Cuspidal modules over Superconformal algebras of rank \geq 1
Consuelo Martinez, Olivier Mathieu, Efim Zelmanov
TL;DR
The paper classifies cuspidal modules over all known rank≥1 superconformal algebras (W(n), S(n;γ), K(N), K^(2)(2m), CK(6)) and their universal central extensions, revealing that central charge is typically trivial except for a unique extension of K(4). It develops a unified highest‑weight framework, uses coinduced modules to construct growth‑one representations, and analyzes split extensions to obtain explicit tensor‑density–type modules Tens(λ,δ,u) as the building blocks of all cuspidal modules. The results interrelate the algebras via embeddings K(3)⊂K̂(4)⊂CK(6) and clarify the role of the exceptional twisted cases, including the twisted K^(2)(2m). Overall, the work advances the representation theory of superconformal algebras by establishing a comprehensive growth‑one classification and detailing the central‑charge phenomena for cuspidal modules.
Abstract
According to V. Kac and J. van de Leur, the superconformal algebras are the simple $\Z$-graded Lie superalgebras of growth one which contains the Witt algebra. We describe an explicit classification of all cuspidal modules over the known supercuspidal algebras of rank $\geq 1$, and their central extensions. Our approach reveals some unnoticed phenomena. Indeed the central charge of cuspidal modules is trivial, except for one specific central extension of the contact algebra $\K(4)$. As shown in the paper, this fact also impacts the representation theory of $\K(3)$, $\CK(6)$ and $\K^{(2)}(4)$. Besides these four cases, the classification relies on general methods based on highest weight theory.