Conformal transformations and equivariance in unbounded KK-theory
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866918349868367872 |
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| author | Masters, Ada Rennie, Adam |
| author_facet | Masters, Ada Rennie, Adam |
| contents | We extend unbounded Kasparov theory to encompass conformal group and quantum group equivariance. This new framework allows us to treat conformal actions on both manifolds and noncommutative spaces. As examples, we present unbounded representatives of Kasparov's $γ$-element for the real and complex Lorentz groups and display the conformal $SL_q(2)$-equivariance of the standard spectral triple of the Podleś sphere. In pursuing descent for conformally equivariant cycles, we are led to a new framework for representing Kasparov classes. Our new representatives are unbounded, possess a dynamical quality, and also include known twisted spectral triples. We define an equivalence relation on these new representatives whose classes form an abelian group surjecting onto KK. The technical innovation which underpins these results is a novel multiplicative perturbation theory. By these means, we obtain Kasparov classes from the bounded transform with minimal side conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_17220 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Conformal transformations and equivariance in unbounded KK-theory Masters, Ada Rennie, Adam Operator Algebras Differential Geometry K-Theory and Homology Quantum Algebra 58B34, 46L87, 19K35, 53C18, 58B32 We extend unbounded Kasparov theory to encompass conformal group and quantum group equivariance. This new framework allows us to treat conformal actions on both manifolds and noncommutative spaces. As examples, we present unbounded representatives of Kasparov's $γ$-element for the real and complex Lorentz groups and display the conformal $SL_q(2)$-equivariance of the standard spectral triple of the Podleś sphere. In pursuing descent for conformally equivariant cycles, we are led to a new framework for representing Kasparov classes. Our new representatives are unbounded, possess a dynamical quality, and also include known twisted spectral triples. We define an equivalence relation on these new representatives whose classes form an abelian group surjecting onto KK. The technical innovation which underpins these results is a novel multiplicative perturbation theory. By these means, we obtain Kasparov classes from the bounded transform with minimal side conditions. |
| title | Conformal transformations and equivariance in unbounded KK-theory |
| topic | Operator Algebras Differential Geometry K-Theory and Homology Quantum Algebra 58B34, 46L87, 19K35, 53C18, 58B32 |
| url | https://arxiv.org/abs/2412.17220 |