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Yucai Su

Publications and source records attributed to Yucai Su.

At least 19 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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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.

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arXiv2025

Representations of the N=1 Heisenberg-Virasoro superalgebra

We first define a class of non-weight modules over the N=1 Heisenberg-Virasoro superalgebra $\mathfrak{g}$, which are reducible modules. Then we give all submodules of such modules, and present the corresponding irreducible quotient modules which were exactly studied in \cite{DL}. Also, we prove that those modules constitute a complete classification of $U(\mathfrak{h})$-free modules of rank $2$ over $\mathfrak{g}$, where $\mathfrak{h}=\C L_{0}\oplus\C H_{0}$ is the degree-0 part of $\mathfrak{g}$. As an application, we obtain a class of weight $\mathfrak{g}$-modules from those non-weight $\mathfrak{g}$-modules by weighting functor. Furthermore, we study the non-weight modules over the four subalgebras of $\mathfrak{g}$: (i) the Heisenberg-Virasoro algebra; (ii) the Neveu-Schwarz algebra; (iii) the Fermion-Virasoro algebra; (iv) the Heisenberg-Clifford superalgebra. As far as we know, those non-weight Heisenberg-Virasoro modules were constructed in \cite{HCS}, but the structure of submodules was not clear. In this paper, we determine all submodules of them, and we show the corresponding irreducible quotient modules which were exactly defined in \cite{CG}.

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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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arXiv2023

Restricted representations of the twisted $N=2$ superconformal algebra

In this paper, we construct a large class of new simple modules over the twisted $N=2$ superconformal algebra. These new simple modules are restricted modules based on the simple modules over certain finite-dimensional solvable Lie superalgebras, including various versions of Whittaker modules. We elaborate that they are also the twisted modules for the universal $N=2$ superconformal vertex algebra. On the other hand, we give an explicit characterization of the simple restricted modules over the twisted $N=2$ superconformal algebra $\mathcal{T}$ under the condition that $T_t$ in $\mathcal{T}$ acts injectively for some $t\in \frac{1}{2}+\Z_+$.

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arXiv2022

A class of graded conformal algebras which is induced by Heisenberg-Virasoro conformal algebra

In this paper, we obtain a class of $\mathbb{Z}$-graded conformal algebras which is induced by Heisenberg-Virasoro conformal algebra. More precisely, we classify $\mathbb{Z}$-graded conformal algebras $\mathcal{A} = \oplus^\infty_{i=-1}\mathcal{A}_i$ satisfying the following conditions, (C1) $\mathcal{A}_0$ is the Heisenberg-Virasoro conformal algebra; C2) Each $\mathcal{A}_i$ for $i\in\mathbb{Z}_{\ge-1}^*$ is an $\mathcal{A}_0$-module of rank one; (C3) $[{X_{-1}}_λX_i]\neq 0$ for $i\ge 0$, where $X_i$ is any one of $\mathbb{C}[\partial]$-generators of $\mathcal{A}_i$ for $i\in \mathbb{Z}_{\ge -1}$. Further, we prove that all finite nontrivial irreducible modules of these algebras under some special conditions are free of rank one as a $\mathbb{C}[\partial]$-module. The conformal derivations of this class of graded Lie conformal algebras are also determined.

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arXiv2022

New irreducible non-weight Virasoro modules from tensor products

In this paper, we present a class of non-weight Virasoro modules $\mathcal{M}\big(V,Ω(λ_0,α_0)\big)\otimes\bigotimes_{i=1}^mΩ(λ_i,α_i)$ where $Ω(λ_i,α_i)$ and $\mathcal{M}\big(V,Ω(λ_0,α_0)\big)$ are irreducible Virasoro modules defined in \cite{LZ2} and \cite{LZ} respectively. The necessary and sufficient conditions for $\mathcal{M}\big(V,Ω(λ_0,α_0)\big)\otimes\bigotimes_{i=1}^mΩ(λ_i,α_i)$ to be irreducible are obtained. Then we determine the necessary and sufficient conditions for two such irreducible Virasoro modules to be isomorphic. At last, we show that the irreducible modules in this class are new.

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arXiv2021

Finite irreducible conformal modules over the Lie conformal superalgebra $\mathcal{S}(p)$

In the present paper, we introduce a class of infinite Lie conformal superalgebras $\mathcal{S}(p)$, which are closely related to Lie conformal algebras of extended Block type defined in \cite{CHS}. Then all finite non-trivial irreducible conformal modules over $\mathcal{S}(p)$ for $p\in\C^*$ are completely classified. As an application, we also present the classifications of finite non-trivial irreducible conformal modules over finite quotient algebras $\mathfrak{s}(n)$ for $n\geq1$ and $\mathfrak{sh}$ which is isomorphic to a subalgebra of Lie conformal algebra of $N=2$ superconformal algebra. Moreover, as a generalized version of $\mathcal{S}(p)$, the infinite Lie conformal superalgebras $\mathcal{GS}(p)$ are constructed, which have a subalgebra isomorphic to the finite Lie conformal algebra of $N=2$ superconformal algebra.

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arXiv2021

Simple non-weight modules over Lie superalgebras of Block type

In this paper, a family of non-weight modules over Lie superalgebras $S(q)$ of Block type are studied. Free $U(η)$-modules of rank $1$ over Ramond-Block algebras and free $U(\mathfrak{h})$-modules of rank $2$ over Neveu-Schwarz-Block algebras are constructed and classified. Moreover, the sufficient and necessary conditions for such modules to be simple are presented, and their isomorphism classes are also determined. The results cover some existing results.

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arXiv2021

Another class of simple graded Lie conformal algebras that cannot be embedded into general Lie conformal algebras

In a previous paper by the authors, we obtain the first example of a finitely freely generated simple $\mathbb Z$-graded Lie conformal algebra of linear growth that cannot be embedded into any general Lie conformal algebra. In this paper, we obtain, as a byproduct, another class of such Lie conformal algebras by classifying $\mathbb Z$-graded simple Lie conformal algebras ${\cal G}=\oplus_{i=-1}^\infty{\cal G}_i$ satisfying the following, (1) ${\cal G}_0\cong{\rm Vir}$, the Virasoro conformal algebra; (2) Each ${\cal G}_i$ for $i\ge-1$ is a ${\rm Vir}$-module of rank one. These algebras include some Lie conformal algebras of Block type.

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arXiv2020

Finite irreducible conformal modules over the extended Block type Lie conformal algebra $\mathfrak{B}(α,β,p)$

In this paper, we introduce a class of infinite Lie conformal algebras $\mathfrak{B}(α,β,p)$, which are the semi-direct sums of Block type Lie conformal algebra $\mathfrak{B}(p)$ and its non-trivial conformal modules of $\Z$-graded free intermediate series. The annihilation algebras are a class of infinite-dimensional Lie algebras, which include a lot of interesting subalgebras: Virasoro algebra, Block type Lie algebra, twisted Heisenberg-Virasoro algebra and so on. We give a complete classification of all finite non-trivial irreducible conformal modules of $\mathfrak{B}(α,β,p)$ for $α,β\in\C, p\in\C^*$. As an application, the classifications of finite irreducible conformal modules over a series of finite Lie conformal algebras $\mathfrak{b}(n)$ for $n\geq1$ are given.

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arXiv2020

Submodule structures of $\mathbb C[s,t]$ over $W(0,b)$ and a new class of irreducible modules over the Virasoro algebra

For any $a,b\in\mathbb C$, $W(a,b)$ is the Lie algebra with basis $\{L_m,M_m\,|\,m\in\mathbb Z\}$ and relations $[L_m,L_n]=(n-m)L_{m+n},$ $[L_m,W_n]=(a+n+bm)W_{m+n}$, $[W_m,W_n]=0$ for $m,n\in\mathbb Z$. For any $λ\in\mathbb C^*,$ $α\in\mathbb C$, $h:=h(t)\in\mathbb C[t]$, there exists a non-weight module over $W(0,b)$ (resp., $W(0,1)$), denoted by $Φ(λ,α,h)$ (resp. $Θ(λ,h)$), which is defined on the space $\mathbb C[s,t]$ of polynomials on variables $s,t$ and is free of rank one over the enveloping algebra $U(\mathbb C L_0\oplus\mathbb C W_0)$ of $\mathbb C L_0\oplus\mathbb C W_0$. In the present paper, by introducing two sequences of useful operators on $\mathbb C[s,t]$, we determine all submodules of $\mathbb C[s,t]$. We also study submodules of $\mathbb C[s,t]$ regarded as modules over the Virasoro algebra $\mathscr V\!$ (with the trivial action of the center), and prove that these submodules are finitely generated if and only if ${\rm deg}\,h(t)\geq1$. In addition, it is proven that $Φ(λ, α,h)$ is an irreducible $\mathscr V\!$-module if and only if $b=-1$, ${\rm deg}\, h(t)=1$, $α\neq0$. Finally, we obtain a large family of new irreducible modules over the Virasoro algebra $\mathscr V\!$, by taking various tensor products of a finite number of irreducible modules $Φ(λ_i,α_i, h_i)$ for $λ_i,α_i\in\mathbb C^*,$ $h_i\in\mathbb C[t]$ with an irreducible $\mathscr V\!$-module $V$, where $V$ satisfies that there exists a nonnegative integer $R_V$ such that $L_m$ acts locally finitely on $V$ for $m\geq R_V$.

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arXiv2019

Irreducible twisted Heisenberg-Virasoro modules from tensor products

In this paper, we realize polynomial $\H$-modules $Ω(λ,α,β)$ from irreducible twisted Heisenberg-Virasoro modules $\A_{α,β}$. It follows from $\H$-modules $Ω(λ,α,β)$ and $\mathrm{Ind}(M)$ that we obtain a class of natural non-weight tensor product modules $\big(\bigotimes_{i=1}^mΩ(λ_i,α_i,β_i)\big)\otimes \mathrm{Ind}(M)$. Then we give the necessary and sufficient conditions under which these modules are irreducible and isomorphic, and also give that the irreducible modules in this class are new.

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arXiv2019

Mixed cohomology of Lie superalgebras

We investigate a new cohomology of Lie superalgebras, which may be compared to a de Rham cohomology of Lie supergroups involving both differential and integral forms. It is defined by a BRST complex of Lie superalgebra modules, which is formulated in terms of a Weyl superalgebra and incorporates inequivalent representations of the bosonic Weyl subalgebra. The new cohomology includes the standard Lie superalgebra cohomology as a special case. Examples of new cohomology groups are computed.

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arXiv2018

A proof of Comes-Kujawa's conjecture

Let $κ$ be a commutative ring containing $2^{-1}$. In this paper, we prove the Comes-Kujawa's conjecture on a $κ$-basis of cyclotomic oriented Brauer-Clifford supercategory. As a by-product, we prove that the cyclotomic walled Brauer-Clifford superalgebra defined by Comes and Kujawa and ours are isomorphic if $κ$ is an algebraically closed field with characteristic not two.

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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 $n\ge1$.

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