Related papers: A new characterization of Kac-type discrete quantu…
The purpose of this paper is to describe some conjectures and results on the existence and uniqueness of invariant measures on formal completions of Kac-Moody groups and associated homogeneous spaces. Existence is rigorously established in…
Kazhdan's property (T) has been studied for several discrete group-like structures, including standard invariants of Jones' subfactors and discrete quantum groups. We prove a Zuk-type spectral gap criterion for property (T) in this setting.…
We describe quantum enveloping algebras of symmetric Kac-Moody Lie algebras via a finite field Hall algebra construction involving Z_2-graded complexes of quiver representations.
A group theoretical description of basic discrete symmetries (space inversion P, time reversal T and charge conjugation C) is given. Discrete subgroups of orthogonal groups of multidimensional spaces over the fields of real and complex…
The meaning of quantum group transformation properties is discussed in some detail by comparing the (co)actions of the quantum group with those of the corresponding Lie group, both of which have the same algebraic (matrix) form of the…
We define the notion of invariant derivation of a C*-algebra under a compact quantum group action and prove that in certain conditions, such derivations are generators of one parameter automorphism groups.
We show that finite index quantum subgroups of a discrete quantum group are induced from finite index quantum subgroups of the unimodularization. As an application, we classify all finite index quantum subgroups of free products of the…
Discrete quantum mechanics is here defined to be a quantum theory of wave functions defined on integers P_i and Q_i, while canonical quantum mechanics is assumed to be based on wave functions on the real numbers, R^n. We study reversible…
The notion of an open quantum subgroup of a locally compact quantum group is introduced and given several equivalent characterizations in terms of group-like projections, inclusions of quantum group C*-algebras and properties of respective…
We determine the quantum automorphism groups of finite graphs. These are quantum subgroups of the quantum permutation groups defined by Wang. The quantum automorphism group is a stronger invariant for finite graphs than the usual one. We…
In this paper we use the recent developments in the representation theory of locally compact quantum groups, to assign, to each locally compact quantum group $\mathbb{G}$, a locally compact group $\tilde \mathbb{G}$ which is the quantum…
We outline the recent classification of differential structures for all main classes of quantum groups. We also outline the algebraic notion of `quantum manifold' and `quantum Riemannian manifold' based on quantum group principal bundles, a…
Double coverings of the orthogonal groups of the real and complex spaces are considered. The relation between discrete transformations of these spaces and fundamental automorphisms of Clifford algebras is established, where an isomorphism…
We investigate compact quantum group actions on unital $C^*$-algebras by analyzing invariant subsets and invariant states. In particular, we come up with the concept of compact quantum group orbits and use it to show that countable compact…
In this paper, we extend past work done on the application of the mathematics of category theory to quantum information science. Specifically, we present a realization of a dagger-compact category that can model finite-dimensional quantum…
A notion of a quantum automorphism group of a finite quantum group, generalising that of a classical automorphism group of a finite group, is proposed and a corresponding existence result proved.
We reconstruct a quantum group associated with any Lie algebra together with its representation theory from twisted homologies of generalized configuration spaces of disks. Along the way it brings new combinatorics to the theory, but our…
The quantum deformed (1+1) Poincare' algebra is shown to be the kinematical symmetry of the harmonic chain, whose spacing is given by the deformation parameter. Phonons with their symmetries as well as multiphonon processes are derived from…
This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples…
To each graph without loops and multiple edges we assign a family of rings. Categories of projective modules over these rings categorify $U^-_q(\mathfrak{g})$, where $\mathfrak{g}$ is the Kac-Moody Lie algebra associated with the graph.