Searcharxiv⌕ Search

arXiv subjects

Huichi Huang

Publications and source records attributed to Huichi Huang.

15 recordsLinked to original sources
arXiv2026

A Concise Course in Galois Theory

Born from years of teaching undergraduate and graduate algebra courses at Chongqing University, this text is designed to introduce Galois theory while minimizing prerequisites. It seeks to reconnect the abstract machinery of modern algeba: groups, rings, and fields with the historical problem that inspired its creation: determining when a polynomial can be solved by radicals. By anchoring abstract concepts in concrete motivation, we hope to illuminate both the ``how" and the ``why" of algebraic structures.

math.HO↗
arXiv2024

A Wiener-type theorem for arcs in the unit cirle

We prove a Wiener-type theorem for arcs in the unit circle which concerns express the measure of an arc in the unit circle via the measure's Fourier coefficients. Then we use it to give the Fourier series of the Cantor and to compute the local dimension of a measure satisfying certain conditions of Fourier coefficients.

math.FA↗
arXiv2021

Amenable fusion algebraic actions of discrete quantum groups on compact quantum spaces

In this paper, we introduce actions of fusion algebras on unital $C^*$-algebras, and define amenability for fusion algebraic actions. Motivated by S.\ Neshveyev et al.'s work, considering the co-representation ring of a compact quantum group as a fusion algebra, we define the canonical fusion algebraic (for short, CFA) form of a discrete quantum group action on a compact quantum space. Furthermore, through the CFA form, we define FA-amenability of discrete quantum group actions, and present some basic connections between FA-amenable actions and amenable discrete quantum groups. As an application, thinking of a state on a unital $C^*$-algebra as a "probability measure" on a compact quantum space, we show that amenability for a discrete quantum group is equivalent to both of FA-amenability for an action of this discrete quantum group on a compact quantum space and the existence of this kind of "probability measure" that is FA-invariant under this action.

math.OA↗
arXiv2017

The quantum group fixing a sequence of finite subsets

Motivated by generalizing Szemerédi's theorem, we the elements in a discrete quantum group fixing a sequence of finite subsets and prove that the set of these elements is a quantum subgroup. Using this we obtain a version of mean ergodic theorem for discrete quantum groups.

math.OA↗
arXiv2017

Fourier coefficients of $\times p$-invariant measures

We consider densities $D_Σ(A)$, $\overline{D}_Σ(A)$ and $\underline{D}_Σ(A)$ for a subset $A$ of $\mathbb{N}$ with respect to a sequence $Σ$ of finite subsets of $\mathbb{N}$ and study Fourier coefficients of ergodic, weakly mixing and strongly mixing $\times p$-invariant measures on the unit circle $\mathbb{T}$. Combining these, we prove the following measure rigidity results: on $\mathbb{T}$, the Lebesgue measure is the only non-atomic $\times p$-invariant measure satisfying one of the following: (1) $μ$ is ergodic and there exist a Følner sequence $Σ$ in $\mathbb{N}$ and a nonzero integer $l$ such that $μ$ is $\times (p^j+l)$-invariant for all $j$ in a subset $A$ of $\mathbb{N}$ with $D_Σ(A)=1$; (2) $μ$ is weakly mixing and there exist a Følner sequence $Σ$ in $\mathbb{N}$ and a nonzero integer $l$ such that $μ$ is $\times (p^j+l)$-invariant for all $j$ in a subset $A$ of $\mathbb{N}$ with $\overline{D}_Σ(A)>0$; (3) $μ$ is strongly mixing and there exists a nonzero integer $l$ such that $μ$ is $\times (p^j+l)$-invariant for infinitely many $j$. Moreover, a $\times p$-invariant measure satisfying (2) or (3) is either a Dirac measure or the Lebesgue measure. As an application we prove that for every increasing function $τ$ defined on positive integers with $\lim_{n\to\infty}τ(n)=\infty$, there exists a multiplicative semigroup $S_τ$ of $\mathbb{Z}^+$ containing $p$ such that $|S_τ\cap[1,n]|\leq (\log_p n)^{τ(n)}$ and the Lebesgue measure is the only non-atomic ergodic $\times p$-invariant measure which is $\times q$-invariant for all $q$ in $S_τ$.

math.DS↗
arXiv2016

Continuous $\times p,\times q$-invariant measures on the unit circle

We express continuous $\times p,\times q$-invariant measures on the unit circle via some simple forms. On one hand, a continuous $\times p,\times q$-invariant measure is the weak-$*$ limit of average of Dirac measures along an irrational orbit. On the other hand, a continuous $\times p,\times q$-invariant measure is a continuous function on $[0,1]$ satisfying certain function equations.

math.DS↗
arXiv2016

Ergodic invariant states and irreducible representations of crossed product $C^*$-algebras

Motivated by reformulating Furstenberg's $\times p,\times q$ conjecture via representations of a crossed product $C^*$-algebra, we show that in a discrete $C^*$-dynamical system $(A,Γ)$, the space of (ergodic) $Γ$-invariant states on $A$ is homeomorphic to a subspace of (pure) state space of $A\rtimesΓ$. Various applications of this in topological dynamical systems and representation theory are obtained. In particular, we prove that the classification of ergodic $Γ$-invariant regular Borel probability measures on a compact Hausdorff space $X$ is equivalent to the classification a special type of irreducible representations of $C(X)\rtimes Γ$.

math.OA↗
arXiv2014

Invariant subsets under compact quantum group actions

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 metrizable spaces with infinitely many points are not quantum homogeneous spaces.

math.OA↗