Editing Witt algebra

{{Short description|Algebra of meromorphic vector fields on the Riemann sphere}}
In [[mathematics]], the complex  '''Witt algebra''', named after [[Ernst Witt]], is the [[Lie algebra]] of meromorphic vector fields defined on the [[Riemann sphere]] that are holomorphic except at two fixed points.  It is also the [[complexification]] of the Lie algebra of polynomial vector fields on a circle, and the Lie algebra of derivations of the ring '''C'''[''z'',''z''<sup>&minus;1</sup>].

There are some related Lie algebras defined over finite fields, that are also called Witt algebras.

The complex Witt algebra was first defined by [[Élie Cartan]] (1909), and its analogues over finite fields were studied by Witt in the 1930s.

==Basis==

A basis for the Witt algebra is given by the [[vector field]]s <math>L_n=-z^{n+1} \frac{\partial}{\partial z}</math>, for ''n'' in ''<math>\mathbb Z</math>''.

The [[Lie derivative|Lie bracket]] of two [[Basis (linear algebra)|basis vector]] fields is given by

:<math>[L_m,L_n]=(m-n)L_{m+n}.</math>

This algebra has a [[Group extension#Central extension|central extension]] called the [[Virasoro algebra]] that is important in [[two-dimensional conformal field theory]] and [[string theory]].

Note that by restricting ''n'' to 1,0,-1, one gets a [[subalgebra]]. Taken over the field of complex numbers, this is just the Lie algebra <math>\mathfrak{sl}(2,\mathbb{C})</math> of the [[Lorentz group]] <math>\mathrm{SO}(3,1)</math>. Over the reals, it is the algebra [[SL(2,R)|'''''sl'''''(2,R)]] = '''''su'''''(1,1).
Conversely, '''''su'''''(1,1) suffices to reconstruct the original algebra in a presentation.<ref> D Fairlie, J Nuyts, and C Zachos (1988). ''Phys Lett'' '''B202''' 320-324.  {{doi|10.1016/0370-2693(88)90478-9}}</ref>

==Over finite fields==

Over a field ''k'' of characteristic ''p''&gt;0, the Witt algebra is defined to be the Lie algebra of derivations of the ring
:''k''[''z'']/''z''<sup>''p''</sup>
The Witt algebra is spanned by ''L''<sub>''m''</sub> for &minus;1&le; ''m'' &le; ''p''&minus;2.

== Images ==
{{multiple image
 | align = center

 | image1 = N = -1 Witt vector field.png
 | width1 = 400
 | caption1 = n = -1 Witt vector field

 | image2 = N = 0 Witt vector field.png
 | width2 = 400
 | caption2 = n = 0 Witt vector field

 | image3 = N = 1 Witt vector field.png
 | width3 = 400
 | caption3 = n = 1 Witt vector field
}}

{{multiple image
 | align = center

 | image1 = Witt minus 2.png
 | width1 = 400
 | caption1 = n = -2 Witt vector field

 | image2 = Witt 2.png
 | width2 = 400
 | caption2 = n = 2 Witt vector field

 | image3 = Witt minus 3.png
 | width3 = 400
 | caption3 = n = -3 Witt vector field
}}

==See also==
*[[Virasoro algebra]]
*[[Heisenberg algebra]]

==References==
{{reflist}}
* [[Élie Cartan]], [http://www.numdam.org/numdam-bin/fitem?id=ASENS_1909_3_26__93_0 ''Les groupes de transformations continus, infinis, simples.''] Ann. Sci. Ecole Norm. Sup. 26, 93-161 (1909).
* {{springer|author= |title=Witt algebra|id=W/w098060}}


[[Category:Conformal field theory]]
[[Category:Lie algebras]]