There are very famous Coleman–Mandula theorem and Haag–Łopuszański–Sohnius theorem , see also this and this.

It states that "space-time and internal symmetries cannot be combined in any but a trivial way".

But this theorem didn't give any restrictions on internal symmetry group. Which symmetries can be realised in QFT? What about Quantum group, Hopf algebra? Or maybe more exotic algebras?

Which internal symmetries one can realise in QFT?

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    $\begingroup$ If Lie, zero-form symmetries are reductive. Higher-form symmetries are finitely-generated abelian groups. Symmetries of different rank may mix into higher categorical structures, cf. e.g. arxiv.org/abs/1802.04790. So if you have, say, both zero-form and one-form symmetries, these may interact in a way that doesn't close to a group, in the classical sense. Is this what you have in mind? $\endgroup$ Jul 25, 2020 at 19:32
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    $\begingroup$ Also in two dimensions symmetries can be shown to form a fusion category arxiv.org/abs/1704.02330 $\endgroup$ Jul 25, 2020 at 20:14
  • $\begingroup$ @AccidentalFourierTransform, thank you! Yes, your answers are very useful! Really, I wanna to understand, which symmetries can be realised in QFT. If you know some other examples, please, write here! $\endgroup$
    – Nikita
    Jul 25, 2020 at 20:41
  • $\begingroup$ @ɪdɪətstrəʊlə, thank you! If you know some other unusual symmetries in QFT, please, write here. $\endgroup$
    – Nikita
    Jul 25, 2020 at 20:42


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