Quantum Symmetries of Higher Rank Graphs and their C*-algebras

Sharma, Jitender (2026) Quantum Symmetries of Higher Rank Graphs and their C*-algebras. PhD thesis, Indian Institute of Science Education and Research Kolkata.

Text (PhD thesis of Jitender Sharma (20IP007)) 20IP007.pdf - Submitted Version Restricted to Repository staff only Download (1MB)

Abstract

In this work we developed the quantum aspect of higher rank graph, especially 2- graph. In particular we formulate the quantum automorphism group for rank 2-graph and carried out the computations for several example. Additionally, we formulate the notion of quantum isomorphism between the pair of 2-graph and generalized various result of 1-graph to rank 2-graph. In particular we have shown that if a pair of 2-graph ∧1 and ∧2 are not quantum isomorphic, then quantum automorphism group Aut⁺(∧1 ⨆ ∧2) is the free product of Aut⁺(∧1) and Aut⁺(∧2). Moreover we have shown that quantum automorphism group of finite disjoint union of a 2-graph ∧ is free wreath product of quantum automorphism group Aut⁺(∧) by quantum permutation group S⁺n . This work enrich the quantum symmetries aspect for higher rank graphs. In the second part of the thesis, we studied the isometric action of higher rank graph C*-algebras C*(∧), using the language of spectral triple and compact quantum group. In particular we have shown that for a strongly connected finite 1-graph ∧, the quantum automorphism group Aut⁺(∧) acts isometrically on the spectral triple associated with higher rank graph in the sense of Goswami. This establishes a concrete link between the quantum symmetries of graphs and their noncommutative geometric structures, thereby providing a bridge between quantum theory higherrank graphs and the theory of spectral geometry on noncommutative spaces.

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