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A normalized Witt central cocycle is cubic

Statement

Let $λ : W \times W \to C$ be an alternating bilinear central Lie-algebra two-cocycle for the Witt algebra Witt algebra of Laurent polynomial vector fields, and suppose $λ (d_{m}, d_{n}) = 0$ whenever $m + n \neq 0$ . If $λ (d_{1}, d_{- 1}) = 0$ , then there is a scalar $a \in C$ such that $λ (d_{m}, d_{n}) = a (m^{3} - m) δ_{m + n, 0} .$

Facts & Assumptions

Given: The Witt bracket and the stated support and normalization of $λ$ .

[F1]

Witt algebra of Laurent polynomial vector fields: $[d_{m}, d_{n}] = (m - n) d_{m + n}$ .

Proof

1.1F1algebra

Put $A (m) = λ (d_{m}, d_{- m})$ . Alternatingness gives $A (- m) = - A (m)$ and $A (0) = 0$ . Apply the cocycle identity to $d_{1}, d_{n}, d_{- n - 1}$ and use [F1]. The three terms are $(1 - n) A (n + 1)$ , $- (2 n + 1) A (1)$ , and $(n + 2) A (n)$ , so $(n - 1) A (n + 1) = (n + 2) A (n) - (2 n + 1) A (1) .$ This holds for every integer $n$ .

2.1F1step 1.1algebra∎

Since $A (1) = 0$ , the recurrence for $n \geq 2$ becomes $(n - 1) A (n + 1) = (n + 2) A (n)$ . Set $a = A (2) / 6$ and $P (n) = a (n^{3} - n)$ . Then $P (1) = 0$ , $P (2) = A (2)$ , and direct expansion gives $(n - 1) P (n + 1) = (n + 2) P (n)$ . Induction using $n - 1 \neq 0$ for $n \geq 2$ proves $A (n) = P (n)$ for every positive $n$ ; oddness extends the formula to negative $n$ , and $A (0) = 0$ covers zero. By the support hypothesis, this is exactly $λ (d_{m}, d_{n}) = a (m^{3} - m) δ_{m + n, 0}$ .

Depends on

Witt algebra of Laurent polynomial vector fields

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Antony Wassermann, Kac–Moody and Virasoro Algebras, Chapter IV §1 (standard reference, not scraped)