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Into Multiplier Hopf ($*$-)Graph Algebras

Farrokh Razavinia

TL;DR

The paper develops a framework of multiplier Hopf $*$-graph algebras built from graph/C*-algebra data, starting from Cuntz–Krieger constructions and the notion of magic unitary matrices. It introduces two construction avenues for multiplier Hopf $*$-graph algebras and furnishes explicit finite- and infinite-dimensional examples, notably yielding $\mathcal{C}^*(\pi_2)=M_4(\mathbb{C})$ and, more generally, multiplier Hopf structures derived from coordinate rings $\mathbb{K}[M_q(n)]$. It then elevates the framework to discrete quantum groups by constructing coproducts on operator algebras such as $\mathcal{O}(GL(n^2))$ and proving that $\mathcal{O}(SL(n))$ (and related unitary/orthogonal variants) form discrete quantum groups in the sense of Van Daele, with counit and antipode. The work also sketches potential extensions toward finite groups of Lie type and $q$-deformations, aiming to connect quantum group theory with graph-based quantum information concepts. Overall, the paper advances a program linking graph algebras, multiplier Hopf structures, and discrete quantum groups, with applications to quantum symmetries and possibly finite quantum-group analogues of Lie-type objects.

Abstract

This paper is concerned with the structures introduced recently by the authors of the current paper concerning the multiplier Hopf $*$-graph algebras and also the Cuntz-Krieger algebras and their relations with the $C^*$-graph algebras, and once again by using the $C^*$-graph algebra constructions associated to our toy example, to initiate our first class of examples concerning the multiplier Hopf $*$-graph algebras. At the final part of the paper, we apply our study to the $SL(n)$ case over the field of complex numbers, and prove that $(\mathcal{O}(SL(n),Δ)$ possesses the initial requirements of being a discrete quantum group in the sense of Van Daele, and propose a direction in approaching one step further to the problem raised by Wang, asking ``if finite groups of Lie type have an analogue of $q$-deformations into finite quantum groups?''.

Into Multiplier Hopf ($*$-)Graph Algebras

TL;DR

The paper develops a framework of multiplier Hopf -graph algebras built from graph/C*-algebra data, starting from Cuntz–Krieger constructions and the notion of magic unitary matrices. It introduces two construction avenues for multiplier Hopf -graph algebras and furnishes explicit finite- and infinite-dimensional examples, notably yielding and, more generally, multiplier Hopf structures derived from coordinate rings . It then elevates the framework to discrete quantum groups by constructing coproducts on operator algebras such as and proving that (and related unitary/orthogonal variants) form discrete quantum groups in the sense of Van Daele, with counit and antipode. The work also sketches potential extensions toward finite groups of Lie type and -deformations, aiming to connect quantum group theory with graph-based quantum information concepts. Overall, the paper advances a program linking graph algebras, multiplier Hopf structures, and discrete quantum groups, with applications to quantum symmetries and possibly finite quantum-group analogues of Lie-type objects.

Abstract

This paper is concerned with the structures introduced recently by the authors of the current paper concerning the multiplier Hopf -graph algebras and also the Cuntz-Krieger algebras and their relations with the -graph algebras, and once again by using the -graph algebra constructions associated to our toy example, to initiate our first class of examples concerning the multiplier Hopf -graph algebras. At the final part of the paper, we apply our study to the case over the field of complex numbers, and prove that possesses the initial requirements of being a discrete quantum group in the sense of Van Daele, and propose a direction in approaching one step further to the problem raised by Wang, asking ``if finite groups of Lie type have an analogue of -deformations into finite quantum groups?''.
Paper Structure (9 sections, 14 theorems, 40 equations, 3 figures)

This paper contains 9 sections, 14 theorems, 40 equations, 3 figures.

Table of Contents

  1. Notations
  2. Introduction
  3. Cuntz-Krieger Algebras
  4. Graph $C^*$-algebras
  5. Multiplier Hopf $*$-Graph Algebras
  6. $n$-type discrete quantum groups
  7. Concluding remarks
  8. Acknowledgement
  9. Appendix

Key Result

Theorem 2.3

RV22 For $\Pi$ a locally connected finite graph with vertex set $I$, there exist a (necessarily unique) universal nondegenerate $*$-algebra $\mathcal{A}$ generated by the elements $u_{ij}$ satisfying the relations of the magic unitary matrix in Definition Def:MUM, and a unique nondegenerate $*$-homo

Figures (3)

  • Figure 1: Illustration of the set of $n-1$ graphs $\mathcal{G}_i$
  • Figure 2: Directed locally connected graph related to $\Pi_2$
  • Figure 3: Directed 2-connected graph related to $\pi_2$

Theorems & Definitions (44)

  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Definition 2.8
  • Remark 2.9
  • Definition 2.10
  • Remark 2.11
  • ...and 34 more