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Rigid Tensor Structure on Big Module Categories for Some W-(super)algebras in Type A

Creutzig T, McRae R, Yang J (2023)

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Publication Type: Journal article

Publication year: 2023

Journal

Book Volume: 404

Pages Range: 339-400

Journal Issue: 1

DOI: 10.1007/s00220-023-04843-1

Abstract

We establish rigid tensor category structure on finitely-generated weight modules for the subregular W-algebras of sln at levels -n+nn+1 (the Bn+1 -algebras of Creutzig–Ridout–Wood) and at levels -n+n+1n (the finite cyclic orbifolds of the βγ -vertex algebra), as well as for their Feigin–Semikhatov dual principal W-superalgebras of sln|1 . These categories are neither finite nor semisimple, and in the W-algebra case they contain modules with infinite-dimensional conformal weight spaces and no lower bound on conformal weights. We give complete lists of indecomposable projective modules in these tensor categories and fusion rules for simple modules. All these vertex operator (super)algebras are simple current extensions of singlet algebras tensored with a rank-one Heisenberg algebra, so we more generally study simple current extensions in direct limit completions of vertex algebraic tensor categories. Then our results for W-(super)algebras follow from the known ribbon category structure on modules for the singlet algebras. Our results include and generalize those of Allen-Wood on the βγ -vertex algebra, as well as our own on the affine vertex superalgebra of gl1 | 1 . Our results also include the first examples of ribbon category structure on all finitely-generated weight modules for an affine vertex algebra at a non-integral admissible level, namely for affine sl2 at levels -43 and -12 .

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How to cite

APA:

Creutzig, T., McRae, R., & Yang, J. (2023). Rigid Tensor Structure on Big Module Categories for Some W-(super)algebras in Type A. Communications in Mathematical Physics, 404(1), 339-400. https://doi.org/10.1007/s00220-023-04843-1

MLA:

Creutzig, Thomas, Robert McRae, and Jinwei Yang. "Rigid Tensor Structure on Big Module Categories for Some W-(super)algebras in Type A." Communications in Mathematical Physics 404.1 (2023): 339-400.

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