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Simple-shifted monoidal derived equivalences for tensor categories

  • Jing Yu
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Abstract

In this paper, we investigate monoidal derived equivalences between tensor categories with infinitely many isomorphism classes of simple objects. We introduce the notion of a simple-shifted monoidal derived equivalence and prove that a simple-shifted monoidal derived equivalence gives rise to a tensor equivalence. In addition, we investigate which tensor categories have the property that every monoidal derived equivalence between them is necessarily simple-shifted, and we establish some sufficient conditions for this property. As a corollary, we prove that if a tensor category is monoidally derived equivalent to the finite-dimensional representation category of either the quantized enveloping algebra Uq(g), the quantized Borel subalgebra Uq≥ 0(g), or the quantized coordinate ring 𝒪q(G) (for a suitable q), it is tensor equivalent to the corresponding category.

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Cite

BibTeXRIS
BibTeX
@misc{yu2026simpleshifted,
  title = {{Simple-shifted monoidal derived equivalences for tensor categories}},
  author = {Jing Yu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.26196},
  eprint = {2610.26196},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.26196},
}
RIS
TY  - UNPB
TI  - Simple-shifted monoidal derived equivalences for tensor categories
AU  - Jing Yu
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.26196
UR  - https://arxiv.org/abs/2610.26196
AB  - In this paper, we investigate monoidal derived equivalences between tensor categories with infinitely many isomorphism classes of simple objects. We introduce the notion of a simple-shifted monoidal derived equivalence and prove that a simple-shifted monoidal derived equivalence gives rise to a tensor equivalence. In addition, we investigate which tensor categories have the property that every monoidal derived equivalence between them is necessarily simple-shifted, and we establish some sufficient conditions for this property. As a corollary, we prove that if a tensor category is monoidally derived equivalent to the finite-dimensional representation category of either the quantized enveloping algebra U_q(g), the quantized Borel subalgebra U_q^≥ 0(g), or the quantized coordinate ring 𝒪_q(G) (for a suitable q), it is tensor equivalent to the corresponding category.
ER  -