arXiv subjects

Ujjal Karmakar

Publications and source records attributed to Ujjal Karmakar.

4 recordsLinked to original sources

arXiv2025

Quantum Automorphism Group of Direct Sum of Cuntz 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.

math.OA↗

arXiv2025

Some Results on Bichon's Quantum Automorphism Group of Graphs

The notion of the quantum automorphism group of a graph was introduced by J. Bichon in 2003 and T. Banica in 2005 respectively. This article explores primarily the quantum automorphism group of a graph $Γ$, denoted by $QAut_{Bic}(Γ)$, in Bichon's framework. First, we provide a sufficient condition for non-commutativity of Bichon's quantum automorphism group and discuss several applications of this criterion. Although it is known that $QAut_{Bic}(Γ) \cong QAut_{Bic}(Γ^c)$ does not hold in general, we identify a family of graphs for which this isomorphism enforces that the graph has no quantum symmetry. Moreover, we describe a few families of graphs having quantum symmetries whose quantum automorphism groups in Bichon's sense are commutative. Finally, we show that free product, tensor product and free wreath product constructions can arise as Bichon's quantum automorphism groups of connected graphs in the following sense: For a finite family of compact matrix quantum groups $\{Q_i\}_{i=1}^{m}$ arising as Bichon's quantum automorphism groups of certain graphs, there exist connected graphs $Γ_{free}$, $Γ_{ten}$ and $Γ_{wr}$ whose quantum automorphism groups are $*_{i=1}^{m} Q_{i}$, $\otimes_{i=1}^{m} Q_i$ and $Q_1 \wr_{*} Q_2$ respectively.

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arXiv2025

Quantum Symmetries of Graph C*-algebras Having Maximal Permutational Symmetry

Quantum symmetry of a graph $C^{*}$-algebra $C^{*}(Γ)$ corresponding to a finite graph $Γ$ has been explored by several mathematicians within different categories in the past few years. In this article, we establish that there are exactly three families of compact matrix quantum groups, containing the symmetric group on the set of edges of the underlying graph $Γ$, that can be achieved as the quantum symmetries of graph $C^*$-algebras in the category introduced by Joardar and Mandal. Moreover, we demonstrate that there does not exist any graph $C^*$-algebra associated with a finite graph $Γ$ without isolated vertices having $A_{u^t}(F^Γ)$ as the quantum automorphism group of $C^*(Γ)$ for a non-scalar matrix $F^Γ$.

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arXiv2024

Rigidity on Quantum Symmetry for a Certain Class of Graph C*-algebras

Quantum symmetry of graph $C^{*}$-algebras has been studied, under the consideration of different formulations, in the past few years. It is already known that the compact quantum group $(\underbrace{C(S^{1})*C(S^{1})*\cdots *C(S^{1})}_{|E(Γ)|-times}, Δ) $ always acts on a graph $C^*$-algebra for a finite, connected, directed graph $Γ$ in the category introduced by Joardar and Mandal, where $|E(Γ)|:=$ number of edges in $Γ$. In this article, we show that for a certain class of graphs including Toeplitz algebra, quantum odd sphere, matrix algebra etc. the quantum symmetry of their associated graph $C^*$-algebras remains $(\underbrace{C(S^{1})*C(S^{1})*\cdots *C(S^{1})}_{|E(Γ)|-times}, Δ) $ in the category as mentioned before. More precisely, if a finite, connected, directed graph $Γ$ satisfies the following graph theoretic properties : (i) there does not exist any cycle of length $\geq$ 2 (ii) there exists a path of length $(|V(Γ)|-1)$ which consists all the vertices, where $|V(Γ)|:=$ number of vertices in $Γ$ (iii) given any two vertices (may not be distinct) there exists at most one edge joining them, then the universal object coincides with $(\underbrace{C(S^{1})*C(S^{1})*\cdots *C(S^{1})}_{|E(Γ)|-times}, Δ) $. Furthermore, we have pointed out a few counter examples whenever the above assumptions are violated.

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