =\; 0$.

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, $\backslash mathfrak\{h\}\_3(\backslash mathbb\{R\})$ is the Lie algebra of the Heisenberg group $\backslash mathrm\{H\}\_3(\backslash mathbb\{R\})$, that is, the group of matrices

:$\backslash left(\; \backslash begin\{array\}\{ccc\}$

1&a&c\\ 0&1&b\\ 0&0&1 \end{array}\right)

under matrix multiplication.

For any field F, the center of $\backslash mathfrak\{h\}\_3(F)$ is the 1-dimensional ideal $F\backslash cdot\; Z$, and the quotient $\backslash mathfrak\{h\}\_3(F)/(F\backslash cdot\; Z)$ is abelian, isomorphic to $F^2$. In the terminology below, it follows that $\backslash mathfrak\{h\}\_3(F)$ is nilpotent (though not abelian).

• The Lie algebra $\backslash mathfrak\{so\}(3)$ of the rotation group SO(3) is the space of skew-symmetric 3 x 3 matrices over $\backslash mathbb\{R\}$. 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

:$F\_1,\; =\; F\_3,$

:$F\_2,\; =\; F\_1,$

:$F\_3,\; =\; F\_2.$

The cross product of vectors in $\backslash mathbb\{R\}^3$ is given by the same formula in terms of the standard basis; so that Lie algebra is isomorphic to $\backslash mathfrak\{so\}(3)$. Also, $\backslash mathfrak\{so\}(3)$ is equivalent to the Spin (physics) angular-momentum component operators for spin-1 particles in quantum mechanics.

The Lie algebra $\backslash mathfrak\{so\}(3)$ 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 $\backslash mathfrak\{so\}(3)$.

• Another simple Lie algebra of dimension 3, in this case over $\backslash mathbb\{C\}$, is the space $\backslash mathfrak\{sl\}(2,\backslash mathbb\{C\})$ of 2 x 2 matrices of trace zero. A basis is given by the three matrices

( \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:

:$H,\; =\; 2E,$

:$H,\; =\; -2F,$

:$E,\; =\; H.$

Using these formulas, one can show that the Lie algebra $\backslash mathfrak\{sl\}(2,\backslash mathbb\{C\})$ is simple, and classify its finite-dimensional representations (defined below). In the terminology of quantum mechanics, one can think of E and F as ladder operator. Indeed, for any representation of $\backslash mathfrak\{sl\}(2,\backslash mathbb\{C\})$, 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.

The Lie algebra $\backslash mathfrak\{sl\}(2,\backslash mathbb\{C\})$ is isomorphic to the complexification of $\backslash mathfrak\{so\}(3)$, meaning the tensor product $\backslash mathfrak\{so\}(3)\backslash otimes\_\{\backslash mathbb\{R\}\}\backslash mathbb\{C\}$. The formulas for the Lie bracket are easier to analyze in the case of $\backslash mathfrak\{sl\}(2,\backslash mathbb\{C\})$. As a result, it is common to analyze complex representations of the group $\backslash mathrm\{SO\}(3)$ by relating them to representations of the Lie algebra $\backslash mathfrak\{sl\}(2,\backslash mathbb\{C\})$.

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Infinite dimensions

• The Lie algebra of vector fields on a smooth manifold of positive dimension is an infinite-dimensional Lie algebra over $\backslash mathbb\{R\}$.
• The Kac–Moody algebras are a large class of infinite-dimensional Lie algebras, say over $\backslash mathbb\{C\}$, with structure much like that of the finite-dimensional simple Lie algebras (such as $\backslash mathfrak\{sl\}(n,\backslash C)$).
• The Moyal bracket 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 $V\backslash mapsto\; L(V)$, 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.

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Representations

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Definitions Given a vector space V, let $\backslash mathfrak\{gl\}(V)$ 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 $\backslash mathfrak\{g\}$ on V is a Lie algebra homomorphism

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

That is, $\backslash pi$ sends each element of $\backslash mathfrak\{g\}$ to a linear map from V to itself, in such a way that the Lie bracket on $\backslash mathfrak\{g\}$ 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 $\backslash mathfrak\{gl\}(n,F)$ for some positive integer n.

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Adjoint representation For any Lie algebra $\backslash mathfrak\{g\}$, the adjoint representation is the representation

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

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

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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 $\backslash mathfrak\{g\}$. Indeed, in the semisimple case, the adjoint representation is already faithful. Rather, the goal is to understand all possible representations of $\backslash mathfrak\{g\}$. For a semisimple Lie algebra over a field of characteristic zero, Weyl's theorem says that every finite-dimensional representation is a direct sum of irreducible representations (those with no nontrivial invariant subspaces). The finite-dimensional irreducible representations are well understood from several points of view; see the representation theory of semisimple Lie algebras and the Weyl character formula.

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Universal enveloping algebra The functor that takes an associative algebra A over a field F to A as a Lie algebra (by $X,Y:=XY-YX$) has a left adjoint $\backslash mathfrak\{g\}\backslash mapsto\; U(\backslash mathfrak\{g\})$, called the universal enveloping algebra. To construct this: given a Lie algebra $\backslash mathfrak\{g\}$ over F, let

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

be the tensor algebra on $\backslash mathfrak\{g\}$, also called the free associative algebra on the vector space $\backslash mathfrak\{g\}$. Here $\backslash otimes$ denotes the tensor product of F-vector spaces. Let I be the two-sided ideal in $T(\backslash mathfrak\{g\})$ generated by the elements $XY-YX-X,Y$ for $X,Y\backslash in\backslash mathfrak\{g\}$; then the universal enveloping algebra is the quotient ring $U(\backslash mathfrak\{g\})\; =\; T(\backslash mathfrak\{g\})\; /\; I$. It satisfies the Poincaré–Birkhoff–Witt theorem: if $e\_1,\backslash ldots,e\_n$ is a basis for $\backslash mathfrak\{g\}$ as an F-vector space, then a basis for $U(\backslash mathfrak\{g\})$ is given by all ordered products $e\_1^\{i\_1\}\backslash cdots\; e\_n^\{i\_n\}$ with $i\_1,\backslash ldots,i\_n$ natural numbers. In particular, the map $\backslash mathfrak\{g\}\backslash to\; U(\backslash mathfrak\{g\})$ is injective.

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

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Representation theory in physics The representation theory of Lie algebras plays an important role in various parts of theoretical physics. There, one considers operators on the space of states that satisfy certain natural commutation relations. These commutation relations typically come from a symmetry of the problem—specifically, they are the relations of the Lie algebra of the relevant symmetry group. An example is the angular momentum operators, whose commutation relations are those of the Lie algebra $\backslash mathfrak\{so\}(3)$ of the rotation group $\backslash mathrm\{SO\}(3)$. 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 the hydrogen atom, for example, quantum mechanics textbooks classify (more or less explicitly) the finite-dimensional irreducible representations of the Lie algebra $\backslash mathfrak\{so\}(3)$.

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Structure theory and classification

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

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Abelian, nilpotent, and solvable

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

A Lie algebra $\backslash mathfrak\{g\}$ is abelian if the Lie bracket vanishes; that is, x, y = 0 for all x and y in $\backslash mathfrak\{g\}$. In particular, the Lie algebra of an abelian Lie group (such as the group $\backslash mathbb\{R\}^n$ under addition or the torus $\backslash mathbb\{T\}^n$) is abelian. Every finite-dimensional abelian Lie algebra over a field $F$ is isomorphic to $F^n$ for some $n\backslash geq\; 0$, meaning an n-dimensional vector space with Lie bracket zero.

A more general class of Lie algebras is defined by the vanishing of all commutators of given length. First, the commutator subalgebra (or derived subalgebra) of a Lie algebra $\backslash mathfrak\{g\}$ is $\backslash mathfrak\{g\},\backslash mathfrak\{g\}$, meaning the linear subspace spanned by all brackets $x,y$ with $x,y\backslash in\backslash mathfrak\{g\}$. The commutator subalgebra is an ideal in $\backslash mathfrak\{g\}$, in fact the smallest ideal such that the quotient Lie algebra is abelian. It is analogous to the commutator subgroup of a group.

A Lie algebra $\backslash mathfrak\{g\}$ is nilpotent if the lower central series

$\backslash mathfrak\{g\}\; \backslash supseteq\; \backslash mathfrak\{g\},\backslash mathfrak\{g\}\; \backslash supseteq\; [\backslash mathfrak\{g\},\backslash mathfrak\{g\},\backslash mathfrak\{g\}]\; \backslash supseteq\; [[\backslash mathfrak\{g\},\backslash mathfrak\{g\},\backslash mathfrak\{g\}],\backslash mathfrak\{g\}]\; \backslash supseteq\; \backslash cdots$

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

$0=\backslash mathfrak\{a\}\_0\; \backslash subseteq\; \backslash mathfrak\{a\}\_1\; \backslash subseteq\; \backslash cdots\; \backslash subseteq\; \backslash mathfrak\{a\}\_r\; =\; \backslash mathfrak\{g\},$

such that $\backslash mathfrak\{a\}\_j/\backslash mathfrak\{a\}\_\{j-1\}$ is central in $\backslash mathfrak\{g\}/\backslash mathfrak\{a\}\_\{j-1\}$ for each j. By Engel's theorem, a Lie algebra over any field is nilpotent if and only if for every u in $\backslash mathfrak\{g\}$ the adjoint endomorphism

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

is nilpotent.

More generally, a Lie algebra $\backslash mathfrak\{g\}$ is said to be solvable if the derived series:

$\backslash mathfrak\{g\}\; \backslash supseteq\; \backslash mathfrak\{g\},\backslash mathfrak\{g\}\; \backslash supseteq\; [\backslash mathfrak\{g\},\backslash mathfrak\{g\},\backslash mathfrak\{g\},\backslash mathfrak\{g\}]\; \backslash supseteq\; [[\backslash mathfrak\{g\},\backslash mathfrak\{g\},\backslash mathfrak\{g\},\backslash mathfrak\{g\}],[\backslash mathfrak\{g\},\backslash mathfrak\{g\},\backslash mathfrak\{g\},\backslash mathfrak\{g\}]]\; \backslash supseteq\; \backslash cdots$

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

$0=\backslash mathfrak\{m\}\_0\; \backslash subseteq\; \backslash mathfrak\{m\}\_1\; \backslash subseteq\; \backslash cdots\; \backslash subseteq\; \backslash mathfrak\{m\}\_r\; =\; \backslash mathfrak\{g\},$

such that $\backslash mathfrak\{m\}\_\{j-1\}$ is an ideal in $\backslash mathfrak\{m\}\_\{j\}$ with $\backslash mathfrak\{m\}\_\{j\}/\backslash mathfrak\{m\}\_\{j-1\}$ abelian for each j.

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

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

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Simple and semisimple

A Lie algebra $\backslash mathfrak\{g\}$ is called simple if it is not abelian and the only ideals in $\backslash mathfrak\{g\}$ are 0 and $\backslash mathfrak\{g\}$. (In particular, a one-dimensional—necessarily abelian—Lie algebra $\backslash mathfrak\{g\}$ is by definition not simple, even though its only ideals are 0 and $\backslash mathfrak\{g\}$.) A finite-dimensional Lie algebra $\backslash mathfrak\{g\}$ is called semisimple if the only solvable ideal in $\backslash mathfrak\{g\}$ is 0. In characteristic zero, a Lie algebra $\backslash mathfrak\{g\}$ is semisimple if and only if it is isomorphic to a product of simple Lie algebras, $\backslash mathfrak\{g\}\; \backslash cong\; \backslash mathfrak\{g\}\_1\; \backslash times\; \backslash cdots\; \backslash times\; \backslash mathfrak\{g\}\_r$.

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

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

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

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

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

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

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Cartan's criterion

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

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

where tr denotes the trace of a linear operator. Namely: a Lie algebra $\backslash mathfrak\{g\}$ is semisimple if and only if the Killing form is nondegenerate. A Lie algebra $\backslash mathfrak\{g\}$ is solvable if and only if $K(\backslash mathfrak\{g\},\backslash mathfrak\{g\},\backslash mathfrak\{g\})=0.$

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Classification

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

The simple Lie algebras of finite dimension over an algebraically closed field F of characteristic zero were classified by Killing and Cartan in the 1880s and 1890s, using . 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 _n is $\backslash mathfrak\{sl\}(n+1,F)$, B _n is $\backslash mathfrak\{so\}(2n+1,F)$, C _n is $\backslash mathfrak\{sp\}(2n,F)$, and D _n is $\backslash mathfrak\{so\}(2n,F)$. The other five are known as the exceptional Lie algebras.

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

In the years leading up to 2004, the finite-dimensional simple Lie algebras over an algebraically closed field of characteristic $p>3$ were classified by Richard Earl Block, Robert Lee Wilson, Alexander Premet, and Helmut Strade. (See restricted Lie algebra#Classification of simple Lie algebras.) It turns out that there are many more simple Lie algebras in positive characteristic than in characteristic zero.

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Relation to Lie groups

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

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

For simply connected Lie groups, there is a complete correspondence: taking the Lie algebra gives an equivalence of categories from simply connected Lie groups to Lie algebras of finite dimension over $\backslash mathbb\{R\}$.

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

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

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

Lie theory also does not work so neatly for infinite-dimensional representations of a finite-dimensional group. Even for the additive group $G=\backslash mathbb\{R\}$, 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 of Harish-Chandra modules is a more subtle relation between infinite-dimensional representations for groups and Lie algebras.

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Real form and complexification

Given a complex Lie algebra $\backslash mathfrak\; g$, a real Lie algebra $\backslash mathfrak\{g\}\_0$ is said to be a real form of $\backslash mathfrak\; g$ if the complexification $\backslash mathfrak\{g\}\_0\; \backslash otimes\_\{\backslash mathbb\; R\}\; \backslash mathbb\{C\}$ is isomorphic to $\backslash mathfrak\{g\}$. A real form need not be unique; for example, $\backslash mathfrak\{sl\}(2,\backslash mathbb\{C\})$ has two real forms up to isomorphism, $\backslash mathfrak\{sl\}(2,\backslash mathbb\{R\})$ and $\backslash mathfrak\{su\}(2)$.

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

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Lie algebra with additional structures

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

For example, the of a simply connected topological space form a graded Lie algebra, using the Whitehead product. In a related construction, Daniel Quillen used differential graded Lie algebras over the $\backslash mathbb\{Q\}$ to describe rational homotopy theory in algebraic terms.

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Lie ring

The definition of a Lie algebra over a field extends to define a Lie algebra over any commutative ring R. Namely, a Lie algebra $\backslash mathfrak\{g\}$ over R is an R-module with an alternating R-bilinear map $\backslash \backslash colon\; \backslash mathfrak\{g\}\; \backslash times\; \backslash mathfrak\{g\}\; \backslash to\; \backslash mathfrak\{g\}$ that satisfies the Jacobi identity. A Lie algebra over the ring $\backslash mathbb\{Z\}$ of is sometimes called a Lie ring. (This is not directly related to the notion of a Lie group.)

Lie rings are used in the study of finite (for a prime number p) through the Lazard correspondence. The lower central factors of a finite p-group are finite abelian p-groups. The direct sum of the lower central factors is given the structure of a Lie ring by defining the bracket to be the commutator of two coset representatives; see the example below.

p-adic Lie groups are related to Lie algebras over the field $\backslash mathbb\{Q\}\_p$ of as well as over the ring $\backslash mathbb\{Z\}\_p$ of . Part of Claude Chevalley's construction of the finite groups of Lie type involves showing that a simple Lie algebra over the complex numbers comes from a Lie algebra over the integers, and then (with more care) a group scheme over the integers.

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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 $G\; =\; G\_1\; \backslash supseteq\; G\_2\; \backslash supseteq\; G\_3\; \backslash supseteq\; \backslash cdots\; \backslash supseteq\; G\_n\; \backslash supseteq\; \backslash cdots$ be a filtration of 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 of G.) Then

: $L\; =\; \backslash bigoplus\_\{i\backslash geq\; 1\}\; G\_i/G\_\{i+1\}$

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 $G\_i/G\_\{i+1\}\; \backslash times\; G\_j/G\_\{j+1\}\; \backslash to\; G\_\{i+j\}/G\_\{i+j+1\}$ given by commutators in the group:

: $xG\_\{i+1\},\; :=\; x,yG\_\{i+j+1\}.$

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

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Definition using category-theoretic notation

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

For the category-theoretic definition of Lie algebras, two braiding isomorphisms are needed. If is a vector space, the interchange isomorphism $\backslash tau:\; A\backslash otimes\; A\; \backslash to\; A\backslash otimes\; A$ is defined by

$\backslash tau(x\backslash otimes\; y)=\; y\backslash otimes\; x.$

The cyclic-permutation braiding $\backslash sigma:A\backslash otimes\; A\backslash otimes\; A\; \backslash to\; A\backslash otimes\; A\backslash otimes\; A$ is defined as

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

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

$\backslash sigma(x\backslash otimes\; y\backslash otimes\; z)=\; y\backslash otimes\; z\backslash otimes\; x.$

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

$\backslash cdot,\backslash cdot\backslash colon\; A\backslash otimes\; A\backslash rightarrow\; A$

that satisfies the two morphism equalities

$\backslash cdot,\backslash cdot\backslash circ(\backslash mathrm\{id\}+\backslash tau)=0,$

and

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

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Generalization

Several generalizations of a Lie algebra have been proposed, many from physics. Among them are graded Lie algebras, , ,

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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
• Orthogonal symmetric Lie algebra
• Poisson algebra
• Pre-Lie algebra
• Quantum groups
• Moyal bracket
• Quasi-Frobenius Lie algebra
• Quasi-Lie algebra
• Restricted Lie algebra
• Serre relations
• Spencer cohomology

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Remarks

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Sources

• Nicolas Bourbaki (1989). 9783540642428, Springer. . ISBN 9783540642428
• (2025). 9781846280405, Springer. ISBN 9781846280405
• Brian C. Hall (2025). 9783319134666, Springer. ISBN 9783319134666
• James E. Humphreys (1978). 9780387900537, Springer-Verlag. . ISBN 9780387900537
• Nathan Jacobson (1979). 9780486638324, Dover. ISBN 9780486638324
• Jean-Pierre Serre (2025). 9783540550082, Springer. ISBN 9783540550082
• Veeravalli S. Varadarajan (1984). 9780387909691, Springer. ISBN 9780387909691

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External links

Categories: Lie Groups

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