Lie algebras MOC

Virasoro algebra

The Virasoro algebra 𝔳 over 𝕂¹ is a Lie algebra given by the unique nontrival 1-dimensional ^central of the Witt algebra 𝔡.² lie The Lie bracket is defined by

[𝑐,𝔳]=0[𝐿𝑚,𝐿𝑛]=(𝑚−𝑛)𝐿𝑚+𝑛+𝑐12(𝑚3−𝑚)𝛿𝑚+𝑛

where 𝜋(𝐿𝑛) =𝑑𝑛 =𝑡𝑛𝑡𝑑𝑑𝑡 is a basis element of 𝔡, and 𝛿∙ is the Kronecker delta. Thus we have the ^central

0→𝕂𝑐↪𝔳𝜋↠𝔡→0

Shifted equivalent extensions

Letting 𝐿′𝑛 =𝐿𝑛 +𝛽𝑛𝑐 for 𝛽𝑛 ∈𝕂 and 𝑛 ∈ℤ, and 𝜋′ :𝐿′𝑛 ↦𝑑𝑛, we get an ^equivalent extension, with the bracket now given by

[𝐿′𝑚,𝐿′𝑛]=(𝑚−𝑛)𝐿′𝑚+𝑛+𝑐12(𝑚3−𝑚)𝛿𝑚+𝑛+𝑐(𝑚−𝑛)𝛽𝑚+𝑛

In particular, 𝐿′𝑛 =𝐿𝑛 −124𝛿𝑛𝑐 removes the linear term.

Proof of uniqueness

Let 𝔳′ be a central extension of 𝔡 such that

𝔳=𝖵𝖾𝖼𝗍𝕂𝔡⊕𝕂𝑐

and

[𝑑𝑚,𝑑𝑛]=(𝑚−𝑛)𝑑𝑚+𝑛+𝛾𝑚,𝑛𝑐[𝑐,𝔳′]=0

It follows from the alternating property and the ^Jacobi that

𝛾𝑚,𝑛+𝛾𝑛,𝑚=0(𝑚−𝑛)𝛾𝑚+𝑛,𝑝+(𝑛−𝑝)𝛾𝑛+𝑝,𝑚+(𝑝−𝑚)𝛾𝑝+𝑚,𝑛=0

for 𝑚,𝑛,𝑝 ∈ℤ. Assume 𝑝 =0 and 𝑚 +𝑛 ≠0, so

−(𝑚+𝑛)𝛾𝑚,𝑛+(𝑚−𝑛)𝛾𝑚+𝑛,0=0

whence

𝛾𝑚,𝑛=𝑚−𝑛𝑚+𝑛𝛾𝑚+𝑛,0

By considering the equivalent shifted extension

𝑑′𝑛=𝑑𝑛+𝑐(1−𝛿𝑛)1𝑛𝛾𝑛,0

we can take 𝛾𝑚,𝑛 =0 for 𝑚 +𝑛 ≠0 without loss of generality. The general solution to the constraints on 𝛾𝑚,𝑛 given 𝑚 +𝑛 +𝑝 =0 is then

𝛾𝑚,−𝑚=𝛼𝑚3+𝛽𝑚

where 𝛼,𝛽 ∈𝕂, since any solution is determined by 𝛾1,−1 and 𝛾2,−2. By shifting we can change 𝛽 arbitrarily and by rescaling 𝑐 we can multiply 𝛼 by any nonzero scalar. Thus the extension of 𝔡 given by 𝔳′ is either  ^equivalent to 𝔳 or the ^trivial.

Properties

1. The extension is the ^trivial restricted to 𝔭 =𝕂𝑑−1 +𝕂𝑑0 +𝕂𝑑1, since the central term becomes zero.

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Footnotes

1. char⁡𝕂 =0 ↩

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