DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Virasoro algebra in the L_m convention

Definition

The Virasoro algebra is the vector space $W \oplus C C$ , with $W$ from Witt algebra of Laurent polynomial vector fields and $C$ a central basis vector, and bracket $[L_{m}, L_{n}] = (m - n) L_{m + n} + \frac{1}{12} (m^{3} - m) δ_{m + n, 0} C,$ where $L_{m}$ denotes the lift of $d_{m}$ . The bracket with $C$ is zero. The cocycle identity and alternatingness proved in The cubic Witt cocycle is a Lie-algebra two-cocycle make this central extension a Lie algebra. A representation on $V$ has central charge $c$ if $C$ acts as $c {id}_{V}$ ; in its operator relation, $C$ is then replaced by $c {id}_{V}$ . Centrality alone does not require $C$ to act as a scalar in an arbitrary representation.

Depends on

Used by

Conformal vertex algebra of CFT type Definition
Highest-weight vector and module for Virasoro Definition
Sugawara modes satisfy Virasoro at every noncritical complex level Theorem

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)