On some algebraic and geometric aspects of the quantum unitary group
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915946366500864 |
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| author | Jana, Debabrata |
| author_facet | Jana, Debabrata |
| contents | Consider the compact quantum group $U_q(2)$, where $q$ is a non-zero complex deformation parameter such that $|q|\neq 1$. Let $C(U_q(2))$ denote the underlying $C^*$-algebra of the compact quantum group $U_q(2)$. We prove that if $q$ is a non-real complex number and $q^\prime$ is real, then the underlying $C^*$-algebras $C(U_q(2))$ and $C(U_{q^\prime}(2))$ are non-isomorphic. This is in sharp contrast with the case of braided $SU_q(2)$, introduced earlier by Woronowicz et al., where $q$ is a non-zero complex deformation parameter. In another direction, on a geometric aspect of $U_q(2)$, we introduce torus action on the $C^*$-algebra $C(U_q(2))$ and obtain a $C^*$-dynamical system $(C(U_q(2)),\mathbb{T}^3,α)$. We construct a $\mathbb{T}^3$-equivariant spectral triple for $U_q(2)$ that is even and $3^+$-summable. It is shown that the Dirac operator is K-homologically nontrivial. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_17863 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On some algebraic and geometric aspects of the quantum unitary group Jana, Debabrata Operator Algebras 58B32, 58B34, 46L89, 16T20 Consider the compact quantum group $U_q(2)$, where $q$ is a non-zero complex deformation parameter such that $|q|\neq 1$. Let $C(U_q(2))$ denote the underlying $C^*$-algebra of the compact quantum group $U_q(2)$. We prove that if $q$ is a non-real complex number and $q^\prime$ is real, then the underlying $C^*$-algebras $C(U_q(2))$ and $C(U_{q^\prime}(2))$ are non-isomorphic. This is in sharp contrast with the case of braided $SU_q(2)$, introduced earlier by Woronowicz et al., where $q$ is a non-zero complex deformation parameter. In another direction, on a geometric aspect of $U_q(2)$, we introduce torus action on the $C^*$-algebra $C(U_q(2))$ and obtain a $C^*$-dynamical system $(C(U_q(2)),\mathbb{T}^3,α)$. We construct a $\mathbb{T}^3$-equivariant spectral triple for $U_q(2)$ that is even and $3^+$-summable. It is shown that the Dirac operator is K-homologically nontrivial. |
| title | On some algebraic and geometric aspects of the quantum unitary group |
| topic | Operator Algebras 58B32, 58B34, 46L89, 16T20 |
| url | https://arxiv.org/abs/2404.17863 |