The tensor category for W(2,2)-vertex algebra

Abstract

This paper studies the category 𝒞 of grading-restricted C₁-cofinite generalized modules for the vertex operator algebra associated to the W-algebra W(2,2). We first show that 𝒞 is the same as the category of finite length modules whose simple composition factors are the irreducible highest weight W(2,2)--modules L[r] of highest weight ((1−r)/2, 0) for r ∈ ℤ_{> 0}, and hence 𝒞 carries a braided tensor category structure. Then we prove the fusion rules for the simple objects L[r] are governed by the sl₂ Clebsch--Gordan rule. In particular, we prove L[r]boxtimes L[s]≅ ⨁_i=0^min{r,s}−1 L[r+s−1−2i]. Using the fusion rules and a recent result of Etingof--Penneys, we establish the rigidity of 𝒞. We also show the semisimple subcategory generated by the simple objects L[r] is tensor equivalent to the category Rep sl₂ of finite dimensional sl₂-modules.

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BibTeX RIS

BibTeX

@misc{adamovic2026tensor,
  title = {{The tensor category for W(2,2)-vertex algebra}},
  author = {Drazen Adamovic and Mingjie Peng and Gordan Radobolja and Jinwei Yang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.02202},
  eprint = {2609.02202},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.02202},
}

RIS

TY  - UNPB
TI  - The tensor category for W(2,2)-vertex algebra
AU  - Drazen Adamovic
AU  - Mingjie Peng
AU  - Gordan Radobolja
AU  - Jinwei Yang
PY  - 2026
DA  - 2026/09/02
PB  - arXiv
DO  - 10.48550/arxiv.2609.02202
UR  - https://arxiv.org/abs/2609.02202
AB  - This paper studies the category 𝒞 of grading-restricted C_1-cofinite generalized modules for the vertex operator algebra associated to the W-algebra W(2,2). We first show that 𝒞 is the same as the category of finite length modules whose simple composition factors are the irreducible highest weight W(2,2)--modules L[r] of highest weight ((1−r)/2, 0) for r ∈ ℤ_> 0, and hence 𝒞 carries a braided tensor category structure. Then we prove the fusion rules for the simple objects L[r] are governed by the sl_2 Clebsch--Gordan rule. In particular, we prove L[r]boxtimes L[s]≅ ⨁_i=0^min{r,s}−1 L[r+s−1−2i]. Using the fusion rules and a recent result of Etingof--Penneys, we establish the rigidity of 𝒞. We also show the semisimple subcategory generated by the simple objects L[r] is tensor equivalent to the category Rep sl_2 of finite dimensional sl_2-modules.
ER  -

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