Virasoro algebra

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

[𝑐, 𝔳] = 0 [𝐿 𝑚, 𝐿 𝑛] = (𝑚 - 𝑛) 𝐿 𝑚 + 𝑛 + 𝑐 12 (𝑚 3 - 𝑚) 𝛿 𝑚 + 𝑛

where $𝜋 (𝐿 𝑛) = 𝑑 𝑛 = 𝑡 𝑛 𝑡 𝑑 𝑑 𝑡$ is a basis element of $𝔡$ , and $𝛿 ∙$ is the Kronecker delta. Thus we have the central extension

0 \to 𝕂 𝑐 ↪ 𝔳 𝜋 ↠ 𝔡 \to 0

Shifted equivalent extensions

Letting $𝐿' 𝑛 = 𝐿 𝑛 + 𝛽 𝑛 𝑐$ for $𝛽 𝑛 \in 𝕂$ and $𝑛 \in ℤ$ , and $𝜋' : 𝐿' 𝑛 \mapsto 𝑑 𝑛$ , we get an equivalent extension, with the bracket now given by

[𝐿' 𝑚, 𝐿' 𝑛] = (𝑚 - 𝑛) 𝐿' 𝑚 + 𝑛 + 𝑐 12 (𝑚 3 - 𝑚) 𝛿 𝑚 + 𝑛 + 𝑐 (𝑚 - 𝑛) 𝛽 𝑚 + 𝑛

In particular, $𝐿' 𝑛 = 𝐿 𝑛 - 124 𝛿 𝑛 𝑐$ removes the linear term.

Proof of uniqueness

Let $𝔳'$ be a central extension of $𝔡$ such that

𝔳 = 𝖵 𝖾 𝖼 𝗍 𝕂 𝔡 \oplus 𝕂 𝑐

and

[𝑑 𝑚, 𝑑 𝑛] = (𝑚 - 𝑛) 𝑑 𝑚 + 𝑛 + 𝛾 𝑚, 𝑛 𝑐 [𝑐, 𝔳'] = 0

It follows from the alternating property and the Jacobi identity that

𝛾 𝑚, 𝑛 + 𝛾 𝑛, 𝑚 = 0 (𝑚 - 𝑛) 𝛾 𝑚 + 𝑛, 𝑝 + (𝑛 - 𝑝) 𝛾 𝑛 + 𝑝, 𝑚 + (𝑝 - 𝑚) 𝛾 𝑝 + 𝑚, 𝑛 = 0

for $𝑚, 𝑛, 𝑝 \in ℤ$ . Assume $𝑝 = 0$ and $𝑚 + 𝑛 \neq 0$ , so

- (𝑚 + 𝑛) 𝛾 𝑚, 𝑛 + (𝑚 - 𝑛) 𝛾 𝑚 + 𝑛, 0 = 0

whence

𝛾 𝑚, 𝑛 = 𝑚 - 𝑛 𝑚 + 𝑛 𝛾 𝑚 + 𝑛, 0

By considering the equivalent shifted extension

𝑑' 𝑛 = 𝑑 𝑛 + 𝑐 (1 - 𝛿 𝑛) 1 𝑛 𝛾 𝑛, 0

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

𝛾 𝑚, - 𝑚 = 𝛼 𝑚 3 + 𝛽 𝑚

where $𝛼, 𝛽 \in 𝕂$ , since any solution is determined by $𝛾 1, - 1$ and $𝛾 2, - 2$ . 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 extension.

Properties

The extension is the trivial extension restricted to $𝔭 = 𝕂 𝑑 - 1 + 𝕂 𝑑 0 + 𝕂 𝑑 1$ , since the central term becomes zero.

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

$c h a r 𝕂 = 0$ ↩
1988. Vertex operator algebras and the Monster, §1.9 pp. 32ff. ↩

Virasoro algebra

Properties

Footnotes