Virasoro-Algebra

In mathematics, the Virasoro algebra is an infinite-dimensional Lie algebra. It is used in mathematical physics, such as in string theory and conformal field theory. There she is treated as algebra over the complex numbers, but can be carried out also for any field of characteristic 0. It was introduced in 1970 by Miguel Virasoro within the string theory, but it also plays an important role in internal mathematical construction of the monster group.

Construction

The starting point is the Witt algebra over a field of characteristic 0 (for example) that will generated by elements with the commutation relations. A Virasoro algebra is defined as a central extension of the Witt algebra. This means there is a short exact sequence of Lie algebras

.

Is a one-dimensional vector space, which may be thought of in included. It is in the center of lie is also referred to sometimes as "central charge " of the Virasoro algebra. The Virasoro algebra is generated by elements and that the archetypes are. For the commutation relations we have some choices. A common choice is

• For all, as is in the center of,
• For everyone.

Where represents the Kronecker delta. This is called the central portion of the Kommutatorrelation; this percentage can be chosen in the most general case, as with. This choice is motivated by the fact that for disappears and, therefore, in the above sequence is isomorphic to ready, the latter being one of SL (2, K) isomorphic Lie algebra. The factor explained by the fact that there are certain representations of the Virasoro algebra, in which this factor disappears; this is only a convenient convention.

An alternative convention

An alternative choice of commutation relations are obtained when going from to. A short calculation shows

• ,

That is, one can get the linear term of the central portion of the commutation relations disappear.

Equivalences

Two central extensions of the Witt algebra and are called equivalent if there is a Lie algebras with isomorphism and there.

It can be shown that there is only one central extension up to equivalence, which is equivalent to a semi- direct sum is not, namely, the above-introduced 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 groups