Witt-Algebra

The Witt algebra is investigated in mathematics, it is a special Lie algebra. It is used in mathematical physics, such as in string theory and conformal field theory. It is named after the German mathematician Ernst Witt.

Definition

Be with integer index as a basis of a vector space. The by Kommutatorrelation

Defined Lie algebra is called Witt algebra. Such algebras are obtained as derivations algebra over the ring of Laurent polynomials.

Realization by vector fields

In most applications, it is considered over derivative ions. It is the Witt algebra as follows realized by complex-valued vector fields:

Sl (2, K) as a subalgebra

From the above commutation relations gives immediately that for those of generated sub ​​- Lie algebra is the same. These three -dimensional Lie algebra is isomorphic to sl (2, K).

Local expansion

If the Witt algebra by the cocycle

Centrally extended, we obtain the Virasoro algebra.

Swell

Igor Frenkel, James Lepowsky, Arne Meurman: Vertex Operator Algebras and the Monster, Academic Press, New York ( 1988) ISBN 0-12-267065-5

• Lie algebra
• Theory of Lie algebras