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{{dablink|The Witt algebra is not directly related to the [[Witt ring (forms)|Witt ring]] of quadratic forms, or to the algebra of [[Witt vector]]s.}} | |||
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>−1</sup>]. Witt algebras occur in the study of [[conformal field theory]]. | |||
There are some related Lie algebras defined over finite fields, that are also called Witt algebras. | |||
The complex Witt algebra was first defined by 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 vector fields is given by | |||
:<math>[L_m,L_n]=(m-n)L_{m+n}.</math> | |||
This algebra has a [[Group extension%23Central extension|central extension]] called the [[Virasoro algebra]] that is important in [[conformal field theory]] and [[string theory]]. | |||
==Over finite fields== | |||
Over a field ''k'' of characteristic ''p''>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 −1≤ ''m'' ≤ ''p''−2. | |||
==See also== | |||
*[[Virasoro algebra]] | |||
*[[Heisenberg algebra]] | |||
==References== | |||
*E. 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]] | |||
Revision as of 21:10, 31 January 2014
Template:Dablink 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−1]. Witt algebras occur in the study of conformal field theory.
There are some related Lie algebras defined over finite fields, that are also called Witt algebras.
The complex Witt algebra was first defined by 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 fields , for n in .
The Lie bracket of two vector fields is given by
This algebra has a central extension called the Virasoro algebra that is important in conformal field theory and string theory.
Over finite fields
Over a field k of characteristic p>0, the Witt algebra is defined to be the Lie algebra of derivations of the ring
- k[z]/zp
The Witt algebra is spanned by Lm for −1≤ m ≤ p−2.
See also
References
- E. Cartan, Les groupes de transformations continus, infinis, simples. Ann. Sci. Ecole Norm. Sup. 26, 93-161 (1909).
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