Projective corepresentations and cohomology of compact quantum groups
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arXiv
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| Format: | Preprint |
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2026
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| author | Goswami, Debashish Maity, Kiran |
| author_facet | Goswami, Debashish Maity, Kiran |
| contents | We study projective unitary (co)representations of compact quantum groups and the associated second cohomology theory.
We introduce left/right/bi/strongly projective corepresentations and study them in details. In particular, we prove that given any compact quantum group $\q$, there are compact quantum groups $\tilde{\q_l}, \tilde{\q_r}, {\tilde \q}_{bi}, {\tilde \q}_{stp}$, each of which contains $\q$ as a Woronowicz subalgebra and every left/right/bi/strongly projective unitary corepresentation of $\q$ lifts to a linear corepresentation of these quantum groups respectively. We observe that the strongly projective corepresentations are associated with the second invariant ($S^1$-valued) cohomology $H^2_{uinv}(\cdot)$ of the quantum group. We define a suitable analogue of normalizer of a compact quantum group in a bigger compact quantum group and using this, associate a canonical discrete group $Γ_\q$ to a compact quantum group $\q$ which is an alternative generalization of the second group cohomology and we show by an example that $Γ_\q$ in general may be different from $H^2_{uinv}(\q,S^1) $. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_16373 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Projective corepresentations and cohomology of compact quantum groups Goswami, Debashish Maity, Kiran Quantum Algebra Operator Algebras We study projective unitary (co)representations of compact quantum groups and the associated second cohomology theory. We introduce left/right/bi/strongly projective corepresentations and study them in details. In particular, we prove that given any compact quantum group $\q$, there are compact quantum groups $\tilde{\q_l}, \tilde{\q_r}, {\tilde \q}_{bi}, {\tilde \q}_{stp}$, each of which contains $\q$ as a Woronowicz subalgebra and every left/right/bi/strongly projective unitary corepresentation of $\q$ lifts to a linear corepresentation of these quantum groups respectively. We observe that the strongly projective corepresentations are associated with the second invariant ($S^1$-valued) cohomology $H^2_{uinv}(\cdot)$ of the quantum group. We define a suitable analogue of normalizer of a compact quantum group in a bigger compact quantum group and using this, associate a canonical discrete group $Γ_\q$ to a compact quantum group $\q$ which is an alternative generalization of the second group cohomology and we show by an example that $Γ_\q$ in general may be different from $H^2_{uinv}(\q,S^1) $. |
| title | Projective corepresentations and cohomology of compact quantum groups |
| topic | Quantum Algebra Operator Algebras |
| url | https://arxiv.org/abs/2602.16373 |