Lie algebras MOC

Virasoro algebra

The Virasoro algebra over 1 is a Lie algebra given by the unique nontrival 1-dimensional central extension of the Witt algebra .2 #m/def/lie The Lie bracket is defined by

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

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 identity 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 extension.

Properties

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

#state/tidy | #lang/en | #SemBr

Footnotes

  1. ↩

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