Witt algebra

Let $𝕂 [𝑡, 𝑡 - 1]$ denote the algebra of Laurent polynomials over a field $𝕂$ . The Witt algebra $𝔡 = 𝔡 𝔢 𝔯 (𝕂 [𝑡, 𝑡 - 1])$ over $𝕂$ is the derivation subalgebra of the Lie algebra $E n d 𝕂 𝕂 [𝑡, 𝑡 - 1]$ . #m/def/lie It is equivalently characterized as follows:¹ For each $𝑝 (𝑡) \in 𝕂 [𝑡, 𝑡 - 1]$ , define the derivation

𝑇 𝑝 (𝑡) = 𝑝 (𝑡) 𝑑 𝑑 𝑡

Then $𝔡 = {𝑇 𝑝 (𝑡) : 𝑝 (𝑡) \in 𝕂 [𝑡, 𝑡 - 1]}$ is the Lie algebra of such derivations. The bracket may be expressed as

[𝑇 𝑝 (𝑡), 𝑇 𝑞 (𝑡)] = 𝑇 𝑝 (𝑡) 𝑞' (𝑡) - 𝑞 (𝑡) 𝑝' (𝑡)

and a basis is given by

𝑑 𝑛 = - 𝑡 𝑛 + 1 𝑑 𝑑 𝑡 = - 𝑡 𝑛 𝑡 𝑑 𝑑 𝑡 [𝑑 𝑚, 𝑑 𝑛] = (𝑚 - 𝑛) 𝑑 𝑚 + 𝑛 𝑚, 𝑛 \in ℤ

Proof of equivalence

Let $𝔡$ denote the first characterization. Let $𝑇 \in 𝔡$ , and set $𝑝 (𝑡) = 𝑇 (𝑡)$ . Then

𝑇 (1) = 𝑇 (1 \cdot 1) = 𝑇 (1) + 𝑇 (1)

whence $𝑇 (1) = 0$ , furthermore

0 = 𝑇 (𝑡 𝑡 - 1) = 𝑇 (𝑡) 𝑡 - 1 + 𝑡 𝑇 (𝑡 - 1)

whence $𝑇 (𝑡 - 1) = - 𝑡 - 2 𝑇 (𝑡)$ . Since these results hold for $𝑇 𝑝 (𝑡)$ in place of $𝑇$ , these two operators concur for all powers of $𝑡$ .

In $c h a r 𝕂 = 0$ , the Witt algebra admits a unique nontrivial 1-dimensional central extension, the Virasoro algebra.

Properties

For any $𝑛 \in ℕ$ , $s p a n {𝑑 - 𝑛, 𝑑 0, 𝑑 𝑛}$ forms a Lie subalgebra isomorphic to $𝔰 𝔩 2 𝕂$ , in particular

𝔭 = 𝕂 𝑑 - 1 + 𝕂 𝑑 0 + 𝕂 𝑑 1 ≅ 𝖫 𝗂 𝖾 𝕂 𝔰 𝔩 2 𝕂

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

1988. Vertex operator algebras and the Monster, §1.9, pp. 31–32 ↩

Witt algebra

Properties

Footnotes