Editing Lie algebra (section)

== Examples ==
=== Matrix Lie algebras ===
A [[Linear group|matrix group]] is a Lie group consisting of invertible matrices, <math>G\subset \mathrm{GL}(n,\mathbb{R})</math>, where the group operation 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{R})</math>: this consists of derivatives of smooth curves in ''G'' at the [[identity matrix]] <math>I</math>:
:<math>\mathfrak{g} = \{ X = c'(0) \in M_n(\mathbb{R}) : \text{ smooth } c: \mathbb{R}\to G, \ c(0) = I \}.</math>
The Lie bracket of <math>\mathfrak{g}</math> is given by the commutator of matrices, <math>[X,Y]=XY-YX</math>. Given a Lie algebra <math>\mathfrak{g}\subset \mathfrak{gl}(n,\mathbb{R})</math>, one can recover the Lie group as the subgroup generated by the [[matrix exponential]] of elements of <math>\mathfrak{g}</math>.<ref>{{harvnb|Varadarajan|1984|loc=section 2.10, Remark 2.}}</ref> (To be precise, this gives the [[identity component]] of ''G'', if ''G'' is not connected.) Here the exponential mapping <math>\exp: M_n(\mathbb{R})\to M_n(\mathbb{R})</math> is defined by <math>\exp(X) = I + X + \tfrac{1}{2!}X^2 + \tfrac{1}{3!}X^3 + \cdots</math>, which converges for every matrix <math>X</math>.

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

Here are some matrix Lie groups and their Lie algebras.<ref>{{harvnb|Hall|2015|loc=§3.4.}}</ref>

* For a positive integer ''n'', the [[special linear group]] <math>\mathrm{SL}(n,\mathbb{R})</math> consists of all real {{math|''n''&nbsp;×&nbsp;''n''}} matrices with determinant 1. This is the group of linear maps from <math>\mathbb{R}^n</math> to itself that preserve volume and [[orientability|orientation]]. More abstractly, <math>\mathrm{SL}(n,\mathbb{R})</math> is the [[commutator subgroup]] of the general linear group <math>\mathrm{GL}(n,\R)</math>. Its Lie algebra <math>\mathfrak{sl}(n,\mathbb{R})</math> consists of all real {{math|''n''&nbsp;×&nbsp;''n''}} matrices with [[Trace (linear algebra)|trace]] 0. Similarly, one can define the analogous complex Lie group <math>{\rm SL}(n,\mathbb{C})</math> and its Lie algebra <math>\mathfrak{sl}(n,\mathbb{C})</math>.
* The [[orthogonal group]] <math>\mathrm{O}(n)</math> plays a basic role in geometry: it is the group of linear maps from <math>\mathbb{R}^n</math> to itself that preserve the length of vectors. For example, rotations and reflections belong to <math>\mathrm{O}(n)</math>. Equivalently, this is the group of ''n'' x ''n'' orthogonal matrices, meaning that <math>A^{\mathrm{T}}=A^{-1}</math>, where <math>A^{\mathrm{T}}</math> denotes the [[transpose]] of a matrix. The orthogonal group has two connected components; the identity component is called the ''special orthogonal group'' <math>\mathrm{SO}(n)</math>, consisting of the orthogonal matrices with determinant 1. Both groups have the same Lie algebra <math>\mathfrak{so}(n)</math>, the subspace of skew-symmetric matrices in <math>\mathfrak{gl}(n,\mathbb{R})</math> (<math>X^{\rm T}=-X</math>). See also [[Skew-symmetric matrix#Infinitesimal rotations|infinitesimal rotations with skew-symmetric matrices]].
:The complex orthogonal group <math>\mathrm{O}(n,\mathbb{C})</math>, its identity component <math>\mathrm{SO}(n,\mathbb{C})</math>, and the Lie algebra <math>\mathfrak{so}(n,\mathbb{C})</math> are given by the same formulas applied to ''n'' x ''n'' complex matrices. Equivalently, <math>\mathrm{O}(n,\mathbb{C})</math> is the subgroup of <math>\mathrm{GL}(n,\mathbb{C})</math> that preserves the standard [[symmetric bilinear form]] on <math>\mathbb{C}^n</math>.
* The [[unitary group]] <math>\mathrm{U}(n)</math> is the subgroup of <math>\mathrm{GL}(n,\mathbb{C})</math> that preserves the length of vectors in <math>\mathbb{C}^n</math> (with respect to the standard [[Hermitian inner product]]). Equivalently, this is the group of ''n''&nbsp;×&nbsp;''n'' unitary matrices (satisfying <math>A^*=A^{-1}</math>, where <math>A^*</math> denotes the [[conjugate transpose]] of a matrix). Its Lie algebra <math>\mathfrak{u}(n)</math> consists of the skew-hermitian matrices in <math>\mathfrak{gl}(n,\mathbb{C})</math> (<math>X^*=-X</math>). This is a Lie algebra over <math>\mathbb{R}</math>, not over <math>\mathbb{C}</math>. (Indeed, ''i'' times a skew-hermitian matrix is hermitian, rather than skew-hermitian.) Likewise, the unitary group <math>\mathrm{U}(n)</math> is a real Lie subgroup of the complex Lie group <math>\mathrm{GL}(n,\mathbb{C})</math>. For example, <math>\mathrm{U}(1)</math> is the [[circle group]], and its Lie algebra (from this point of view) is <math>i\mathbb{R}\subset \mathbb{C}=\mathfrak{gl}(1,\mathbb{C})</math>.
* The [[special unitary group]] <math>\mathrm{SU}(n)</math> is the subgroup of matrices with determinant 1 in <math>\mathrm{U}(n)</math>. Its Lie algebra <math>\mathfrak{su}(n)</math> consists of the skew-hermitian matrices with trace zero.
*The [[symplectic group]] <math>\mathrm{Sp}(2n,\R)</math> is the subgroup of <math>\mathrm{GL}(2n,\mathbb{R})</math> that preserves the standard [[symplectic vector space|alternating bilinear form]] on <math>\mathbb{R}^{2n}</math>. Its Lie algebra is the [[symplectic Lie algebra]] <math>\mathfrak{sp}(2n,\mathbb{R})</math>.
*The [[classical Lie algebra]]s are those listed above, along with variants over any field.

=== 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 <math>\mathfrak{g}</math> of dimension 2 over any field ''F'', up to isomorphism.<ref>{{harvnb|Erdmann|Wildon|2006|loc=Theorem 3.1.}}</ref> Here <math>\mathfrak{g}</math> has a basis <math>X,Y</math> for which the bracket is given by <math> \left [X, Y\right ] = Y</math>. (This determines the Lie bracket completely, because the axioms imply that <math>[X,X]=0</math> and <math>[Y,Y]=0</math>.) Over the real numbers, <math>\mathfrak{g}</math> can be viewed as the Lie algebra of the Lie group <math>G=\mathrm{Aff}(1,\mathbb{R})</math> of [[Affine group|affine transformations]] of the real line, <math>x\mapsto ax+b</math>.

:The affine group ''G'' can be identified with the group of matrices
::<math> \left( \begin{array}{cc} a & b\\ 0 & 1 \end{array} \right) </math>
:under matrix multiplication, with <math>a,b \in \mathbb{R} </math>, <math>a \neq 0</math>. Its Lie algebra is the Lie subalgebra <math>\mathfrak{g}</math> of <math>\mathfrak{gl}(2,\mathbb{R})</math> consisting of all matrices
::<math> \left( \begin{array}{cc} c & d\\ 0 & 0 \end{array}\right). </math>
:In these terms, the basis above for <math>\mathfrak{g}</math> is given 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>

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

=== Three dimensions ===
* The [[Heisenberg algebra]] <math>\mathfrak{h}_3(F)</math> over a field ''F'' is the three-dimensional Lie algebra with a basis <math>X,Y,Z</math> such that<ref>{{harvnb|Erdmann|Wildon|2006|loc=section 3.2.1.}}</ref>
::<math>[X,Y] = Z,\quad [X,Z] = 0, \quad [Y,Z] = 0</math>.
:It can be viewed as the Lie algebra 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>

:Over the real numbers, <math>\mathfrak{h}_3(\mathbb{R})</math> is the Lie algebra of the [[Heisenberg group]] <math>\mathrm{H}_3(\mathbb{R})</math>, that is, the group of matrices
::<math>\left( \begin{array}{ccc}
1&a&c\\
0&1&b\\
0&0&1
\end{array}\right)
</math>
:under matrix multiplication.

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

* The Lie algebra <math>\mathfrak{so}(3)</math> of the [[rotation group SO(3)]] is the space of skew-symmetric 3 x 3 matrices over <math>\mathbb{R}</math>. A basis is given 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 cross product of vectors in <math>\mathbb{R}^3</math> is given by the same formula in terms of the standard basis; so that Lie algebra is isomorphic to <math>\mathfrak{so}(3)</math>. Also, <math>\mathfrak{so}(3)</math> is equivalent to the [[Spin (physics)]] angular-momentum component operators for spin-1 particles in [[quantum mechanics]].<ref name="quantum">{{harvnb|Wigner|1959|loc=Chapters 17 and 20.}}</ref>

:The Lie algebra <math>\mathfrak{so}(3)</math> 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 <math>\mathfrak{so}(3)</math>.

* Another simple Lie algebra of dimension 3, in this case over <math>\mathbb{C}</math>, is the space <math>\mathfrak{sl}(2,\mathbb{C})</math> of 2 x 2 matrices of trace zero. A basis is given by the three matrices
:<math>H= \left( \begin{array}{cc} 1 & 0\\ 0 & -1 \end{array} \right),\ E =\left
( \begin{array}{cc} 0 & 1\\ 0 & 0 \end{array} \right),\ F =\left( \begin{array}{cc} 0 & 0\\ 1 & 0 \end{array} \right).</math>
{{multiple image
 | width = 220
 | footer = The action of <math>\mathfrak{sl}(2,\mathbb{C})</math> on the [[Riemann sphere]] <math>\mathbb{CP}^1</math>. In particular, the Lie brackets of the vector fields shown are: <math>[H,E]=2E</math>, <math>[H,F]=-2F</math>, <math>[E,F]=H</math>.
 | image1 = Vector field H.png
 | alt1 = Vector field H
 | caption1 = H
 | image2 = Vector field E.png
 | alt2 = Vector field E
 | caption2 = E
 | image3 = Vector field F.png
 | alt3 = Vector field F
 | caption3 = F
}}
:The Lie bracket is given by:
::<math>[H, E] = 2E,</math>
::<math>[H, F] = -2F,</math>
::<math>[E, F] = H.</math>

:Using these formulas, one can show that the Lie algebra <math>\mathfrak{sl}(2,\mathbb{C})</math> is simple, and classify its finite-dimensional representations (defined below).<ref>{{harvnb|Erdmann|Wildon|2006|loc=Chapter 8.}}</ref> In the terminology of quantum mechanics, one can think of ''E'' and ''F'' as [[ladder operator|raising and lowering operators]]. Indeed, for any representation of <math>\mathfrak{sl}(2,\mathbb{C})</math>, the relations above imply that ''E'' maps the ''c''-[[eigenspace]] of ''H'' (for a complex number ''c'') into the <math>(c+2)</math>-eigenspace, while ''F'' maps the ''c''-eigenspace into the <math>(c-2)</math>-eigenspace.

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

=== Infinite dimensions ===
* The Lie algebra of vector fields on a smooth manifold of positive dimension is an infinite-dimensional Lie algebra over <math>\mathbb{R}</math>.
* The [[Kac–Moody algebra]]s are a large class of infinite-dimensional Lie algebras, say over <math>\mathbb{C}</math>, with structure much like that of the finite-dimensional simple Lie algebras (such as <math>\mathfrak{sl}(n,\C)</math>).
* The [[Moyal bracket|Moyal algebra]] is an infinite-dimensional Lie algebra that contains all the [[Classical Lie groups#Relationship with bilinear forms|classical Lie algebra]]s 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]] <math>V\mapsto L(V)</math>, 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 <math>L(V)</math> is infinite-dimensional for ''V'' of dimension at least 2.<ref>{{harvnb|Serre|2006|loc=Part I, Chapter IV.}}</ref>