Quantum Automorphism Group of Direct Sum of Cuntz Algebras
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911297096908800 |
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| author | Karmakar, Ujjal Mandal, Arnab |
| author_facet | Karmakar, Ujjal Mandal, Arnab |
| contents | In this article, we explore the quantum symmetry of the direct sum of a finite family of Cuntz algebras $\{\mathcal{O}_{n_i} \}_{i=1}^{m}$, viewing them as graph $C^*$-algebras associated to the graphs $\{L_{n_i}\}_{i=1}^{m}$ (where $L_n$ denotes the graph containing $n$ loops based at a single vertex), in the category introduced by Joardar and Mandal. It has been shown that the quantum automorphism group of the direct sum of non-isomorphic Cuntz algebras is ${U}_{n_1}^{+}*{U}_{n_2}^{+}* \cdots *{U}_{n_m}^{+}$ for distinct $n_i$'s, i.e.
\begin{equation*}
Q_τ^{Lin}(\sqcup_{i=1}^{m} ~ L_{n_i}) \cong *_{i=1}^{m} ~~ Q_τ^{Lin}(L_{n_i}) \cong {U}_{n_1}^{+}*{U}_{n_2}^{+}* \cdots *{U}_{n_m}^{+}, \end{equation*}
where $Q_τ^{Lin}(Γ)$ denotes the quantum automorphism group of the graph $C^*$-algebra associated to $Γ$.
Also, the quantum automorphism group of the direct sum of $m$ copies of isomorphic Cuntz algebra $\mathcal{O}_n$ is $U_n^+ \wr_* S_m^+$, i.e.
\begin{equation*}
Q_τ^{Lin}(\sqcup_{i=1}^{m} ~ L_n) \cong Q_τ^{Lin}(L_n) \wr_* S_m^+ \cong U_n^+ \wr_* S_m^+.
\end{equation*}
Furthermore, we have provided counter-examples to demonstrate that the isomorphisms mentioned above cannot be generalized to arbitrary graph $C^*$-algebras, whereas analogous relations can be extended in the context of quantum automorphism groups of graphs in the sense of Banica and Bichon. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_06241 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantum Automorphism Group of Direct Sum of Cuntz Algebras Karmakar, Ujjal Mandal, Arnab Operator Algebras In this article, we explore the quantum symmetry of the direct sum of a finite family of Cuntz algebras $\{\mathcal{O}_{n_i} \}_{i=1}^{m}$, viewing them as graph $C^*$-algebras associated to the graphs $\{L_{n_i}\}_{i=1}^{m}$ (where $L_n$ denotes the graph containing $n$ loops based at a single vertex), in the category introduced by Joardar and Mandal. It has been shown that the quantum automorphism group of the direct sum of non-isomorphic Cuntz algebras is ${U}_{n_1}^{+}*{U}_{n_2}^{+}* \cdots *{U}_{n_m}^{+}$ for distinct $n_i$'s, i.e. \begin{equation*} Q_τ^{Lin}(\sqcup_{i=1}^{m} ~ L_{n_i}) \cong *_{i=1}^{m} ~~ Q_τ^{Lin}(L_{n_i}) \cong {U}_{n_1}^{+}*{U}_{n_2}^{+}* \cdots *{U}_{n_m}^{+}, \end{equation*} where $Q_τ^{Lin}(Γ)$ denotes the quantum automorphism group of the graph $C^*$-algebra associated to $Γ$. Also, the quantum automorphism group of the direct sum of $m$ copies of isomorphic Cuntz algebra $\mathcal{O}_n$ is $U_n^+ \wr_* S_m^+$, i.e. \begin{equation*} Q_τ^{Lin}(\sqcup_{i=1}^{m} ~ L_n) \cong Q_τ^{Lin}(L_n) \wr_* S_m^+ \cong U_n^+ \wr_* S_m^+. \end{equation*} Furthermore, we have provided counter-examples to demonstrate that the isomorphisms mentioned above cannot be generalized to arbitrary graph $C^*$-algebras, whereas analogous relations can be extended in the context of quantum automorphism groups of graphs in the sense of Banica and Bichon. |
| title | Quantum Automorphism Group of Direct Sum of Cuntz Algebras |
| topic | Operator Algebras |
| url | https://arxiv.org/abs/2402.06241 |