Equivariant Tannaka-Krein reconstruction and quantum automorphism groups of discrete structures

TL;DR

This paper introduces a generalized Tannaka-Krein reconstruction method to define quantum automorphism groups for various discrete structures, including quantum Cayley graphs, expanding the understanding of symmetries in quantum and discrete mathematics.

Contribution

It develops a broad framework for constructing quantum automorphism groups of discrete structures using a generalized Tannaka-Krein approach, applicable to complex algebraic and combinatorial objects.

Findings

Constructed algebraic quantum groups acting on matrix algebra sums

Established equivalence between categories of equivariant corepresentations and bimodule categories

Defined quantum automorphism groups for quantum Cayley graphs

Abstract

We define quantum automorphism groups of a wide range of discrete structures. The central tool for their construction is a generalisation of the Tannaka-Krein reconstruction theorem. For any direct sum of matrix algebras $M$, and any concrete unitary 2-category of finite type Hilbert-$M$-bimodules $\mathcal{C}$, under reasonable conditions, we construct an algebraic quantum group $\mathbb{G}$ which acts on $M$ by $\alpha$, such that the category of $\alpha$-equivariant corepresentations of $\mathbb{G}$ on finite type Hilbert-$M$-bimodules is equivalent to $\mathcal{C}$. Moreover, we explicitly describe how to get such categories from connected locally finite discrete structures. As an example, we define the quantum automorphism group of a quantum Cayley graph.