×

Monoidal abelian envelopes with a quotient property. (English) Zbl 1504.18014

A fully faithful embedding of a pseudo-tensor category into a tensor category with an appropriate universal property is known as an abelian envelope of the pseudo-tensor category. The question of which pseudo-tensor categories do admit an abelian envelope has attracted a lot of attention recently [D. Benson and P. Etingof, Adv. Math. 351, 967–999 (2019; Zbl 1430.18013); D. Benson et al., “New incompressible symmetric tensor categories in positive characteristic”, Preprint, arXiv:2003.10499; J. Comes and V. Ostrik, Algebra Number Theory 8, No. 2, 473–496 (2014; Zbl 1305.18019); K. Coulembier et al., Algebra Number Theory 16, No. 9, 2099–2117 (2022; Zbl 1503.18006); K. Coulembier, “Additive Grothendieck pretopologies and presentations of tensor categories”, Preprint, arXiv:2011.02137; P. Deligne, in: Algebraic groups and homogeneous spaces. Proceedings of the international colloquium, Mumbai, India, January 6–14, 2004. New Delhi: Narosa Publishing House/Published for the Tata Institute of Fundamental Research. 209–273 (2007; Zbl 1165.20300); I. Entova-Aizenbud et al., Int. Math. Res. Not. 2020, No. 15, 4602–4666 (2020; Zbl 1477.18034)], leading to various interesting applications. It is observed empirically that every object in the tensor category is a quotient of an object in the pseudo-tensor category. On the other hand, it was observed in [D. Benson et al., “New incompressible symmetric tensor categories in positive characteristic”, Preprint, arXiv:2003.10499] that when a pseudo-tensor category \(\boldsymbol{D}\) can be embedded in a tensor category \(\boldsymbol{T}\) such that every object in \(\boldsymbol{T}\) is a quotient of an object in \(\boldsymbol{D}\), \(\boldsymbol{T}\) must be the abelian envelope of \(\boldsymbol{D}\).
This paper aims to further investigate abelian envelopes, in particular in relation to the above quotient property. The first main result establishes an intrinsic criterion on a pseudo-tensor category determining whether there exists an abelian envelope with the quotient property. The other main results establish new ways to interpret known tensor categories as abelian envelopes. It is also investigated whether the Deligne product and extension of scalars are always tensor categories. It is finally demonstrated that if the extension of scalars is always a tensor category, then so is Deligne’s product of tensor categories.

MSC:

18M05 Monoidal categories, symmetric monoidal categories
17B37 Quantum groups (quantized enveloping algebras) and related deformations
14A20 Generalizations (algebraic spaces, stacks)
14L15 Group schemes
20G05 Representation theory for linear algebraic groups
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] H. H. Andersen, Tensor products of quantized tilting modules, Comm. Math. Phys. 149 (1992), no. 1, 149-159. · Zbl 0760.17004
[2] H. H. Andersen, The strong linkage principle for quantum groups at roots of 1, J. Algebra 260 (2003), 2-15. · Zbl 1043.17005
[3] Y. André and B. Kahn, Nilpotence, radicaux et structures monoïdales, Rend. Semin. Mat. Univ. Padova 108 (2002), 107-291. · Zbl 1165.18300
[4] D. Benson and P. Etingof, Symmetric tensor categories in characteristic 2, Adv. Math. 351 (2019), 967-999. · Zbl 1430.18013
[5] D. Benson, P. Etingof and V. Ostrik, New incompressible symmetric tensor categories in positive characteristic, preprint (2020), https://arxiv.org/abs/2003.10499.
[6] A. Białynicki Birula, G. Hochschild and G. D. Mostow, Extensions of representations of algebraic linear groups, Amer. J. Math. 85 (1963), 131-144. · Zbl 0116.02302
[7] J. Comes and V. Ostrik, On Deligne’s category \rm Re{\rm p}^{ab}(S_d), Algebra Number Theory 8 (2014), no. 2, 473-496. · Zbl 1305.18019
[8] K. Coulembier, Additive Grothendieck pretopologies and presentations of tensor categories, preprint (2020), https://arxiv.org/abs/2011.02137.
[9] K. Coulembier, Monoidal abelian envelopes, Compos. Math. 157 (2021), no. 7, 1584-1609. · Zbl 1471.18020
[10] K. Coulembier, I. Entova-Aizenbud and T. Heidersdorf, Monoidal abelian envelopes and a conjecture of Benson-Etingof, preprint (2019), https://arxiv.org/abs/1911.04303.
[11] K. Coulembier, R. Street and M. van den Bergh, Freely adjoining monoidal duals, Math. Structures Comput. Sci. 31 (2021), no. 7, 748-768. · Zbl 1495.18019
[12] P. Deligne, Catégories tannakiennes, The Grothendieck Festschrift. Vol. II, Progr. Math. 87, Birkhäuser, Boston (1990), 111-195. · Zbl 0727.14010
[13] P. Deligne, La catégorie des représentations du groupe symétrique S_t, lorsque t n’est pas un entier naturel, Algebraic groups and homogeneous spaces, Tata Inst. Fund. Res. Stud. Math. 19, Tata Institute of Fundamental Research, Mumbai (2007), 209-273. · Zbl 1165.20300
[14] P. Deligne, Semi-simplicité de produits tensoriels en caractéristique p, Invent. Math. 197 (2014), no. 3, 587-611. · Zbl 1315.14062
[15] P. Deligne and J. S. Milne, Tannakian categories, Hodge cycles, motives, and Shimura varieties, Lecture Notes in Math. 900, Springer, Berlin (1982), 101-228. · Zbl 0477.14004
[16] M. Demazure and P. Gabriel, Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs, Masson & Cie, Paris 1970.
[17] S. Donkin, On tilting modules for algebraic groups, Math. Z. 212 (1993), no. 1, 39-60. · Zbl 0798.20035
[18] I. Entova-Aizenbud, V. Hinich and V. Serganova, Deligne categories and the limit of categories Rep(\operatorname{GL}(m|n)), Int. Math. Res. Not. IMRN 2020 (2020), no. 15, 4602-4666. · Zbl 1477.18034
[19] P. Etingof, S. Gelaki, D. Nikshych and V. Ostrik, Tensor categories, Math. Surveys Monogr. 205, American Mathematical Society, Providence 2015.
[20] P. Etingof and V. Ostrik, On the Frobenius functor for symmetric tensor categories in positive characteristic, J. reine angew. Math. 773 (2021), 165-198. · Zbl 1478.18019
[21] F. D. Grosshans, Algebraic homogeneous spaces and invariant theory, Lecture Notes in Math. 1673, Springer, Berlin 1997. · Zbl 0886.14020
[22] J. I. Hutchinson, On a remarkable class of entire functions, Trans. Amer. Math. Soc. 25 (1923), no. 3, 325-332. · JFM 49.0217.02
[23] U. Jannsen, Motives, numerical equivalence, and semi-simplicity, Invent. Math. 107 (1992), no. 3, 447-452. · Zbl 0762.14003
[24] J. C. Jantzen, Representations of algebraic groups, 2nd ed., Math. Surveys Monogr. 107, American Mathematical Society, Providence 2003. · Zbl 1034.20041
[25] C. Jones, Triangle presentations and tilting modules for SL_{2k+1}, J. Comb. Algebra 5 (2021), no. 1, 59-92. · Zbl 1476.18006
[26] I. López Franco, Tensor products of finitely cocomplete and abelian categories, J. Algebra 396 (2013), 207-219. · Zbl 1305.18042
[27] A. Neeman, Triangulated categories, Ann. of Math. Stud. 148, Princeton University, Princeton 2001. · Zbl 0974.18008
[28] J. E. Roos, Locally Noetherian categories and generalized strictly linearly compact rings. Applications, Category theory, homology theory and their applications. II, Springer, Berlin (1969), 197-277. · Zbl 0211.32602
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.