Editing Lie algebra (section)

=== 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>.