Lie algebras MOC

Virasoro algebra

The Virasoro algebra over ¹ is a Lie algebra given by the unique nontrival 1-dimensional ^central of the Witt algebra .² lie The Lie bracket is defined by

where is a basis element of , and is the Kronecker delta. Thus we have the ^central

Shifted equivalent extensions

Letting for and , and , we get an ^equivalent extension, with the bracket now given by

In particular, removes the linear term.

Proof of uniqueness

Let be a central extension of such that

and

It follows from the alternating property and the ^Jacobi that

for . Assume and , so

whence

By considering the equivalent shifted extension

we can take for without loss of generality. The general solution to the constraints on given is then

where , since any solution is determined by and . By shifting we can change arbitrarily and by rescaling we can multiply by any nonzero scalar. Thus the extension of given by is either  ^equivalent to or the ^trivial.

Properties

1. The extension is the ^trivial restricted to , since the central term becomes zero.

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Footnotes

1. ↩

2. 1988. Vertex operator algebras and the Monster, §1.9 pp. 32ff. ↩