Rationality in the admissible-module convention

In this page, a conformal vertex algebra of CFT type $V$ is rational if every admissible $V$-module $M={⨁}_{n\ge 0}M(n)$, as defined in Admissible module for a vertex operator algebra, is an internal direct sum $M={⨁}_{\alpha \in I}{S}_{\alpha }$ of irreducible admissible submodules. Each $S_{\alpha }$ is simple as a $V$-module in the sense of Module for an ordinary vertex algebra and carries the inherited grading $S_{\alpha }={⨁}_{n\ge 0}(S_{\alpha }\cap M(n))$; thus the decomposition respects the specified admissible grading. The zero module is the empty direct sum. This is the admissible-module semisimplicity convention. The definition does not include a finite-number-of-simple-modules condition, $C_2$-cofiniteness, self-contragredience, or a tensor-category conclusion; each is a separate property unless a theorem explicitly assumes or proves it.

Terminology varies across the literature. In particular, the “finite reductivity” conditions used in some tensor-category treatments include semisimplicity, finitely many irreducible modules, and finite fusion rules in their chosen module class. No equivalence between rationality and any Zhu-algebra, $C_2$-cofiniteness, fusion, or modularity condition is part of this definition.