Witt algebra
Algebra of meromorphic vector fields on the Riemann sphere
title: "Witt algebra" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["conformal-field-theory", "lie-algebras"] description: "Algebra of meromorphic vector fields on the Riemann sphere" topic_path: "general/conformal-field-theory" source: "https://en.wikipedia.org/wiki/Witt_algebra" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::summary 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−1].
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 fields L_n=-z^{n+1} \frac{\partial}{\partial z}, for n in \mathbb Z.
The Lie bracket of two basis vector fields is given by
:[L_m,L_n]=(m-n)L_{m+n}.
This algebra has a 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 \mathfrak{sl}(2,\mathbb{C}) of the Lorentz group \mathrm{SO}(3,1). Over the reals, it is the algebra sl(2,R) = su(1,1). Conversely, su(1,1) suffices to reconstruct the original algebra in a presentation.
Over finite fields
Over a field k of characteristic p0, the Witt algebra is defined to be the Lie algebra of derivations of the ring :k[z]/z**p The Witt algebra is spanned by L**m for −1≤ m ≤ p−2.
Images
| 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
| 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
References
- Élie Cartan, Les groupes de transformations continus, infinis, simples. Ann. Sci. Ecole Norm. Sup. 26, 93-161 (1909).
References
- D Fairlie, J Nuyts, and C Zachos (1988). ''Phys Lett'' '''B202''' 320-324. {{doi. 10.1016/0370-2693(88)90478-9
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