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In [[Mathematics|mathematics]], a '''Lie algebra''' (pronounced {{IPAc-en|l|iː}} {{respell|LEE}}) is a [[Vector space|vector space]] <math>\mathfrak g</math> together with an [[Binary operation|operation]] called the '''Lie bracket''', an [[Alternating multilinear map|alternating bilinear map]] <math>\mathfrak g \times \mathfrak g \rightarrow \mathfrak g</math>, that satisfies the [[Jacobi identity]]. In other words, a Lie algebra is an [[Algebra over a field|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 <math>x</math> and <math>y</math> is denoted <math>[x,y]</math>. A Lie algebra is typically a [[Non-associative algebra|non-associative algebra]]. However, every [[Associative algebra|associative algebra]] gives rise to a Lie algebra, with the Lie bracket defined as the [[Commutator|commutator]] <math>[x,y] = x y - yx </math>.

Lie algebras are closely related to [[Lie group]]s, which are [[Group (mathematics)|group]]s that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its [[Tangent space|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 space|connected]] Lie group unique up to coverings ([[Lie's third theorem]]). This [[Lie group–Lie algebra correspondence|correspondence]] allows one to study the structure and [[List of simple Lie groups|classification]] of Lie groups in terms of Lie algebras.

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 [[Physics:Quantum mechanics|quantum mechanics]] and particle physics.

An elementary example (not directly coming from an associative algebra) is the space of three dimensional vectors <math>\mathfrak{g}=\mathbb{R}^3</math> with the Lie bracket operation defined by the [[Cross product|cross product]] <math>[x,y]=x\times y.</math> This is skew-symmetric since <math>x\times y = -y\times x</math>, and instead of associativity it satisfies the Jacobi identity:
:<math> x\times(y\times z) \ =\ (x\times y)\times z \ +\  y\times(x\times z). </math>
This is the Lie algebra of the Lie group of [[3D rotation group|rotations of space]], and each vector <math>v\in\R^3</math> may be pictured as an infinitesimal rotation around the axis <math>v</math>, with velocity equal to the magnitude of <math>v</math>. The Lie bracket is a measure of the non-commutativity between two rotations: since a rotation commutes with itself, one has the alternating property <math>[x,x]=x\times x = 0</math>.

== History ==
Lie algebras were introduced to study the concept of [[Infinitesimal transformation|infinitesimal transformation]]s by [[Biography:Sophus Lie|Marius Sophus Lie]] in the 1870s,<ref>{{harvnb|O'Connor|Robertson|2000}}</ref> and independently discovered by Wilhelm Killing<ref>{{harvnb|O'Connor|Robertson|2005}}</ref> in the 1880s. The name ''Lie algebra'' was given by [[Biography:Hermann Weyl|Hermann Weyl]] in the 1930s; in older texts, the term ''infinitesimal group'' is used.

== Definitions ==
===Definition of a Lie algebra===
A Lie algebra is a [[Vector space|vector space]] <math>\,\mathfrak{g}</math> over some [[Field (mathematics)|field]] <math>F</math> together with a [[Binary operation|binary operation]] <math>[\,\cdot\,,\cdot\,]: \mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}</math> called the Lie bracket satisfying the following axioms:{{efn|{{harvtxt|Bourbaki|1989|loc=Section 2.}} allows more generally for a [[Module (mathematics)|module]] over a [[Commutative ring|commutative ring]]; in this article, this is called a Lie ring.}}

* Bilinearity,
::<math> [a x + b y, z] = a [x, z] + b [y, z], </math>
::<math> [z, a x + b y] = a[z, x] + b [z, y] </math>
:for all scalars <math>a</math>, <math>b</math> in <math>F</math> and all elements <math>x</math>, ''<math>y</math>'', ''<math>z</math>'' in <math>\mathfrak{g}</math>.

* Alternativity,
::<math> [x,x]=0\ </math>
:for all <math>x</math> in <math>\mathfrak{g}</math>.

* The [[Jacobi identity]],
:: <math> [x,[y,z]] + [y,[z,x]] + [z,[x,y]]  = 0 \ </math>
:for all <math>x</math>, ''<math>y</math>'', ''<math>z</math>'' in <math>\mathfrak{g}</math>.

Using bilinearity to expand the Lie bracket <math> [x+y,x+y] </math> and using alternativity shows that <math> [x,y] + [y,x]=0\ </math> for all elements  <math>x</math>, ''<math>y</math>'' in <math>\mathfrak{g}</math>, showing that bilinearity and alternativity together imply 
* [[Anticommutativity]],
:: <math> [x,y] = -[y,x],\ </math> 
:for all elements  <math>x</math>, ''<math>y</math>'' in <math>\mathfrak{g}</math>. If the field's [[Characteristic (algebra)|characteristic]] is not 2 then anticommutativity implies alternativity, since it implies <math>[x,x]=-[x,x].</math><ref>{{harvnb|Humphreys|1978|p=1}}</ref>

It is customary to denote a Lie algebra by a lower-case fraktur letter such as <math>\mathfrak{g, h, b, n}</math>. If a Lie algebra is associated with a [[Lie group]], then the algebra is denoted by the fraktur version of the group: for example the Lie algebra of [[Special unitary group|SU(''n'')]] is <math>\mathfrak{su}(n)</math>.

===Generators and dimension===
Elements of a Lie algebra <math>\mathfrak{g}</math> are said to [[Generator (mathematics)|generate]] it if the smallest subalgebra containing these elements is <math>\mathfrak{g}</math> itself. The ''dimension'' of a Lie algebra is its [[Dimension (vector space)|dimension as a vector space]] over ''<math>F</math>''. The cardinality of a minimal generating set of a Lie algebra is always less than or equal to its dimension.

See the [[Classification of low-dimensional real Lie algebras|classification of low-dimensional real Lie algebras]] for other small examples.

=== Subalgebras, ideals and homomorphisms ===
The Lie bracket is not required to be associative, meaning that <math>x,y],z]</math> need not equal <math>[x,[y,z</math>. Nonetheless, much of the terminology of associative [[Ring (mathematics)|rings]] and [[Associative algebra|algebra]]s has analogs for Lie algebras. A ''Lie subalgebra''  is a linear subspace <math>\mathfrak{h} \subseteq \mathfrak{g}</math> which is closed under the Lie bracket.  An ''ideal'' <math>\mathfrak i\subseteq\mathfrak{g}</math> is a subalgebra satisfying the stronger condition:<ref>Due to the anticommutativity of the commutator, the notions of a left and right ideal in a Lie algebra coincide.</ref>

:<math>[\mathfrak{g},\mathfrak i]\subseteq \mathfrak i.</math>

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

:<math> \phi: \mathfrak{g}\to\mathfrak{g'}, \quad \phi([x,y])=[\phi(x),\phi(y)] \ \text{for all}\ 
x,y \in \mathfrak g. </math>

As for associative rings, ideals are precisely the [[Kernel (algebra)|kernels]] of homomorphisms; given a Lie algebra <math>\mathfrak{g}</math> and an ideal <math>\mathfrak i</math> in it, one constructs the ''factor algebra'' or ''quotient algebra'' <math>\mathfrak{g}/\mathfrak i</math>, and the first isomorphism theorem holds for Lie algebras.

Since the Lie bracket is a kind of infinitesimal [[Commutator|commutator]] of the corresponding Lie group, two elements <math>x,y\in\mathfrak g</math> are said to ''commute'' if their bracket vanishes: <math>[x,y]=0</math>.

The centralizer subalgebra of a subset <math>S\subset \mathfrak{g}</math> is the set of elements commuting with ''<math>S</math>'': that is, <math>\mathfrak{z}_{\mathfrak g}(S) = \{x\in\mathfrak g\ \mid\ [x, s] = 0 \ \text{ for all } s\in S\}</math>. The centralizer of <math>\mathfrak{g}</math> itself is the ''center''  <math>\mathfrak{z}(\mathfrak{g})</math>. Similarly, for a subspace ''S'', the normalizer subalgebra of ''<math>S</math>'' is <math>\mathfrak{n}_{\mathfrak g}(S) = \{x\in\mathfrak g\ \mid\ [x,s]\in S \ \text{ for all}\ s\in S\}</math>.<ref>{{harvnb|Jacobson|1962|p=28}}</ref> Equivalently, if <math>S</math> is a Lie subalgebra, <math>\mathfrak{n}_{\mathfrak g}(S)</math> is the largest subalgebra such that <math>S</math> is an ideal of <math>\mathfrak{n}_{\mathfrak g}(S)</math>.

==== Examples ====
For <math>\mathfrak{d}(2) \subset \mathfrak{gl}(2)</math>, the commutator of two elements <math>g \in \mathfrak{gl}(2)</math> and <math>d \in \mathfrak{d}(2)</math>:<blockquote><math>\begin{align}
\left[
\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}, 
\begin{bmatrix}
x & 0 \\
0 & y
\end{bmatrix}
\right] &= \begin{bmatrix}
ax & by\\
cx & dy \\
\end{bmatrix} - \begin{bmatrix}
ax & bx\\
cy & dy \\
\end{bmatrix} \\
&= \begin{bmatrix}
0 & b(y-x) \\
c(x-y) & 0
\end{bmatrix}

\end{align}</math></blockquote>shows <math>\mathfrak{d}(2)</math> is a subalgebra, but not an ideal. In fact, every one-dimensional linear subspace of a Lie algebra has an induced abelian Lie algebra structure, which is generally not an ideal. For any simple Lie algebra, all abelian Lie algebras can never be ideals.

=== Direct sum and semidirect product ===
For two Lie algebras <math>\mathfrak{g^{}}</math> and <math>\mathfrak{g'}</math>, their [[Direct sum of modules|direct sum]] Lie algebra is the vector space <math>\mathfrak{g}\oplus\mathfrak{g'}</math>consisting of all pairs <math>\mathfrak{}(x,x'), \,x\in\mathfrak{g}, \ x'\in\mathfrak{g'}</math>, with the operation

:<math> [(x,x'),(y,y')]=([x,y],[x',y']),</math>

so that the copies of <math>\mathfrak g, \mathfrak g'</math> commute with each other: <math>[(x,0), (0,x')] = 0.</math>

Let <math>\mathfrak{g}</math> be a Lie algebra and <math>\mathfrak{i}</math> an ideal of <math>\mathfrak{g}</math>. If the canonical map <math>\mathfrak{g} \to \mathfrak{g}/\mathfrak{i}</math> splits (i.e., admits a section), then <math>\mathfrak{g}</math> is said to be a [[Semidirect product|semidirect product]] of <math>\mathfrak{i}</math> and <math>\mathfrak{g}/\mathfrak{i}</math>, <math>\mathfrak{g}=\mathfrak{g}/\mathfrak{i}\ltimes\mathfrak{i}</math>. See also [[Physics:Lie algebra extension#By semidirect sum|semidirect sum of Lie algebras]].

Levi's theorem says that a finite-dimensional Lie algebra over a field of characteristic zero is a semidirect product of its radical and the complementary subalgebra (Levi subalgebra).

=== Derivations ===
For an [[Algebra over a field|algebra]] ''A'' over a field ''F'', a ''derivation'' of ''A'' is a [[Linear map|linear map]] <math>D\colon A\to A</math> that satisfies the [[General Leibniz rule|Leibniz rule]]
:<math>D(xy) = D(x)y + xD(y)</math>
for all <math>x,y\in A</math>. (The definition makes sense for a possibly [[Non-associative algebra|non-associative algebra]].) Given two derivations <math>D_1</math> and <math>D_2</math>, their commutator <math>[D_1,D_2]:=D_1D_2-D_2D_1</math> is again a derivation. This operation makes the space <math>\text{Der}_k(A)</math> of all derivations of ''A'' over ''F'' into a Lie algebra.<ref>{{harvnb|Humphreys|1978|p=4}}</ref> 

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 very natural way to construct Lie algebras: they are the "infinitesimal automorphisms" of ''A''. Indeed, writing out the condition that
:<math>(1+\epsilon D)(xy)=(1+\epsilon D)(x)\cdot (1+\epsilon D)(y) \pmod{\epsilon^2}</math>
(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 <math>C^{\infty}(X)</math> of smooth functions on a smooth manifold ''X''. Then a derivation of ''A'' over '''R''' is equivalent to a [[Vector field|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 <math>\text{Vect}(X)</math> of vector fields into a Lie algebra (see [[Lie bracket of vector fields]]).<ref>{{harvnb|Varadarajan|2004|p=49}}</ref> Informally speaking, <math>\text{Vect}(X)</math> 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|action]] of a Lie group ''G'' on a manifold ''X'' determines a homomorphism of Lie algebras <math>\mathfrak{g}\to \text{Vect}(X)</math>.

A Lie algebra can be viewed as a non-associative algebra, and so each Lie algebra <math>\mathfrak{g}</math> over a field ''F'' determines its Lie algebra of derivations, <math>\text{Der}_F(\mathfrak{g})</math>. That is, a derivation of <math>\mathfrak{g}</math> is a linear map <math>D\colon \mathfrak{g}\to \mathfrak{g}</math> such that
:<math>D([x,y])=[D(x),y]+[x,D(y)]</math>.
The ''inner derivation'' associated to any <math>x\in\mathfrak g</math> is the adjoint mapping <math>\mathrm{ad}_x</math> defined by <math>\mathrm{ad}_x(y):=[x,y]</math>. (This is a derivation as a consequence of the Jacobi identity.) That gives a homomorphism of Lie algebras, <math>\operatorname{ad}\colon\mathfrak{g}\to \text{Der}_F(\mathfrak{g})</math>. The image <math>\text{Inn}_F(\mathfrak{g})</math> is an ideal in <math>\text{Der}_F(\mathfrak{g})</math>, and the Lie algebra of '''outer derivations''' is defined as the quotient Lie algebra, <math>\text{Out}_F(\mathfrak{g})=\text{Der}_F(\mathfrak{g})/\text{Inn}_F(\mathfrak{g})</math>. (This is exactly analogous to the [[Outer automorphism group|outer automorphism group]] of a group.) For a [[Semisimple Lie algebra|semisimple Lie algebra]] over a field of characteristic zero, every derivation is inner.

==== Examples ====
For example, given a Lie algebra ideal <math>\mathfrak{i} \subset \mathfrak{g}</math> the adjoint representation <math>\mathfrak{ad}_\mathfrak {g}</math> of <math>\mathfrak{g}</math> acts as outer derivations on <math>\mathfrak{i}</math> since <math>[x,i] \subset \mathfrak{i}</math> for any <math>x \in \mathfrak{g}</math> and <math>i \in \mathfrak{i}</math>. For the Lie algebra <math>\mathfrak{b}_n</math> of upper triangular matrices in <math>\mathfrak{gl}(n)</math>, it has an ideal <math>\mathfrak{n}_n</math> of strictly upper triangular matrices (where the only non-zero elements are above the diagonal of the matrix). For instance, the commutator of elements in <math>\mathfrak{b}_3</math> and <math>\mathfrak{n}_3</math> gives<blockquote><math>\begin{align}
\left[
\begin{bmatrix}
a & b & c \\
0 & d & e \\
0 & 0 & f
\end{bmatrix}, 
\begin{bmatrix}
0 & x & y \\
0 & 0 & z \\
0 & 0 & 0
\end{bmatrix}
\right] &= \begin{bmatrix}
0 & ax & ay+bz \\
0 & 0 & dz \\
0 & 0 & 0
\end{bmatrix} - \begin{bmatrix}
0 & dx & ex+yf \\
0 & 0  & fz \\
0 & 0 & 0
\end{bmatrix} \\
&= \begin{bmatrix}
0 & (a-d)x & (a-f)y-ex+bz \\
0 & 0 & (d-f)z \\
0 & 0 & 0
\end{bmatrix}
\end{align}</math></blockquote>shows there exist outer derivations from <math>\mathfrak{b}_3</math> in <math>\text{Der}(\mathfrak{n}_3)</math>.

=== Split Lie algebra ===
Let ''V'' be a finite-dimensional vector space over a field ''F'', <math>\mathfrak{gl}(V)</math> the Lie algebra of linear transformations and <math>\mathfrak{g} \subseteq \mathfrak{gl}(V)</math> a Lie subalgebra. Then <math>\mathfrak{g}</math> is said to be '''split''' if the roots of the characteristic polynomials of all linear transformations in <math>\mathfrak{g}</math> are in the base field ''F''.<ref>{{harvnb|Jacobson|1962|p=42}}</ref> More generally, a finite-dimensional Lie algebra <math>\mathfrak{g}</math> is said to be split if it has a Cartan subalgebra whose image under the [[Adjoint representation|adjoint representation]] <math>\operatorname{ad}: \mathfrak{g} \to \mathfrak{gl}(\mathfrak g)</math> is a split Lie algebra. A split real form of a complex semisimple Lie algebra (cf. #Real form and complexification) is an example of a split real Lie algebra. See also [[Split Lie algebra|split Lie algebra]] for further information.

=== Vector space basis ===
For practical calculations, it is often convenient to choose an explicit vector space basis for the algebra.  A common construction for this basis is sketched in the article structure constants.

===Definition using category-theoretic notation===
Although the definitions above are sufficient for a conventional understanding of Lie algebras, once this is understood, additional insight can be gained by using notation common to [[Category theory|category theory]], that is, by defining a Lie algebra in terms of [[Linear map|linear map]]s—that is, [[Morphism|morphism]]s of the category of vector spaces—without considering individual elements. (In this section, the [[Field (mathematics)|field]] over which the algebra is defined is supposed to be of [[Characteristic (algebra)|characteristic]] different from two.)

For the category-theoretic definition of Lie algebras, two [[Tensor product#Tensor powers and braiding|braiding isomorphisms]] are needed. If {{mvar|A}} is a vector space, the ''interchange isomorphism'' <math>\tau: A\otimes A \to A\otimes A</math> is defined by 
:<math>\tau(x\otimes y)= y\otimes x.</math>
The ''cyclic-permutation braiding'' <math>\sigma:A\otimes A\otimes A \to A\otimes A\otimes A </math> is defined as
:<math>\sigma=(\mathrm{id}\otimes \tau)\circ(\tau\otimes \mathrm{id}),</math>
where <math>\mathrm{id}</math> is the identity morphism.
Equivalently, <math>\sigma</math> is defined by
:<math>\sigma(x\otimes y\otimes z)= y\otimes z\otimes x.</math>

With this notation, a Lie algebra can be defined as an object <math>A</math> in the category of vector spaces together with a [[Morphism|morphism]] 
:<math>[\cdot,\cdot]:A\otimes A\rightarrow A</math> 
that satisfies the two morphism equalities
:<math>[\cdot,\cdot]\circ(\mathrm{id}+\tau)=0,</math>
and
:<math>[\cdot,\cdot]\circ ([\cdot,\cdot]\otimes \mathrm{id}) \circ (\mathrm{id} +\sigma+\sigma^2)=0.</math>

== Examples ==

=== Vector spaces ===
Any vector space <math>V</math> endowed with the identically zero Lie bracket becomes a Lie algebra. Such Lie algebras are called [[Abelian Lie algebra|abelian]], cf. below. Any one-dimensional Lie algebra over a field of characteristic different from 2 is abelian, by the alternating property of the Lie bracket.

=== Associative algebra with commutator bracket ===
* On an [[Associative algebra|associative algebra]] <math>A</math> over a field <math>F</math> with multiplication <math>(x, y) \mapsto xy</math>, a Lie bracket may be defined by the [[Commutator#Ring theory|commutator]] <math>[x,y] = xy - yx</math>. With this bracket, <math>A</math> is a Lie algebra.<ref>{{harvnb|Bourbaki|1989|loc=§1.2. Example 1.}}</ref>  The associative algebra ''A'' is called an ''enveloping algebra'' of the Lie algebra <math>(A, [\,\cdot\, , \cdot \,])</math>. Every Lie algebra can be embedded into one that arises from an associative algebra in this fashion; see [[Universal enveloping algebra|universal enveloping algebra]].
* The [[Endomorphism ring|associative algebra of the endomorphisms]] of an ''F''-vector space <math>V</math> with the above Lie bracket is denoted <math>\mathfrak{gl}(V)</math>.
*For a finite dimensional vector space <math>V = F^n</math>, the previous example is exactly the Lie algebra of ''n'' × ''n'' matrices, denoted <math>\mathfrak{gl}(n, F)</math> or <math>\mathfrak{gl}_n(F)</math>,<ref>{{harvnb|Bourbaki|1989|loc=§1.2. Example 2.}}</ref> and with bracket <math>[X,Y]=XY-YX</math> where adjacency indicates matrix multiplication. This is the Lie algebra of the [[General linear group|general linear group]], consisting of invertible matrices.

=== Special matrices ===
Two important subalgebras of <math>\mathfrak{gl}_n(F)</math> are:

* The matrices of [[Trace (linear algebra)|trace]] zero form the [[Special linear Lie algebra|special linear Lie algebra]] <math>\mathfrak{sl}_n(F)</math>, the Lie algebra of the [[Special linear group|special linear group]] <math>\mathrm{SL}_n(F)</math>.<ref>{{harvnb|Humphreys|1978|p=2}}</ref>
*The skew-hermitian matrices form the unitary Lie algebra <math>\mathfrak u(n)</math>, the Lie algebra of the [[Unitary group|unitary group]]&nbsp;''U''(''n'').

=== Matrix Lie algebras ===
A complex [[Linear group|matrix group]] is a Lie group consisting of matrices, <math>G\subset M_n(\mathbb{C})</math>, where the multiplication of ''G'' is matrix multiplication. The corresponding Lie algebra <math>\mathfrak g</math> is the space of matrices which are tangent vectors to ''G'' inside the linear space <math>M_n(\mathbb{C})</math>: this consists of derivatives of smooth curves in ''G'' at the identity: <blockquote><math>\mathfrak{g} = \{ X = c'(0) \in M_n(\mathbb{C}) \ \mid\ \text{ smooth } c : \mathbb{R}\to G, \ c(0) = I \}.</math></blockquote>The Lie bracket of <math>\mathfrak{g}</math> is given by the commutator of matrices, <math>[X,Y]=XY-YX</math>. Given the Lie algebra, one can recover the Lie group as the image of the [[Matrix exponential|matrix exponential]] mapping <math>\exp: M_n(\mathbb{C})\to M_n(\mathbb{C})</math> defined by <math>\exp(X) = I + X + \tfrac{1}{2!}X^2+\cdots</math>, which converges for every matrix <math>X</math>: that is, <math>G=\exp(\mathfrak g)</math>. 
The following are examples of Lie algebras of matrix Lie groups:<ref>{{harvnb|Hall|2015|loc=§3.4}}</ref>

* The [[Special linear group|special linear group]] <math>{\rm SL}_n(\mathbb{C})</math>, consisting of all {{math|''n''&nbsp;×&nbsp;''n''}} matrices with determinant 1. Its Lie algebra <math>\mathfrak{sl}_n(\mathbb{C})</math> consists of all {{math|''n''&nbsp;×&nbsp;''n''}} matrices with complex entries and trace 0. Similarly, one can define the corresponding real Lie group <math>{\rm SL}_n(\mathbb{R})</math> and its Lie algebra <math>\mathfrak{sl}_n(\mathbb{R})</math>.
* The [[Unitary group|unitary group]] <math>U(n)</math> consists of ''n''&nbsp;×&nbsp;''n'' unitary matrices (satisfying <math>U^*=U^{-1}</math>). Its Lie algebra <math>\mathfrak{u}(n)</math> consists of skew-self-adjoint matrices (<math>X^*=-X</math>).
* The special [[Orthogonal group|orthogonal group]] <math>\mathrm{SO}(n)</math>, consisting of real determinant-one orthogonal matrices (<math>A^{\mathrm{T}}=A^{-1}</math>). Its Lie algebra <math>\mathfrak{so}(n)</math> consists of real skew-symmetric matrices (<math>X^{\rm T}=-X</math>). The full orthogonal group <math>\mathrm{O}(n)</math>, without the determinant-one condition, consists of <math>\mathrm{SO}(n)</math> and a separate connected component, so it has the ''same'' Lie algebra as <math>\mathrm{SO}(n)</math>. See also [[Skew-symmetric matrix#Infinitesimal rotations|infinitesimal rotations with skew-symmetric matrices]]. Similarly, one can define a complex version of this group and algebra, simply by allowing complex matrix entries.

=== Two dimensions ===

* On any field <math>F</math> there is, up to isomorphism, a single two-dimensional nonabelian Lie algebra. With generators ''x, y,''  its bracket is defined as <math> \left [x, y\right ] = y</math>. It generates the [[Affine group#Matrix representation|affine group in one dimension]].

:This can be realized by the matrices: 
::<math> 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).   </math>
Since
:<math> \left( \begin{array}{cc} 1 & c\\ 0 & 0 \end{array}\right)^{n+1} = \left( \begin{array}{cc} 1 & c\\ 0 & 0 \end{array}\right)</math>
for any natural number <math>n</math> and any <math>c</math>, one sees that the resulting Lie group elements are upper triangular 2×2 matrices with unit lower diagonal:
::<math> \exp(a\cdot{}x+b\cdot{}y)= \left( \begin{array}{cc} e^a & \tfrac{b}{a}(e^a-1)\\ 0 & 1 \end{array}\right) = 1 + \tfrac{e^a-1}{a}\left(a\cdot{}x+b\cdot{}y\right). </math>

=== Three dimensions ===
* The Heisenberg algebra <math>{\rm H}_3(\mathbb{R})</math> is a three-dimensional Lie algebra generated by elements {{mvar|x}}, {{mvar|y}}, and {{mvar|z}} with Lie brackets

::<math>[x,y] = z,\quad [x,z] = 0, \quad [y,z] = 0</math>.
:It is usually realized as the space of 3&times;3 strictly upper-triangular matrices, with the commutator Lie bracket and the basis
::<math>
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
</math>

:Any element of the [[Heisenberg group]] has a representation as a product of group generators, i.e., [[Matrix exponential|matrix exponential]]s of these Lie algebra generators, 
::<math>\left( \begin{array}{ccc}
1&a&c\\
0&1&b\\
0&0&1
\end{array}\right)= e^{by} e^{cz} e^{ax}~.
</math>

* The Lie algebra <math>\mathfrak{so}(3)</math> of the group SO(3) is spanned by the three matrices<ref>{{harvnb|Hall|2015|loc=Example 3.27}}</ref>
::<math>
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
</math>
:The commutation relations among these generators are
::<math>[F_1, F_2] = F_3,</math>
:: <math>[F_2, F_3] = F_1,</math>
:: <math>[F_3, F_1] = F_2.</math>

:The three-dimensional [[Euclidean space]] <math>\mathbb{R}^3</math> with the Lie bracket given by the [[Cross product|cross product]] of vectors has the same commutation relations as above: thus, it is isomorphic to <math>\mathfrak{so}(3)</math>. This Lie algebra is unitarily equivalent to the usual [[Physics:Spin|Spin]] angular-momentum component operators for spin-1 particles in [[Physics:Quantum mechanics|quantum mechanics]].

=== Infinite dimensions ===
*[[Kac–Moody algebra]]s are a large class of infinite-dimensional Lie algebras whose structure is very similar to the finite-dimensional cases above.
* The [[Moyal bracket|Moyal algebra]] is an infinite-dimensional Lie algebra that contains all classical Lie algebras as subalgebras.
* The [[Virasoro algebra]] is of paramount importance in [[Physics:String theory|string theory]].

== Representations ==
{{main|Lie algebra representation}}

===Definitions===
Given a vector space ''V'', let <math>\mathfrak{gl}(V)</math> denote the Lie algebra consisting of all linear [[Endomorphism|endomorphism]]s of ''V'', with bracket given by <math>[X,Y]=XY-YX</math>. A ''representation'' of a Lie algebra <math>\mathfrak{g}</math> on ''V'' is a Lie algebra homomorphism
:<math>\pi: \mathfrak g \to \mathfrak{gl}(V).</math>

A representation is said to be ''faithful'' if its kernel is zero. [[Ado's theorem]]<ref>{{harvnb|Jacobson|1962|loc=Ch. VI}}</ref> states that every finite-dimensional Lie algebra over a field of characteristic zero has a faithful representation on a finite-dimensional vector space.

===Adjoint representation===
For any Lie algebra <math>\mathfrak{g}</math>, the adjoint representation is the representation
:<math>\operatorname{ad}\colon\mathfrak{g} \to \mathfrak{gl}(\mathfrak{g})</math>
given by <math>\operatorname{ad}(x)(y) = [x, y]</math>.

===Goals of representation theory===
One important aspect of the study of Lie algebras (especially semisimple Lie algebras) is the study of their representations. (Indeed, most of the books listed in the references section devote a substantial fraction of their pages to representation theory.) 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 <math>\mathfrak{g}</math>. Indeed, in the semisimple case, the adjoint representation is already faithful. Rather the goal is to understand ''all'' possible representations of <math>\mathfrak{g}</math>, up to the natural notion of equivalence. In the semisimple case over a field of characteristic zero, [[Weyl's theorem on complete reducibility|Weyl's theorem]]<ref>{{harvnb|Hall|2015|loc=Theorem 10.9}}</ref> says that every finite-dimensional representation is a direct sum of irreducible representations (those with no nontrivial invariant subspaces). The irreducible representations, in turn, are classified by a [[Lie algebra representation#Classifying finite-dimensional representations of Lie algebras|theorem of the highest weight]].

===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 would be the [[Physics:Angular momentum operator|angular momentum operator]]s, whose commutation relations are those of the Lie algebra <math>\mathfrak{so}(3)</math> of the [[Rotation group SO(3)|rotation group SO(3)]]. Typically, the space of states is very 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 quantum [[Physics:Hydrogen-like atom|hydrogen atom]], for example, quantum mechanics textbooks give (without calling it that) a classification of the irreducible representations of the Lie algebra <math>\mathfrak{so}(3)</math>.

== Structure theory and classification ==

Lie algebras can be classified to some extent. In particular, this has an application to the classification of Lie groups.

=== Abelian, nilpotent, and solvable ===
Analogously to abelian, nilpotent, and solvable groups, defined in terms of the derived subgroups, one can define abelian, nilpotent, and solvable Lie algebras.

A Lie algebra <math>\mathfrak{g}</math> is ''abelian{{anchor|abelian}}'' if the Lie bracket vanishes, i.e. [''x'',''y''] = 0, for all ''x'' and ''y'' in <math>\mathfrak{g}</math>. Abelian Lie algebras correspond to commutative (or [[Abelian group|abelian]]) connected Lie groups such as vector spaces <math>\mathbb{K}^n</math> or [[Torus|tori]] <math>\mathbb{T}^n</math>, and are all of the form <math>\mathfrak{k}^n,</math> meaning an ''n''-dimensional vector space with the trivial Lie bracket.

A more general class of Lie algebras is defined by the vanishing of all commutators of given length. A Lie algebra <math>\mathfrak{g}</math> is ''[[Nilpotent Lie algebra|nilpotent]]'' if the lower central series

:<math> \mathfrak{g} > [\mathfrak{g},\mathfrak{g}] > [[\mathfrak{g},\mathfrak{g}],\mathfrak{g}] > [[[\mathfrak{g},\mathfrak{g}],\mathfrak{g}],\mathfrak{g}] > \cdots</math>

becomes zero eventually. By [[Engel's theorem]], a Lie algebra is nilpotent if and only if for every ''u'' in <math>\mathfrak{g}</math> the adjoint endomorphism

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

is nilpotent.

More generally still, a Lie algebra <math>\mathfrak{g}</math> is said to be ''[[Solvable Lie algebra|solvable]]'' if the derived series:

:<math> \mathfrak{g} > [\mathfrak{g},\mathfrak{g}] > \mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g} > [\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g},\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g}]  > \cdots</math>

becomes zero eventually. 

The second term <math> [\mathfrak{g},\mathfrak{g}]</math> in the derived series (or, equivalently, the second term in the lower central series) is called the commutator subalgebra or commutator ideal of <math>\mathfrak{g}</math>.

Every finite-dimensional Lie algebra has a unique maximal solvable ideal, called its [[Radical of a Lie algebra|radical]]. Under the Lie correspondence, nilpotent (respectively, solvable) connected Lie groups correspond to nilpotent (respectively, solvable) Lie algebras.

=== Simple and semisimple ===
{{main|Semisimple Lie algebra}}
A Lie algebra is "[[Simple Lie algebra|simple]]" if it has no non-trivial ideals and is not abelian. (This implies that a one-dimensional—necessarily abelian—Lie algebra is by definition not simple, even though it has no nontrivial ideals.) A Lie algebra <math>\mathfrak{g}</math> is called ''[[Semisimple Lie algebra|semisimple]]'' if it is isomorphic to a direct sum of simple algebras. There are several equivalent characterizations of semisimple algebras, such as having no nonzero solvable ideals.

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, any finite-dimensional representation of a semisimple Lie algebra is [[Semisimple representation|semisimple]] (i.e., direct sum of irreducible representations). In general, a Lie algebra is called [[Reductive Lie algebra|reductive]] if the adjoint representation is semisimple. Thus, a semisimple Lie algebra is reductive.<!--
 of a Lie algebra <math>\mathfrak{g}</math> over ''F'' is equivalent to the complete reducibility of all finite-dimensional [[Lie algebra representation|representations]] of <math>\mathfrak{g}.</math> An early proof of this statement proceeded via connection with compact groups (Weyl's unitary trick), but later entirely algebraic proofs were found.-->

=== Cartan's criterion ===
[[Cartan's criterion]] gives conditions for a Lie algebra of characteristic zero to be nilpotent, solvable, or semisimple. It is based on the notion of the [[Killing form]], a symmetric bilinear form on <math>\mathfrak{g}</math> defined by the formula
: <math>K(u,v)=\operatorname{tr}(\operatorname{ad}(u)\operatorname{ad}(v)),</math>
where tr denotes the [[Trace (linear algebra)|trace of a linear operator]]. A Lie algebra <math>\mathfrak{g}</math> is semisimple if and only if the Killing form is nondegenerate. A Lie algebra <math>\mathfrak{g}</math> is solvable if and only if <math>K(\mathfrak{g},[\mathfrak{g},\mathfrak{g}])=0.</math>

=== Classification ===
The [[Levi decomposition]] expresses an arbitrary Lie algebra as a semidirect sum of its solvable radical and a semisimple Lie algebra, almost in a canonical way. (Such a decomposition exists for a finite-dimensional Lie algebra over a field of characteristic zero.<ref>{{harvnb|Jacobson|1962|loc=Ch. III, § 9.}}</ref>) Furthermore, semisimple Lie algebras over an algebraically closed field of characteristic zero have been completely classified through their [[Root system|root system]]s.

== Relation to Lie groups ==
{{main|Lie group–Lie algebra correspondence}}
[[Image:Image Tangent-plane.svg|thumb| The tangent space of a [[Sphere|sphere]] at a point <math>x</math>. If <math>x</math> is the identity element, then the tangent space is also a Lie algebra.]]
Although Lie algebras are often studied in their own right, historically they arose as a means to study [[Lie group]]s.

We now briefly outline the relationship between Lie groups and Lie algebras. Any Lie group gives rise to a canonically determined Lie algebra over '''R''' (concretely, ''the tangent space at the identity''). Conversely, for any finite-dimensional Lie algebra <math>\mathfrak g</math>, there exists a corresponding connected Lie group <math>G</math> with Lie algebra <math>\mathfrak g</math>. 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 in particular, have the same universal cover. For instance, the special orthogonal group [[SO(3)]] and the [[Special unitary group|special unitary group]] SU(2) give rise to the same Lie algebra, which is isomorphic to <math>\mathbb{R}^3</math> with the cross-product, but SU(2) is a simply-connected twofold cover of SO(3).

If we consider ''simply connected'' Lie groups, however, we have a one-to-one correspondence: For each (finite-dimensional real) Lie algebra <math>\mathfrak g</math>, there is a unique simply connected Lie group <math>G</math> with Lie algebra <math>\mathfrak g</math>.

The correspondence between Lie algebras and Lie groups is used in several ways, including in the [[List of simple Lie groups|classification of Lie groups]] and the related matter of the [[Representation theory|representation theory]] of Lie groups.  Every representation of a Lie algebra lifts uniquely to a representation of the corresponding simply connected Lie group, and conversely every representation of any Lie group induces a representation of the group's Lie algebra; the representations are in one-to-one correspondence.  Therefore, knowing the representations of a Lie algebra settles the question of representations of the group.

As for classification, it can be shown that any connected Lie group with a given Lie algebra is isomorphic to the universal cover mod a discrete central subgroup.  So classifying Lie groups becomes simply a matter of counting the discrete subgroups of the [[Center (group theory)|center]], once the classification of Lie algebras is known (solved by [[Biography:Élie Cartan|Cartan]] et al. in the [[Semisimple Lie algebra|semisimple]] case).

If the Lie algebra is infinite-dimensional, the issue is more subtle.  In many instances, the exponential map is not even locally a [[Homeomorphism|homeomorphism]] (for example, in Diff('''S'''<sup>1</sup>), one may find diffeomorphisms arbitrarily close to the identity that are not in the image of exp).  Furthermore, some infinite-dimensional Lie algebras are not the Lie algebra of any group.

== Real form and complexification ==
Given a [[Complex Lie algebra|complex Lie algebra]] <math>\mathfrak g</math>, a real Lie algebra <math>\mathfrak{g}_0</math> is said to be a ''real form'' of <math>\mathfrak g</math> if the [[Complexification|complexification]] <math>\mathfrak{g}_0 \otimes_{\mathbb R} \mathbb{C} \simeq \mathfrak{g}</math> is isomorphic to <math>\mathfrak{g}</math>.<ref name="Fulton 26">{{harvnb|Fulton|Harris|1991|loc=§26.1.}}</ref> A real form need not be unique; for example, <math>\mathfrak{sl}_2 \mathbb{C}</math> has two real forms, <math>\mathfrak{sl}_2 \mathbb{R}</math> and <math>\mathfrak{su}_2</math>.<ref name="Fulton 26" />

Given a semisimple finite-dimensional complex Lie algebra <math>\mathfrak g</math>, 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 isomorphisms).<ref name="Fulton 26" /> 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.<ref name="Fulton 26" />

== Lie algebra with additional structures ==
A Lie algebra can be equipped with some additional structures that are assumed to be compatible with the bracket. For example, a [[Graded Lie algebra|graded Lie algebra]] is a Lie algebra with a graded vector space structure. If it also comes with a differential (so that the underlying graded vector space is a [[Chain complex|chain complex]]), then it is called a [[Differential graded Lie algebra|differential graded Lie algebra]].

A [[Simplicial Lie algebra|simplicial Lie algebra]] is a simplicial object in the category of Lie algebras; in other words, it is obtained by replacing the underlying set with a [[Simplicial set|simplicial set]] (so it might be better thought of as a family of Lie algebras).

== Lie ring ==
A ''Lie ring'' arises as a generalisation of Lie algebras, or through the study of the lower central series of [[Group (mathematics)|groups]]. A Lie ring is defined as a nonassociative ring with multiplication that is anticommutative and satisfies the [[Jacobi identity]]. More specifically we can define a Lie ring <math>L</math> to be an [[Abelian group|abelian group]] with an operation <math>[\cdot,\cdot]</math> that has the following properties:

* Bilinearity:

::<math> [x + y, z] = [x, z] + [y, z], \quad  [z, x + y] = [z, x] + [z, y] </math>

:for all ''x'', ''y'', ''z'' &isin; ''L''.

* The ''Jacobi identity'':

:: <math> [x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0 \quad </math>

:for all ''x'', ''y'', ''z'' in ''L''.

* For all ''x'' in ''L'':

::<math> [x,x]=0.</math>

Lie rings need not be [[Lie group]]s under addition.  Any Lie algebra is an example of a Lie ring.  Any associative ring can be made into a Lie ring by defining a bracket operator <math>[x,y] = xy - yx</math>.  Conversely to any Lie algebra there is a corresponding ring, called the [[Universal enveloping algebra|universal enveloping algebra]].

Lie rings are used in the study of finite [[P-group|p-group]]s through the ''Lazard correspondence''.  The lower central factors of a ''p''-group are finite abelian ''p''-groups, so modules over '''Z'''/''p'''''Z'''.  The direct sum of the lower central factors is given the structure of a Lie ring by defining the bracket to be the [[Commutator|commutator]] of two coset representatives.  The Lie ring structure is enriched with another module homomorphism, the ''p''th power map, making the associated Lie ring a so-called restricted Lie ring.

Lie rings are also useful in the definition of a p-adic analytic groups and their endomorphisms by studying Lie algebras over rings of integers such as the p-adic integers.  The definition of finite groups of Lie type due to Chevalley involves restricting from a Lie algebra over the complex numbers to a Lie algebra over the integers, and then reducing modulo ''p'' to get a Lie algebra over a finite field.

=== Examples ===
* Any Lie algebra over a general [[Ring (mathematics)|ring]] instead of a [[Field (mathematics)|field]] is an example of a Lie ring.  Lie rings are ''not'' [[Lie group]]s under addition, despite the name.
* Any associative ring can be made into a Lie ring by defining a bracket operator
:: <math>[x,y] = xy - yx.</math>
* For an example of a Lie ring arising from the study of [[Group (mathematics)|groups]], let <math>G</math> be a group with <math>[x,y]= x^{-1}y^{-1}xy</math> the commutator operation, and let <math>G = G_0 \supseteq G_1 \supseteq G_2 \supseteq \cdots \supseteq G_n \supseteq \cdots</math> be a central series  in <math>G</math> &mdash; that is the commutator subgroup <math>[G_i,G_j]</math> is contained in <math>G_{i+j}</math> for any <math>i,j</math>. Then

:: <math>L = \bigoplus G_i/G_{i+1}</math>

:is a Lie ring with addition supplied by the group operation (which is abelian in each homogeneous part), and the bracket operation given by

:: <math>[xG_i, yG_j] = [x,y]G_{i+j}\ </math>

:extended linearly. The centrality of the series ensures that the commutator <math>[x,y]</math> gives the bracket operation the appropriate Lie theoretic properties.

==See also==
{{div col|colwidth=28em}}
* Adjoint representation of a Lie algebra
* [[Affine Lie algebra]]
* [[Anyonic Lie algebra]]
* [[Automorphism of a Lie algebra]]
* [[Chiral Lie algebra]]
* [[Free Lie algebra]]<!-- Need to be discussed somewhere in this article but not clear where, probably not too early. -->
* [[Index of a Lie algebra]]
* [[Lie algebra cohomology]]
* [[Physics:Lie algebra extension|Lie algebra extension]]
* [[Lie algebra representation]]
* [[Lie bialgebra]]
* [[Lie coalgebra]]
* [[Lie operad]]
* [[Physics:Particle physics and representation theory|Particle physics and representation theory]]
* [[Lie superalgebra]]
* [[Poisson algebra]]
* [[Pre-Lie algebra]]
* Quantum groups
* [[Moyal bracket|Moyal algebra]]
* [[Quasi-Frobenius Lie algebra]]
* [[Quasi-Lie algebra]]
* [[Restricted Lie algebra]]
* Serre relations
* Symmetric Lie algebra <!-- missing article -->
* [[Gelfand–Fuks cohomology]]
{{div col end}}

== Remarks ==
{{notelist|1}}

== References ==
{{reflist|25e}}

== Sources ==
* {{cite book |last=Beltiţă |first= Daniel |title=Smooth Homogeneous Structures in Operator Theory |series=CRC Monographs and Surveys in Pure and Applied Mathematics |volume=137 |publisher=CRC Press |year= 2006 |isbn=978-1-4200-3480-6 |mr=2188389 |url=https://books.google.com/books?id=x8NQc-pWbLQC&q=%22Lie+algebra%22}}
* {{Cite journal |last1=Boza |first1=Luis |last2=Fedriani |first2=Eugenio M. |last3=Núñez |first3=Juan |date=2001-06-01 |title=A new method for classifying complex filiform Lie algebras |journal=Applied Mathematics and Computation |volume=121 |issue=2–3 |pages=169–175 |doi=10.1016/s0096-3003(99)00270-2 |issn=0096-3003}}
* {{cite book |last=Bourbaki |first=Nicolas |title=Lie Groups and Lie Algebras: Chapters 1-3 |year=1989 |isbn=978-3-540-64242-8 |publisher=Springer |url=https://books.google.com/books?id=brSYF_rB2ZcC}}
* {{Cite book |last1=Erdmann |first1=Karin |last2=Wildon |first2=Mark |title=Introduction to Lie Algebras |publisher=Springer |year=2006 |isbn=1-84628-040-0}}
* {{Fulton-Harris}}
* {{cite book |last=Hall |first=Brian C. |title=Lie groups, Lie algebras, and Representations: An Elementary Introduction |edition=2nd |series=Graduate Texts in Mathematics |volume=222 |publisher=Springer |year=2015 |isbn=978-3319134666 |doi=10.1007/978-3-319-13467-3 |issn=0072-5285}}
* {{cite book |last1=Hofmann |first1=Karl H. |last2=Morris |first2=Sidney A |title=The Lie Theory of Connected Pro-Lie Groups |publisher=European Mathematical Society |year=2007 |isbn=978-3-03719-032-6}}
* {{cite book |last=Humphreys |first=James E. |title=Introduction to Lie Algebras and Representation Theory |edition=2nd |series=Graduate Texts in Mathematics |volume=9 |publisher=Springer-Verlag |year=1978 |isbn=978-0-387-90053-7 |url-access=registration |url=https://archive.org/details/introductiontoli00jame}}
* {{cite book |last=Jacobson |first=Nathan |title=Lie algebras |orig-year=1962 |publisher=Dover |year=1979 |isbn=978-0-486-63832-4 |ref={{harvid |Jacobson |1962}}}}
* {{cite book |last1=Kac |first1=Victor G. |display-authors=0 |title=Course notes for MIT 18.745: Introduction to Lie Algebras |url=http://math.mit.edu/~lesha/745lec/ |url-status=bot: unknown |archive-url=https://web.archive.org/web/20100420004313/http://math.mit.edu/~lesha/745lec/ |archive-date=2010-04-20}}
* {{cite journal |last1=Mubarakzyanov |first1=G.M. |title=On solvable Lie algebras |journal=Izv. Vys. Ucheb. Zaved. Matematika |volume=1 |issue=32 |year=1963 |pages=114–123 |mr=153714 |zbl=0166.04104 |url=http://mi.mathnet.ru/eng/ivm2141 |language=ru}}
* {{cite web |last1=O'Connor |first1=J.J |last2=Robertson |first2=E.F. |author-link2=Edmund F. Robertson |title=Biography of Sophus Lie |year=2000 |publisher=MacTutor History of Mathematics Archive |url=http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Lie.html}}
* {{cite web |last1=O'Connor |first1=J.J |last2=Robertson |first2=E.F. |title=Biography of Wilhelm Killing |year=2005 |publisher=MacTutor History of Mathematics Archive |url=http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Killing.html}}
* {{cite journal |last1=Popovych |first1=R.O. |last2=Boyko |first2=V.M. |last3=Nesterenko |first3=M.O. |last4=Lutfullin |first4=M.W. |display-authors=0 |title=Realizations of real low-dimensional Lie algebras |journal=J. Phys. A: Math. Gen. |volume=36 |issue=26 |year=2003 |pages=7337–60 |doi=10.1088/0305-4470/36/26/309 |arxiv=math-ph/0301029 |bibcode=2003JPhA...36.7337P |s2cid=9800361}}
* {{cite book |last=Serre |first= Jean-Pierre |title=Lie Algebras and Lie Groups |edition=2nd |publisher=Springer |year=2006 |isbn=978-3-540-55008-2}}
* {{cite book |last=Steeb |first= Willi-Hans |title=Continuous Symmetries, Lie Algebras, Differential Equations and Computer Algebra |edition=2nd |publisher= World Scientific |year= 2007 |isbn=978-981-270-809-0 |mr=2382250 |doi=10.1142/6515}}
* {{cite book |last=Varadarajan |first=Veeravalli S. |title=Lie Groups, Lie Algebras, and Their Representations |edition=1st |publisher=Springer |year=2004 |isbn=978-0-387-90969-1}}

==External links==
* {{springer|title=Lie algebra|id=p/l058370}}
* {{cite web |last=McKenzie |first=Douglas |year=2015 |url=http://www.liealgebrasintro.com |title=An Elementary Introduction to Lie Algebras for Physicists}}


{{DEFAULTSORT:Lie Algebra}}
[[Category:Lie groups]]
[[Category:Lie algebras| ]]

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