Chengkang Xu
In this paper, we study a class of infinitesimal deformations of the centerless higher rank Heisenberg-Virasoro algebras. Explicitly, the universal central extensions, derivations and isomorphism classes of these algebras are determined.
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Derivations and 2-local derivations on the mirror Heisenberg-Virasoro algebra
本文利用第一上同调确定了镜像Heisenberg-Virasoro代数上的导子，并进一步证明该代数上的任意2-局部导子均为导子。
In this paper, the biderivations without the skew-symmetric condition of the twisted Heisenberg-Virasoro algebra are presented. We find some non-inner and non-skew-symmetric biderivations. As applications, the characterizations of the forms of linear commuting maps and the commutative post-Lie algebra structures on the twisted Heisenberg-Virasoro algebra are given. It also is proved that every biderivation of the graded twisted Heisenberg-Virasoro left-symmetric algebra is trivial.
In this paper we investigate Lie bialgebra structures on the twisted Heisenberg-Virasoro algebra. With the classifications of Lie bialgebra structures on the Virasoro algebra, we determined such structures on the twisted Heisenberg-Virasoro algebra. Moreover, some general and useful results are obtained. With our methods and results we also can easily to determine such structures on some Lie algebras related to the twisted Heisenberg-Virasoro algebra.
In this paper, by studying the maximal good subspaces, we determine the dual Lie coalgebras of the centerless twisted Heisenberg-Virasoro algebra. Based on this, we construct the dual Lie bialgebras structures of the twisted Heisenberg-Virasoro type. As by-products, four new infinite dimensional Lie algebras are obtained.
Derivations, 2-local derivations, biderivations and automorphisms of generalized Loop Heisenberg-Virasoro algebras
本文引入广义Loop Heisenberg-Virasoro代数，确定其导子、证明2-局部导子为导子、双导子为内双导子并给出应用，最后给出自同构群。
Second cohomology groups and left-symmetric algebraic structures of the generalized loop Heisenberg-Virasoro algebra
作为广义圈海森堡-维拉索罗代数系列研究的第二篇，本文利用其Witt李子代数刻画了该代数的二阶上同调群与左对称代数结构。
In this paper, we compute the cohomology of the Heisenberg-Virasoro conformal algebra with coefficients in its modules, and in particular with trivial coefficients both for the basic and reduced complexes.
In this paper, the conjugate-linear anti-involutions and the unitary irreducible modules of the intermediate series over the twisted Heisenberg-Virasoro algebra are classified respectively. We prove that any unitary irreducible module of the intermediate series over the twisted Heisenberg-Virasoro algebra is of the form $\mathcal{A}_{a,b,c}$ for $a\in \mathbb{R}, b\in 1/2+\sqrt{-1}\mathbb{R}, c\in \mathbb{C}.$
Local and Quasi Derivations of the Mirror Heisenberg-Virasoro Algebra
本文确定了镜像Heisenberg-Virasoro代数的广义导子和局部导子，并完整刻画了其$\delta$-(双)导子。
Maps on the mirror Heisenberg-Virasoro algebra, II
本文作为系列研究第二篇，刻画复镜像 Heisenberg-Virasoro 代数上的二元映射，给出其双导子与左对称代数结构。
Whittaker Modules for the twisted Heisenberg-Virasoro Algebra
我们定义的扭曲海森堡Virasoro代数的Whittaker模块，并获得类似的几个结果，从经典的设置，包括简单的Whittaker模块的中心字符的分类。
2-local derivation is a generalized derivation for a Lie algebra, which plays an important role to the study of local properties of the structure of the Lie algebra. In this paper, we prove that every 2-local derivation on the twisted Heisenberg-Virasoro algebra is a derivation.
Let $HV$ be the loop Heisenberg-Virasoro Lie algebra over $\C$ with basis $\{L_{\a,i},H_{\b,j}\,|\,\a,\,\b,i,j\in\Z\}$ and brackets $[L_{\a,i},L_{\b,j}]=(\a-\b)L_{\a+\b,i+j}, [L_{\a,i},H_{\b,j}]=-\b H_{\a+\b,i+j},[H_{\a,i},H_{\b,j}]=0$. In this paper, a formal distribution Lie algebra of $HV$ is constructed. Then the associated conformal algebra $CHV$ is studied, where $CHV$ has a $\C[\partial]$-basis $\{L_i,H_i\,|\,i\in\Z\}$ with $λ$-brackets $[L_i\, {}_λ\, L_j]=(\partial+2λ) L_{i+j}, [L_i\, {}_λ\, H_j]=(\partial+λ) H_{i+j}, [H_i\, {}_λ\, L_j]=λL_{i+j}$ and $[H_i\, {}_λ\, H_j]=0$. In particul…
In this paper, we study the structure theory of a class of not-finitely graded Lie algebras related to generalized Heisenberg-Virasoro algebras. In particular, the derivation algebras, the automorphism groups and the second cohomology groups of these Lie algebras are determined.
Transposed Poisson structures on Virasoro-type algebras
本文研究形变及非有限阶化Heisenberg-Virasoro代数，通过计算其$\frac{1}{2}$-导子，完整分类了这些代数上的转置Poisson结构。
In this paper, we describe nonstandard quantum deformation of the super-Virasoro algebra. Using the Drinfel'd twist quantization technique, we obtain the deformed coproduct and antipode. Hence we get a family of noncommutative and noncocommutative Hopf superalgebras.
Irreducible representations of the twisted Heisenberg-Virasoro algebra of Hom type
本文针对Hom-李代数推广背景，研究Hom型扭Heisenberg-Virasoro代数的不可约表示，提出可由其诱导李代数的不可约表示诱导得到的方法，并构造了若干类该代数的不可约表示。
Two classes of Lie conformal superalgebras related to the Heisenberg--Virasoro Lie conformal algebra
本文构造了两类与Heisenberg-Virasoro李共形超代数相关的李共形超代数$\mathcal{HVS}(α)$和$\mathcal{HVS}(β,γ,τ)$，并完全确定了它们的共形导子、共形双导子、自同构群及秩$(1+1)$的共形模。
In this paper, we study extensions between two finite irreducible conformal modules over the Schrödinger-Virasoro conformal algebra and the extended Schrödinger-Virasoro conformal algebra. Also, we classify all finite nontrivial irreducible conformal modules over the extended Schrödinger-Virasoro conformal algebra. As a byproduct, we obtain a classification of extensions of Heisenberg-Virasoro conformal modules.
Given a scalar parameter $q$, the $q$-deformed Heisenberg algebra $\mathcal{H}(q)$ is the unital associative algebra with two generators $A,B$ that satisfy the $q$-deformed commutation relation $AB-qBA= I$, where $I$ is the multiplicative identity. For $\mathcal{H}(q)$ of torsion-type, that is if $q$ is a root of unity, characterization is obtained for all the Lie polynomials in $A,B$ and basis and graded structure and commutation relations for associated Lie algebras are studied.
Let $\mathbb{F}$ be a field, and fix a $q\in\mathbb{F}$. The $q$-deformed Heisenberg algebra $\mathcal{H}(q)$ is the unital associative algebra over $\mathbb{F}$ with generators $A$, $B$ and a relation which asserts that $AB - qBA$ is the multiplicative identity in $\mathcal{H}(q)$. We extend $\mathcal{H}(q)$ into an algebra $\mathcal{R}(q)$ defined by generators $A$, $B$ and a relation which asserts that $AB-qBA$ is central in $\mathcal{R}(q)$. We identify all elements of $\mathcal{R}(q)$ that are Lie polynomials in $A$, $B$.
In this paper, we classify all finite irreducible conformal modules over a class of Lie conformal algebras $\mathcal{W}(b)$ with $b\in\mathbb{C}$ related to the Virasoro conformal algebra. Explicitly, any finite irreducible conformal module over $\mathcal{W}(b)$ is proved to be isomorphic to $M_{Δ,α,β}$ with $Δ\neq 0$ or $β\neq 0$ if $b=0$, or $M_{Δ,α}$ with $Δ\neq 0$ if $b\neq0$. As a byproduct, all finite irreducible conformal modules over the Heisenberg-Virasoro conformal algebra and the Lie conformal algebra of $\mathcal{W}(2,2)$-type are classified. Finally, the same thing is done for the…
Let $\mathbb{F}$ be a field, and let $q\in\mathbb{F}$. The $q$-deformed Heisenberg algebra is the unital associative $\mathbb{F}$-algebra $\mathcal{H}(q)$ with generators $A,B$ and relation $AB-qBA=I$, where $I$ is the multiplicative identity in $\mathcal{H}(q)$. The set of all Lie polynomials in $A,B$ is the Lie subalgebra $\mathcal{L}(q)$ of $\mathcal{H}(q)$ generated by $A,B$. If $q\neq 1$ or the characteristic of $\mathbb{F}$ is not $2$, then the equation $AB-qBA=I$ cannot be expressed in terms of Lie algebra operations only, yet this equation still has consequences on the Lie algebra stru…
We prove that any irreducible Harish-Chandra modules for a class of Lie algebras, which we call gap-$p$ Virasoro algebras, must be a highest weight module, a lowest weight module, or a module of intermediate series.These algebras are closely related to the Heisenberg-Virasoro algebra and the algebra of derivations over a quantum torus. They also contain subalgebras which are isomorphic to the Virasoro algebra $Vir$, but graded by $p\mathbb Z$(unlike $Vir$ by $\mathbb Z$).
Algebraic curves for commuting elements in the q-deformed Heisenberg algebra
本文将Burchnall和Chaundy关于Heisenberg代数中可交换微分算子的消去式构造推广到q-变形Heisenberg代数，并证明了当q满足一个自然条件时，它又给出了可交换元的零化曲线.作为一个副结果，我们得到的估计的特征空间的元素，这个代数的忠实模的洛朗级数的尺寸。
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