A Schur-Weyl type duality for twisted weak modules over a vertex algebra (Q6583749)

Let \(V\) be a vertex operator algebra, \(G\), a finite group of automorphisms of \(V\), and \(V^G\) the fixed point vertex subalgebra.\N\NThe study of \(V^G\)-modules in terms of \(V\) and \(G\)-modules has led to significant progress, particularly in the context of Schur-Weyl type duality. Dong and Mason initiated this line of work, demonstrating such duality for solvable groups \(G\). This concept has since been extended using the Zhu algebra and related structures, alongside systematic studies of twisted \(V\)-modules. These modules connect \(V^G\)-modules with \(g\)-twisted \(V\)-modules, under certain conjectured conditions, and form a central theme in understanding representations. Challenges arise when dealing with direct sums of modules associated with different automorphisms, prompting the introduction of the ``\(G\)-twisted Zhu algebra'' for unified analysis. To address these issues, Miyamoto and the author a while ago introduced the algebra \(A_\alpha(G, S)\), a finite-dimensional semisimple associative algebra constructed using a finite \(G\)-stable set \(S\) of inequivalent irreducible twisted \(V\)-modules and a 2-cocycle determined by the \(G\)-action. This algebra allowed them to study twisted \(V\)-modules in a unified way and they showed a Schur-Weyl type duality for any finite \(G\)-stable set \(S\) of inequivalent irreducible twisted \(V\)-modules (see [\textit{M. Miyamoto} and \textit{K. Tanabe}, J. Algebra 274, No. 1, 80--96 (2004; Zbl 1046.17009), Theorem 2]).\N\NThis paper generalizes the Schur-Weyl type duality to twisted *weak modules* for vertex algebras, extending the framework beyond ordinary twisted modules. It also introduces the so-called weak \((V, T)\)-modules (see Definition 2.2), a generalization enabling closure under direct sums and a unified study of twisted weak \(V\)-modules.\N\NWith these tools, the main finding -- presented as a Schur-Weyl type duality for \((A_\alpha(G, S), V^G)\) -- are detailed in Theorem 1.1 and Corollary 1.2.