Cartan subproduct systems
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914210544353280 |
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| author | Bhattacharjee, Suvrajit Giselsson, Olof Neshveyev, Sergey |
| author_facet | Bhattacharjee, Suvrajit Giselsson, Olof Neshveyev, Sergey |
| contents | Given a semisimple compact Lie group $G$ and a nonzero dominant integral weight $λ$, the highest weight $G_q$-modules $V_{nλ}$ form a subproduct system of finite dimensional Hilbert spaces. Using a conjectural asymptotic behavior of Clebsch-Gordan coefficients we identify the corresponding Cuntz-Pimsner algebras with algebras of quantized functions on homogeneous spaces of $G$. We also show that the gauge-invariant part of the Toeplitz algebra provides a model for convergence of full matrix algebras to quantum flag manifolds, complementing and generalizing results of Landsman and Rieffel for $q=1$ and results of Vaes-Vergnioux in the rank one case for $q\ne1$.
We verify our conjecture on Clebsch-Gordan coefficients for $G=SU(n)$ and all weights that are either regular or multiples of the fundamental weight $ω_1$. For $λ=ω_1$, we also provide a detailed description of the Toeplitz and Cuntz-Pimsner algebras, generalizing results of Arveson on symmetric subproduct systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_17690 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cartan subproduct systems Bhattacharjee, Suvrajit Giselsson, Olof Neshveyev, Sergey Operator Algebras Quantum Algebra Representation Theory Given a semisimple compact Lie group $G$ and a nonzero dominant integral weight $λ$, the highest weight $G_q$-modules $V_{nλ}$ form a subproduct system of finite dimensional Hilbert spaces. Using a conjectural asymptotic behavior of Clebsch-Gordan coefficients we identify the corresponding Cuntz-Pimsner algebras with algebras of quantized functions on homogeneous spaces of $G$. We also show that the gauge-invariant part of the Toeplitz algebra provides a model for convergence of full matrix algebras to quantum flag manifolds, complementing and generalizing results of Landsman and Rieffel for $q=1$ and results of Vaes-Vergnioux in the rank one case for $q\ne1$. We verify our conjecture on Clebsch-Gordan coefficients for $G=SU(n)$ and all weights that are either regular or multiples of the fundamental weight $ω_1$. For $λ=ω_1$, we also provide a detailed description of the Toeplitz and Cuntz-Pimsner algebras, generalizing results of Arveson on symmetric subproduct systems. |
| title | Cartan subproduct systems |
| topic | Operator Algebras Quantum Algebra Representation Theory |
| url | https://arxiv.org/abs/2512.17690 |