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The cubic Witt cocycle is a Lie-algebra two-cocycle

Statement

For the Witt algebra of Witt algebra of Laurent polynomial vector fields, define $ω (d_{m}, d_{n}) = δ_{m + n, 0} (m^{3} - m) / 12$ and extend bilinearly. Then $ω$ is alternating and satisfies $ω ([x, y], z) + ω ([y, z], x) + ω ([z, x], y) = 0$ for all $x, y, z \in W$ .

Facts & Assumptions

Given: The basis and bracket of the Laurent-polynomial Witt algebra.

[F1]

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

Proof

1.1F1algebra

The scalar $m^{3} - m$ is odd in $m$ . Therefore $ω (d_{n}, d_{m}) = - ω (d_{m}, d_{n})$ . If $m = n$ , the Kronecker condition forces $m = 0$ , and $0^{3} - 0 = 0$ ; hence $ω (d_{m}, d_{m}) = 0$ . Bilinearity extends both identities to all finite sums.

2.1F1step 1.1algebra∎

By trilinearity it suffices to use $d_{m}, d_{n}, d_{p}$ . If $m + n + p \neq 0$ , all three Kronecker factors vanish. If $p = - m - n$ , twelve times the cyclic sum is $N = (m - n) ((m + n)^{3} - (m + n)) + (n - p) ((n + p)^{3} - (n + p)) + (p - m) ((p + m)^{3} - (p + m)) .$ The linear part is $(m - n) p + (n - p) m + (p - m) n = 0$ . The cubic part, after substituting $p = - m - n$ , is $(m - n) p^{3} + (n - p) m^{3} + (p - m) n^{3} = - (m - n) (m + n)^{3} + (m + 2 n) m^{3} - (2 m + n) n^{3} = 0,$ as expansion cancels the $m^{4}$ , $m^{3} n$ , $m n^{3}$ , and $n^{4}$ terms. Thus $N = 0$ in every case, proving the cocycle identity.

Depends on

Witt algebra of Laurent polynomial vector fields

Used by

Virasoro algebra in the Lₘ convention Definition

Dependency tree · two levels

2 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)