arXiv subjects

Xiansheng Dai

Publications and source records attributed to Xiansheng Dai.

9 recordsLinked to original sources

arXiv2026

Classification of Simple Harish-Chandra Modules over the Loop Mirror HeisenbergVirasoro Algebra

The loop mirror Heisenberg-Virasoro algebra, an embedded subalgebra of the loop Heisenberg-Virasoro algebra, admits a family of interesting truncated subalgebras including those of Takiff type and $\mathfrak{bms}_3$ type. We give a complete classification of simple Harish-Chandra modules over the loop mirror Heisenberg-Virasoro algebra, whose simple modules fall into three categories, highest weight modules, lowest weight modules, and evaluation modules of the intermediate series. As a by-product, we classify all simple Harish-Chandra modules over the truncated mirror Heisenberg-Virasoro algebras $\mathcal{L}(n)$ for $n\geq2$. By virtue of shift operators in the $d$-parameter family, we give a more streamlined proof of Theorem 3.3 from the work [Classification of simple $W_n$-modules with finite-dimensional weight spaces, {\it J. Reine Angew. Math.}, {\bf 720} (2016), 199-216] by Y. Billig and V. Futorny, which states the key Billig-Futorny identity. Furthermore, our approach can be extended to the computation of annihilators for uniformly bounded modules over some other Lie (super)algebras.

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arXiv2024

Irreducible Modules for Super-Virasoro Algebras from Algebraic D-Modules

In this paper, we introduce a new family of functors from the category of modules for the Weyl algebra to the category of modules for the super-Virasoro algebras. The properties of these functors are investigated, with an emphasis on irreducibility preservation and natural isomorphisms. By utilizing these functors, we recover some old irreducible super-Virasoro modules, including those from the irreducible intermediate series as well as irreducible $U(\mathfrak{h})$-free modules. Additionally, we provide several families of new irreducible super-Virasoro modules via our constructed functors.

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arXiv2024

Irreducible modules over N=2 superconformal algebras from algebraic D-modules

In this paper, we introduce a family of functors denoted $\mathscr{F}_b$ that act on algebraic D-modules and generate modules over N=2 superconformal algebras. We prove these functors preserve irreducibility for all values of $b$, with a few clear exceptions described. We also establish necessary and sufficient conditions to determine when two such functors are naturally isomorphic. Applying $\mathscr{F}_b$ to N=1 super-Virasoro algebras recovers the functors previously introduced in \cite{CDLP}. Our new functors also facilitate the recovery of specific irreducible modules over N=2 superconformal algebras, including intermediate series and $U(\mathfrak{h})$-free modules. Additionally, our constructed functors produce several new irreducible modules for N=2 superconformal algebras.

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arXiv2023

Nonabelian embedding tensors on 3-Lie algebras and 3-Leibniz-Lie algebras

In this paper, first we introduce the notion of a nonabelian embedding tensor on the 3-Lie algebra. Then, we introduce the notion of a 3-Leibniz-Lie algebra, which is the underlying algebraic structure of a nonabelian embedding tensor on the 3-Lie algebra, and can also be viewed as a nonabelian generalization of a 3-Leibniz algebra. Next we develop the cohomology of nonabelian embedding tensors on 3-Lie algebras with coefficients in a suitable representation and use the first cohomology group to characterize infinitesimal deformations. Finally, we investigate nonabelian embedding tensors on 3-Lie algebras induced by Lie algebras.

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arXiv2023

The Heisenberg-Virasoro Lie conformal superalgebra

In this paper, we introduce a finite Lie conformal superalgebra called the Heisenberg-Virasoro Lie conformal superalgebra $\mathfrak{s}$ by using a class of Heisenberg-Virasoro Lie conformal modules. The super Heisenberg-Virasoro algebra of Ramond type $§$ is defined by the formal distribution Lie superalgebra of $\mathfrak{s}$. Then we construct a class of simple $§$-modules, which are induced from simple modules of some finite dimensional solvable Lie superalgebras. These modules are isomorphic to simple restricted $§$-modules, and include the highest weight modules, Whittaker modules and high order Whittaker modules. As a byproduct, we present a subalgebra of $§$, which is isomorphic to the super Heisenberg-Virasoro algebra of Neveu-Schwarz type.

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arXiv2023

A class of non-weight modules over the super-BMS$_3$ algebra

In the present paper, a class of non-weight modules over the super-BMS$_3$ algebras $§^ε$ ($ε=0$ or $\frac{1}{2}$) are constructed. Assume that $\mathfrak{t}=\C L_0\oplus\C W_0\oplus\C G_0$ and $\mathfrak{T}=\C L_0\oplus\C W_0$ are the Cartan subalgebra (modulo center) of $§^{0}$ and $§^{\frac{1}{2}}$, respectively. These modules over $§^{0}$ when restricted to the $\mathfrak{t}$ are free of rank $1$, while these modules over $§^{\frac{1}{2}}$ when restricted to the $\mathfrak{T}$ are free of rank $2$. Then we determine the necessary and sufficient conditions for these modules being simple, as well as determining the necessary and sufficient conditions for two $§^ε$-modules being isomorphic. %Moreover, we see that the category of free $U(\mathfrak{t})$-modules of rank $1$ over $§^0$ is %equivalent to the category of free $U(\mathfrak{T})$-modules of rank $2$ over %$§^{\frac{1}{2}}$.

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arXiv2021

A family of simple non-weight modules over the twisted $N=2$ superconformal algebra

We construct a class of non-weight modules over the twisted $N=2$ superconformal algebra $\T$. Let $\mathfrak{h}=\C L_0\oplus\C G_0$ be the Cartan subalgebra of $\T$, and let $\mathfrak{t}=\C L_0$ be the Cartan subalgebra of even part $\T_{\bar 0}$. These modules over $\T$ when restricted to the $\mathfrak{h}$ are free of rank $1$ or when restricted to the $\mathfrak{t}$ are free of rank $2$. We provide the sufficient and necessary conditions for those modules being simple, as well as giving the sufficient and necessary conditions for two $\T$-modules being isomorphic. We also compute the action of an automorphism on them. Moreover, based on the weighting functor introduced in \cite{N2}, a class of intermediate series modules $A_σ$ are obtained. As a byproduct, we give a sufficient condition for two $\T$-modules are not isomorphic.

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arXiv2016

Loop super-Virasoro Lie conformal superalgebra

The loop super-Virasoro conformal superalgebra $\mathfrak{cls}$ associated with the loop super-Virasoro algebra is constructed in the present paper. The conformal superderivation algebra of $\mathfrak{cls}$ is completely determined, which is shown to consist of inner superderivations. And nontrivial free and free $\mathbb{Z}$-graded $\mathfrak{cls}$-modules of rank two are classified. We also give a classification of irreducible free $\mathfrak{cls}$-modules of rank two and all irreducible submodules of each free $\mathbb{Z}$-graded $\mathfrak{cls}$-module of rank two.

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arXiv2016

Super-biderivations of Lie superalgebras

In this paper we attempt to investigate the super-biderivations of Lie superalgebras. Furthermore, we prove that all super-biderivations on the centerless super-Virasoro algebras are inner super-biderivations. Finally, we study the linear super commuting maps on the centerless super-Virasoro algebras.

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