Higher Order Measures, Generalized Quantum Mechanics and Hopf Algebras

Quaternions, Spinors and the Hopf Fibration: Hidden Variables in Classical Mechanics

Rotations in 3 dimensional space are equally described by the SU(2) and SO(3) groups. These isomorphic groups generate the same 3D kinematics using different algebraic structures of the unit quaternion. The Hopf Fibration is a projection between the hypersphere $\mathbb{S}^3$ of the quaternion in 4D space, and the unit sphere $\mathbb{S}^2$ in 3D space. Great circles in $\mathbb{S}^3$ are mapped to points in $\mathbb{S}^2$ via the 6 Hopf maps, and are illustrated via the stereographic projection. The higher and lower dimensional spaces are connected via the $\mathbb{S}^1$ fibre bundle which co…

A generic Hopf algebra for quantum statistical mechanics

In this paper, we present a Hopf algebra description of a bosonic quantum model, using the elementary combinatorial elements of Bell and Stirling numbers. Our objective in doing this is as follows. Recent studies have revealed that perturbative quantum field theory (pQFT) displays an astonishing interplay between analysis (Riemann zeta functions), topology (Knot theory), combinatorial graph theory (Feynman diagrams) and algebra (Hopf structure). Since pQFT is an inherently complicated study, so far not exactly solvable and replete with divergences, the essential simplicity of the relationships…

基于Hopf代数的广义cluster态：不可逆对称性与Hopf张量网络表示

Generalized cluster states from Hopf algebras: non-invertible symmetry and Hopf tensor network representation

本文提出了基于Hopf代数的cluster states构造方法，发展了完整的Hopf qudit理论，揭示了其不可逆对称性，并引入了可用于求解该模型的Hopf张量网络表示。

Non-Geometric Fluxes, Quasi-Hopf Twist Deformations and Nonassociative Quantum Mechanics

We analyse the symmetries underlying nonassociative deformations of geometry in non-geometric R-flux compactifications which arise via T-duality from closed strings with constant geometric fluxes. Starting from the non-abelian Lie algebra of translations and Bopp shifts in phase space, together with a suitable cochain twist, we construct the quasi-Hopf algebra of symmetries that deforms the algebra of functions and the exterior differential calculus in the phase space description of nonassociative R-space. In this setting nonassociativity is characterised by the associator 3-cocycle which cont…

广义多重融合弦网模型的弱Hopf对称性与管代数

Weak Hopf symmetry and tube algebra of the generalized multifusion string-net model

本文研究了广义多重融合弦网模型的弱Hopf对称性，构造了其弱Hopf管代数，并利用幺正多重融合范畴刻画了带隙边界与畴壁相的对称性及管代数结构。

Ribbon operators in the generalized Kitaev quantum double model based on Hopf algebras

Kitaev's quantum double model is a family of exactly solvable lattice models that realize two dimensional topological phases of matter. Originally it is based on finite groups, and is later generalized to semi-simple Hopf algebras. We rigorously define and study ribbon operators in the generalized Kitaev quantum double model. These ribbon operators are important tools to understand quasi-particle excitations. It turns out that there are some subtleties in defining the operators in contrast to what one would naively think. In particular, one has to distinguish two classes of ribbons which we ca…