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Witt algebra of Laurent polynomial vector fields

Definition

Let $R = C [z, z^{- 1}]$ be the algebra of finite Laurent polynomials. Define ${Der}_{C} (R)$ as the vector space of complex-linear maps $D : R \to R$ satisfying the Leibniz rule $D (f g) = D (f) g + f D (g)$ . Let $\partial_{z}$ be the linear map with $\partial_{z} (z^{q}) = q z^{q - 1}$ for all $q \in Z$ . For each $m \in Z$ , set $d_{m} = - z^{m + 1} \partial_{z}$ . The Witt algebra $W$ is the complex span of the $d_{m}$ inside ${Der}_{C} (R)$ , with bracket the commutator of derivations. The modes are indexed by all integers; all linear combinations in $W$ are finite, and no completion is intended.

Facts & Assumptions

Given: The Laurent-polynomial algebra $R$ and the displayed derivations $d_{m}$ .

[F1]

Lie algebras over a field: a Lie algebra over $C$ is a complex vector space with an alternating bilinear bracket satisfying Jacobi.

Proof

1.1givenalgebra

For $a, b \in Z$ , $\partial_{z} (z^{a + b}) = (a + b) z^{a + b - 1} = \partial_{z} (z^{a}) z^{b} + z^{a} \partial_{z} (z^{b})$ , so linearity and bilinearity prove the Leibniz rule on all Laurent polynomials. Multiplying a derivation by $- z^{m + 1}$ preserves the Leibniz rule because $R$ is commutative; hence each $d_{m}$ belongs to ${Der}_{C} (R)$ . For every $q \in Z$ , $d_{m} (z^{q}) = - q z^{m + q}$ , so $d_{m} d_{n} (z^{q}) = q (n + q) z^{m + n + q}$ and $d_{n} d_{m} (z^{q}) = q (m + q) z^{m + n + q}$ . Therefore $[d_{m}, d_{n}] (z^{q}) = q (n - m) z^{m + n + q} = (m - n) d_{m + n} (z^{q})$ . Since Laurent monomials span $R$ , this proves $[d_{m}, d_{n}] = (m - n) d_{m + n}$ and closure of $W$ under the commutator.

1.2givenalgebra

If a finite relation $\sum_{m} a_{m} d_{m} = 0$ holds, applying it to $z$ gives $- \sum_{m} a_{m} z^{m + 1} = 0$ . Distinct Laurent monomials are linearly independent in $R$ , so every $a_{m} = 0$ ; thus the $d_{m}$ form a basis of $W$ .

2.1F1step 1.1algebra∎

For endomorphisms $A, B, C$ of any vector space, the commutator is alternating, and expanding $[A, [B, C]] + [B, [C, A]] + [C, [A, B]]$ cancels each of the six triple compositions with its negative. Thus the commutator bracket satisfies [F1] on ${End}_{C} (R)$ ; by the closure in step 1.1, its restriction makes $W$ a Lie algebra.

Depends on

Lie algebras over a field

Used by

Virasoro algebra in the Lₘ convention Definition
A normalized Witt central cocycle is cubic Lemma
The cubic Witt cocycle is a Lie-algebra two-cocycle Lemma

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Christophe Nozaradan, Introduction to Vertex Algebras, §2.7.1 (standard reference, not scraped)
Antony Wassermann, Kac–Moody and Virasoro Algebras, Chapter III §1 (standard reference, not scraped)