Operator Algebras, Topology and Subgroups of Quantum Symmetry – Construction of Subgroups of Quantum Groups –

AbstractThe study of graph C*-algebras has a long history in operator algebras. Surprisingly, their quantum symmetries have not yet been computed. We close this gap by proving that the quantum automorphism group of a finite, directed graph without multiple edges acts maximally on the corresponding graph C*-algebra. This shows that the quantum symmetry of a graph coincides with the quantum symmetry of the graph C*-algebra. In our result, we use the definition of quantum automorphism groups of graphs as given by Banica in 2005. Note that Bichon gave a different definition in 2003; our action is inspired from his work. We review and compare these two definitions and we give a complete table of quantum automorphism groups (with respect to either of the two definitions) for undirected graphs on four vertices.

Download Full-text

Uprolling unrolled quantum groups

Communications in Contemporary Mathematics ◽

10.1142/s0219199721500231 ◽

2021 ◽

pp. 2150023

Author(s):

Thomas Creutzig ◽

Matthew Rupert

Keyword(s):

Quantum Group ◽

Lie Algebra ◽

Quantum Groups ◽

Vertex Operator ◽

Operator Algebras ◽

Vertex Algebra ◽

Vertex Operator Algebras ◽

Roots Of Unity ◽

Simple Lie Algebra ◽

Higher Rank

We construct families of commutative (super) algebra objects in the category of weight modules for the unrolled restricted quantum group [Formula: see text] of a simple Lie algebra [Formula: see text] at roots of unity, and study their categories of local modules. We determine their simple modules and derive conditions for these categories being finite, non-degenerate, and ribbon. Motivated by numerous examples in the [Formula: see text] case, we expect some of these categories to compare nicely to categories of modules for vertex operator algebras. We focus in particular on examples expected to correspond to the higher rank triplet vertex algebra [Formula: see text] of Feigin and Tipunin and the [Formula: see text] algebras.

Download Full-text

Various doublings of Hopf algebras. Operator algebras on quantum groups, complex cobordisms

Russian Mathematical Surveys ◽

10.1070/rm1992v047n05abeh000957 ◽

1992 ◽

Vol 47 (5) ◽

pp. 198-199 ◽

Cited By ~ 4

Author(s):

S P Novikov

Keyword(s):

Quantum Groups ◽

Hopf Algebras ◽

Operator Algebras

Download Full-text

Type classification of extreme quantized characters

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2019.58 ◽

2019 ◽

Vol 41 (2) ◽

pp. 593-605

Author(s):

RYOSUKE SATO

Keyword(s):

Dynamical Systems ◽

Quantum Groups ◽

Asymptotic Representation ◽

Operator Algebras ◽

Complete Solution ◽

Unitary Groups ◽

Inductive System ◽

Von Neumann ◽

Type Classification

The notion of quantized characters was introduced in our previous paper as a natural quantization of characters in the context of asymptotic representation theory for quantum groups. As in the case of ordinary groups, the representation associated with any extreme quantized character generates a von Neumann factor. From the viewpoint of operator algebras (and measurable dynamical systems), it is natural to ask what is the Murray–von Neumann–Connes type of the resulting factor. In this paper, we give a complete solution to this question when the inductive system is of quantum unitary groups $U_{q}(N)$.

Download Full-text

Quantum symmetries on noncommutative complex spheres with partial commutation relations

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025718500285 ◽

2018 ◽

Vol 21 (04) ◽

pp. 1850028

Author(s):

Simeng Wang

Keyword(s):

Quantum Groups ◽

Unitary Groups ◽

Symmetry Groups ◽

Orthogonal Groups ◽

Commutation Relations ◽

Quantum Symmetry ◽

Quantum Symmetries ◽

Free Independence

We introduce the notion of noncommutative complex spheres with partial commutation relations for the coordinates. We compute the corresponding quantum symmetry groups of these spheres, and this yields new quantum unitary groups with partial commutation relations. We also discuss some geometric aspects of the quantum orthogonal groups associated with the mixture of classical and free independence discovered by Speicher and Weber. We show that these quantum groups are quantum symmetry groups on some quantum spaces of spherical vectors with partial commutation relations.

Download Full-text