arXiv subjects

Chunguang Xia

Publications and source records attributed to Chunguang Xia.

14 recordsLinked to original sources

arXiv2025

Regular actions and semisimplicity of conformal modules over the general conformal algebra

We introduce the notion of a regular action in the category of conformal modules over Lie conformal algebras with Virasoro elements. We show that a finite conformal module over the general conformal algebra $\mathfrak{gc}_1$ (resp., $\mathfrak{gc}_N$ with $N\ge2$) is semisimple if and only if there exists a pair of different Virasoro elements (resp., canonical Virasoro elements) with regular actions. Along the way to finding a semisimplicity criteria, we also discuss the classification of Virasoro elements of $\mathfrak{gc}_N$ in-depth, leading us to construct a huge number of new Virasoro conformal modules.

math.RT↗

arXiv2024

Representations of non-finitely graded Heisenberg-Virasoro type Lie algebras

We construct and study non-finitely graded Lie algebras $\mathcal{HV}(a,b;ε)$ related to Heisenberg-Virasoro type Lie algebras, where $a,b$ are complex numbers, and $ε= \pm 1$. Using combinatorial techniques, we completely classify the free $\mathcal{U}(\mathfrak h)$-modules of rank one over $\mathcal{HV}(a,b;ε)$. It turns out that these modules are more varied and complex than those over non-finitely graded Virasoro algebras, and in particular admit infinitely many free parameters if $b=1$ and $ε=-1$. Meanwhile, we also determine the simplicity and isomorphism classes of these modules.

math.RT↗

arXiv2024

Representations of non-finitely graded Lie algebras related to Virasoro algebra

In this paper, we study representations of non-finitely graded Lie algebras $\mathcal{W}(ε)$ related to Virasoro algebra, where $ε= \pm 1$. Precisely speaking, we completely classify the free $\mathcal{U}(\mathfrak h)$-modules of rank one over $\mathcal{W}(ε)$,and find that these module structures are rather different from those of other graded Lie algebras. We also determine the simplicity and isomorphism classes of these modules.

math.RT↗

arXiv2020

Classification of finite irreducible conformal modules over $N=2$ Lie conformal superalgebras of Block type

We introduce the $N=2$ Lie conformal superalgebras ${\frak {K}}(p)$ of Block type, and classify their finite irreducible conformal modules for any nonzero parameter $p$. where $p$ is a nonzero complex number. In particular, we show that such a conformal module admits a nontrivial extension of a finite conformal module $M$ over $K_2$ if $p=-1$ and $M$ has rank $(2+2)$, where $K_2$ is an $N=2$ conformal subalgebra of ${\frak {K}}(p)$. As a byproduct, we obtain the classification of finite irreducible conformal modules over a series of finite Lie conformal superalgebras ${\frak k}(n)$ for $n\ge1$. Composition factors of all the involved reducible conformal modules are also determined.

math.RT↗

arXiv2019

Structure of a class of Lie conformal algebras of Block type

Let $p$ be a nonzero complex number. Recently, a class of infinite rank Lie conformal algebras $\mathfrak{B}(p)$ was introduced in [13]. In this paper, we study the structure theory of this class of Lie conformal algebras. Specifically, we completely determine the conformal derivations, the conformal biderivations and certain second cohomologies of $\mathfrak{B}(p)$.

math.RA↗

arXiv2017

Classification of finite irreducible conformal modules over a class of Lie conformal algebras of Block type

We classify finite irreducible conformal modules over a class of infinite Lie conformal algebras ${\frak {B}}(p)$ of Block type, where $p$ is a nonzero complex number. In particular, we obtain that a finite irreducible conformal module over ${\frak {B}}(p)$ may be a nontrivial extension of a finite conformal module over ${\frak {Vir}}$ if $p=-1$, where ${\frak {Vir}}$ is a Virasoro conformal subalgebra of ${\frak {B}}(p)$. As a byproduct, we also obtain the classification of finite irreducible conformal modules over a series of finite Lie conformal algebras ${\frak b}(n)$ for $n\ge1$.

math.RA↗

arXiv2016

Infinite rank Schrodinger-Virasoro type Lie conformal algebras

Motivated by the structure of certain modules over the loop Virasoro Lie conformal algebra and the Lie structures of Schrodinger-Virasoro algebras, we construct a class of infinite rank Lie conformal algebras CSV (a, b), where a, b are complex numbers. The conformal derivations of CSV (a, b) are uniformly determined. The rank one conformal modules and Z-graded free intermediate series modules over CSV (a, b) are classified. Corresponding results of the conformal subalgebra CHV (a, b) of CSV (a, b) are also presented.

math.RA↗

arXiv2014

Simple Deformed Witt Algebras

For any factorization domain $\cal A$ and an algebra endomorphism $σ$ of $\cal A$, there exists a non-associative algebra $({\cal A},σ,[\cdot,\cdot])$ with multiplication satisfying skew-symmetry and generalized (twisted) Jacobi identities, called a $σ$-deformed Witt algebra. In this paper, we obtain the necessary and sufficient conditions for the algebra $({\cal A},σ,[\cdot,\cdot])$ to be simple.

math.RA↗

arXiv2012

Classification of quasifinite representations of a Lie algebra related to Block type

A well-known theorem of Mathieu's states that a Harish-chandra module over the Virasoro algebra is either a highest weight module, a lowest weight module or a module of the intermediate series. It is proved in this paper that an analogous result also holds for the Lie algebra $\BB$ related to Block type, with basis {L_{\a,i},C|a,i\in\Z, i\ge0} and relations [L_{\a,i},L_{\b,j}]=((i+1)\b-(j+1)\a)L_{\a+\b,i+j}+\d_{\a+\b,0}\d_{i+j,0}\frac{\a^3-\a}{6}C, [C,L_{\a,i}]=0.Namely, an irreducible quasifinite $\BB$-module is either a highest weight module, a lowest weight module or a module of the intermediate series.

math.RT↗

arXiv2011

Structure of a class of Lie algebras of Block type

Let $\BB$ be a class of Lie algebras of Block type with basis $\{L_{\a,i}|\a,i\in\Z, i\geq 0\}$ and relations $[L_{\a,i},L_{\b,j}]=(\b(i+q)-\a(j+q))L_{\a+\b,i+j}$, where $q$ is a positive integer. In this paper, it is shown that $\BB$ are different from each other for distinct positive integers $q$'s. The automorphism groups, the derivation algebras and the central extensions of all $\BB$ are also uniformly and explicitly described, which generalize some previous results.

math.RA↗

arXiv2011

Quasifinite representations of a class of Block type Lie algebras $\BB$

Intrigued by a well-known theorem of Mathieu's on Harish-Chandra modules over the Virasoro algebra, we give an analogous result for a class of Block type Lie algebras $\BB$, where the parameter $q$ is a nonzero complex number. We also classify quasifinite irreducible highest weight $\BB$-modules and irreducible $\BB$-modules of the intermediate series. In particular, we obtain that an irreducible $\BB$-module of the intermediate series may be a nontrivial extension of a $\Vir$-module of the intermediate series if $q$ is half of a negative integer, where $\Vir$ is a subalgebra of $\BB$ isomorphic to the Virasoro algebra.

math.RT↗

arXiv2010

Broadband Acoustic Cloak for Ultrasound Waves

Invisibility devices based on coordinate transformation have opened up a new field of considerable interest. Such a device is proposed to render the hidden object undetectable under the flow of light or sound, by guiding and controlling the wave path through an engineered space surrounding the object. We present here the first practical realization of a low-loss and broadband acoustic cloak for underwater ultrasound. This metamaterial cloak is constructed with a network of acoustic circuit elements, namely serial inductors and shunt capacitors. Our experiment clearly shows that the acoustic cloak can effectively bend the ultrasound waves around the hidden object, with reduced scattering and shadow. Due to the non-resonant nature of the building elements, this low loss (~6dB/m) cylindrical cloak exhibits excellent invisibility over a broad frequency range from 52 to 64 kHz in the measurements. The low visibility of the cloaked object for underwater ultrasound shed a light on the fundamental understanding of manipulation, storage and control of acoustic waves. Furthermore, our experimental study indicates that this design approach should be scalable to different acoustic frequencies and offers the possibility for a variety of devices based on coordinate transformation.

physics.optics↗

arXiv2010

First Jump of Microgel: Actuation Speed Enhancement by Elastic Instability

Swelling-induced snap-buckling in a 3D micro hydrogel device, inspired by the insect-trapping action of Venus flytrap, makes it possible to generate astonishingly fast actuation. We demonstrate that elastic energy is effectively stored and quickly released from the device by incorporating elastic instability. Utilizing its rapid actuation speed, the device can even jump by itself upon wetting.

cond-mat.soft↗

arXiv2010

Derivations and automorphisms of a Lie algebra of Block type

Let $\BB$ be the Lie algebra of Block type with basis $\{L_{\a,i}|\,\a,i\in\Z, i\geq0\}$ and relations $[L_{\a,i},L_{\b,j}]=\left((\a-1)(j+1)-(\b-1)(i+1)\right)L_{\a+\b,i+j}$. In the present paper, the derivation algebra and the automorphism group of $\BB$ are explicitly described. In particular, it is shown that the outer derivation space is 1-dimensional and the inner automorphism group of $\BB$ is trivial.

math.RA↗