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Rencai Lü

Publications and source records attributed to Rencai Lü.

12 recordsLinked to original sources
arXiv2026

Tensor modules over the Lie algebras of divergence zero vector fields on $\mathbb{C}^n$

Let $n\geq 2$ be an integer, $S_n$ be the Lie algebra of vector fields on $\mathbb{C}^n$ with zero divergence, and $D_n$ be the Weyl algebra over the polynomial algebra $A_n=\mathbb{C}[t_1,t_2,\cdots,t_n]$. In this paper, we study the simplicity of the tensor $S_n$-module $F(P,M)$, where $P$ is a simple $D_n$-module and $M$ is a simple $\mathfrak{sl}_n$-module. We obtain the necessary and sufficient conditions for $F(P,M)$ to be an irreducible module, and determine all simple subquotients of $F(P,M)$ when it is reducible.

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arXiv2026

Simple modules over the superconformal algebra $\mathcal{S}^{\prime}(1,n)$

Let $n\geq 2$, and let $\mathcal{S}(1,n)$ be the Lie superalgebra of zero-superdivergence superderivations of $\mathbb{C}[t^{\pm1}]\otimesΛ(n)$. Its derived algebra $\mathcal{S}^\prime(1,n):=[\mathcal{S}(1,n),\mathcal{S}(1,n)]$ is well known as a superconformal algebra. In this paper, we first study Shen-Larsson modules over $\mathcal{S}^\prime(1,n)$. These modules, introduced by G. Shen and T. A. Larsson, are constructed from modules over the Weyl superalgebra $K_{1,n}$ and the special linear Lie superalgebra $\mathfrak{sl}(1,n)$. We establish necessary and sufficient conditions for the simplicity of Shen-Larsson modules and investigate their simple subquotients in the non-simple case. Then as an application, building on the classification of simple cuspidal $\mathcal{S}^\prime(1,n)$-modules by C. Martínez, O. Mathieu and E. Zelmanov, we obtain an explicit construction of all simple cuspidal modules over $\mathcal{S}^\prime(1,n)$.

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arXiv2025

$(d,σ)$-twisted Affine-Virasoro superalgebras

For any finite dimensional Lie superalgebra $\dot{\mathfrak{g}}$ (maybe a Lie algebra) with an even derivation $d$ and a finite order automorphism $σ$ that commutes with $d$, we introduce the $(d,σ)$-twisted Affine-Virasoro superalgebra $\mathfrak{L}=\mathfrak{L}(\dot{\mathfrak{g}},d,σ)$ and determine its universal central extension $\hat{\mathfrak{L}}=\hat{\mathfrak{L}}(\dot{\mathfrak{g}},d,σ)$. This is a huge class of infinite-dimensional Lie superalgebras. Such Lie superalgebras consist of many new and well-known Lie algebras and superalgebras, including the Affine-Virasoro superalgebras, the twisted Heisenberg-Virasoro algebra, the mirror Heisenberg-Virasoro algebra, the W-algebra $W(2,2)$, the gap-$p$ Virasoro algebras, the Fermion-Virasoro algebra, the $N=1$ BMS superalgebra, the planar Galilean conformal algebra. Then we give the classification of cuspidal $A\mathfrak{L}$-modules by using the weighting functor from $U(\mathfrak{h})$-free modules to weight modules. Consequently, we give the classification of simple cuspidal $\mathfrak{L}$-modules by using the $A$-cover method. Finally, all simple quasi-finite modules over $\mathfrak{L}$ and $\hat{\mathfrak{L}}$ are classified. Our results recover many known Lie superalgebra results from mathematics and mathematical physics, and give many new Lie superalgebras.

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arXiv2025

Classification of simple quasifinite modules for contact superconformal algebras with $N\ne4$

In this paper, we classify all simple jet modules for contact superconformal algebras $\mathcal{K}(N;ε)$ with $N\neq4$. Then all simple quasifinite modules for $\widehat{\mathcal{K}}(N;ε)$ ($N\neq4$), the universal central extension of $\mathcal{K}(N;ε)$, are classified. Our results show that Matínez-Zelmanov's conjecture in \cite{MZe1} holds for $\mathcal{K}(N;ε)$ ($N\ne4$).

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arXiv2020

bounded weight modules over the Lie superalgebra of Cartan W-type

Let $A_{m,n}$ be the tensor product of the polynomial algebra in $m$ even variables and the exterior algebra in $n$ odd variables over the complex field $\C$, and the Witt superalgebra $W_{m,n}$ be the Lie superalgebra of superderivations of $A_{m,n}$. In this paper, we classify the non-trivial simple bounded weight $W_{m,n}$ modules with respect to the standard Cartan algebra of $W_{m,n}$. Any such module is a simple quotient of a tensor module $F(P,L(V_1\otimes V_2))$ for a simple weight module $P$ over the Weyl superalgebra $\mathcal K_{m,n}$, a finite-dimensional simple $\gl_m$-module $V_1$ and a simple bounded $\gl_n$-module $V_2$.

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arXiv2020

Weight modules for map (super)algebra related to the Virasoro algebra

We classify Jet modules for the Lie (super)algebras $\mathfrak{L}=W\ltimes(\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}])$, where $W$ is the Witt algebra and $\mathfrak{g}$ is a Lie superalgebra with an even diagonlizable derivation. Then we give a concept method to classify all simple cuspidal modules for $\mathfrak{L}$ and the map superalgebras, which are of the form $\mathfrak{L}\otimes R$, where $R$ is a Noetherian unital supercommutative associative superalgebra.

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arXiv2020

Simple weight modules with finite-dimensional weight spaces over Witt superalgebras

Let $A_{m,n}$ be the tensor product of the Laurient polynomial algebra in $m$ even variables and the exterior algebra in $n$ odd variables over the complex field $\bC$, and the Witt superalgebra $W_{m,n}$ be the Lie superalgebra of superderivations of $A_{m,n}$. In this paper, we classify the simple weight $W_{m,n}$ modules with finite-dimensional weight spaces with respect to the standard Cartan algebra of $W_{m,0}$. Every such module is either a simple quotient of a tensor module or a module of highest weight type.

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arXiv2014

Category O for the Schrödinger algebra

We study category O for the (centrally extended) Schrödinger algebra. We determine the quivers for all blocks and relations for blocks of nonzero central charge. We also describe the quiver and relations for the finite dimensional part of O. We use this to determine the center of the universal enveloping algebra and annihilators of Verma modules. Finally, we classify primitive ideals with nonzero central charge.

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