arXiv subjects

Kaiming Zhao

Publications and source records attributed to Kaiming Zhao.

At least 19 recordsLinked to original sources

arXiv2026

On $U(\mathfrak{h})$-free modules of finite rank over $\mathfrak{sl}(2)$

We study $\mathfrak{sl}(2)$-modules that are free of finite rank over $U(\mathfrak h)$, where $\mathfrak h$ is a fixed Cartan subalgebra of $\mathfrak{sl}(2)$. These modules form a natural class of non-weight modules. The coherent families obtained from this class via the weighting functor are identified. We also study a distinguished class of indecomposable $U(\mathfrak h)$-free modules defined in terms of Jordan blocks and give a recursive description of their socle filtrations. Finally, we apply the general results to exponential modules arising from the first Weyl algebra and obtain simplicity criteria for these modules.

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arXiv2026

A family of simple $U(\mathfrak{h})$-free modules of rank 2 over $\mathfrak{sl} (2)$

We study simple $\mathfrak{sl}(2)$-modules over $\mathbb C$ that are free of finite rank as $U(\mathfrak h)$-modules, where $\mathfrak h$ is a Cartan subalgebra of $\mathfrak{sl}(2)$. Our main result is an explicit classification of the scalar-type simple modules of rank $2$. We also give a criterion for when two such modules are isomorphic. Both the classification and the isomorphism problem reduce to twisted conjugacy classes in $\mbox{GL}_2({\mathbb C}[h])$ and rely on Cohn's standard form.

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arXiv2025

On the boundary Carrollian conformal algebra

We initiate the mathematical study of the boundary Carrollian conformal algebra (BCCA), an infinite-dimensional Lie algebra recently discovered in the context of Carrollian physics. The BCCA is an intriguing object from both physical and mathematical perspectives, since it is a filtered but not graded Lie algebra. In this paper, we first construct some modules for the BCCA and one of its subalgebras, which we call $\mathcal{O}$, by restriction of well-known modules of the BMS$_3$ and Witt algebras respectively. Along the way, we prove the irreducibility criteria for the so-called ``induced modules'' of the BMS$_3$ algebra (which we prefer to call massive modules to avoid ambiguity) and show that this is the same criteria for the irreducibility of the Verma modules of the BMS$_3$ algebra. Interestingly, the modules generated by the action of the BCCA on the generating vector of the massive modules are also irreducible under the same criteria. When this criteria holds, every massive module decomposes into a direct sum of two BCCA-submodules, each of which we conjecture to be indecomposable. Meanwhile, restricting Verma modules to the BCCA and $\mathcal{O}$ leads to free or ``almost free'' modules, which are not particularly interesting from a representation-theoretic viewpoint. This motivates the construction of BCCA modules intrinsically. To do this, we go through some structure theory on the BCCA to define a new basis and a decreasing filtration on the algebra, using which we construct Whittaker modules over the BCCA and the subalgebra $\mathcal{O}$ and prove criteria for their irreducibility.

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arXiv2025

Derivations and Biderivations of affine-Virasoro Lie algebras

In this paper we determine all derivations and biderivations of an affine-Virasoro Lie algebra associated with a finite-dimensional complex simple Lie algebra $\mathfrak{g}$. We prove that all the derivations and biderivations of affine-Virasoro Lie algebras are inner.

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arXiv2025

Quasi-Whittaker modules

In this paper, a general setting is proposed to define a class of modules over nonsemisimple Lie algebras $\mathfrak{g}$ induced by a nonperfect ideal $\mathfrak{p}$. This class of Lie algebras includes many well-known Lie algebras, and some of this class of modules are Whittaker modules and others are not. We call these modules quasi-Whittaker modules. By introducing a new concept: the Whittaker annihilator for universal quasi-Whittaker modules, we are able to determine the necessary and sufficient conditions for the irreducibility of the universal quasi-Whittaker modules. In the reducible case, we can obtain some maximal submodules. In particular, we classify the irreducible quasi-Whittaker modules for many Lie algebras, and obtain a lot of irreducible smooth $\mathcal{W}_n^+$-modules of height $2$.

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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

Simple smooth modules over the Lie algebras of polynomial vector fields

Let $\mathfrak{g}:={\rm Der}(\mathbb{C}[t_1, t_2,\cdots, t_n])$ and $\mathcal{L}:={\rm Der}(\mathbb{C}[[t_1, t_2,\cdots, t_n]])$ be the Witt Lie algebras. Clearly, $\mathfrak{g}$ is a proper subalegbra of $\mathcal{L}$. Surprisingly, we prove that simple smooth modules over $\mathfrak{g}$ are exactly the simple modules over $\mathcal{L}$ studied by Rodakov (no need to take completion). Then we find an easy and elementary way to classify all simple smooth modules over $\mathfrak{g}$. When the height $\ell_{V}\geq2$ or $n=1$, any nontrivial simple smooth $\mathfrak{g}$-module $V$ is isomorphic to an induced module from a simple smooth $\mathfrak{g}_{\geq0}$-module $V^{(\ell_{V})}$. When $\ell_{V}=1$ and $n\geq2$, any such module $V$ is the unique simple quotient of the tensor module $F(P_{0},M)$ for some simple $\gl_{n}$-module $M$, where $P_0$ is a particular simple module over the Weyl algebra $\mathcal{K}^+_n$. We further show that a simple $\mathfrak{g}$-module $V$ is a smooth module if and only if the action of each of $n$ particular vectors in $\mathfrak{g}$ is locally finite on $V$.

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arXiv2025

Simple smooth modules over the Ramond algebra and applications to vertex operator superalgebras

Simple smooth modules over the Virasoro algebra and one of the super-Virasoro algebras, named the Neveu-Schwarz algebra, have been classified. This problem remained unsolved for the other super-Virasoro algebra called the Ramond algebra.In this paper, all simple smooth modules over the Ramond algebra are classified. More precisely, we show that a simple smooth module over the Ramond algebra is either a simple highest weight module or isomorphic to an induced module from a simple module over a finite dimensional solvable Lie superalgebra.As an application we obtain all simple weak $ψ$-twisted modules over some vertex operator superalgebras.

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arXiv2024

Singular vectors, characters, and composition series for the N=1 BMS superalgebra

This paper investigates the structure of Verma modules over the N=1 BMS superalgebra. We provide a detailed classification of singular vectors, establish necessary and sufficient conditions for the existence of subsingular vectors, uncover the structure of maximal submodules, present the composition series of Verma modules, and derive character formulas for irreducible highest weight modules. As a byproduct, we also explicitly determine all singular vectors, subsingular vectors, and the composition series of Verma modules over the algebra W(2,2).

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arXiv2024

Smooth representations of affine Kac-Moody algebras

Smooth modules for affine Kac-Moody algebras have a prime importance for the quantum field theory as they correspond to the representations of the universal affine vertex algebras. But, very little is known about such modules beyond the category of positive energy representations. We construct a new class of smooth modules over affine Kac-Moody algebras. In a particular case, these modules are isomorphic to those induced from generalized Whittaker modules for Takiff Lie algebras. We establish the irreducibility criterion for constructed modules in the case of the affine sl(2) Lie algebra.

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arXiv2024

Biderivations of Lie algebras

In this paper, we first introduce the concept of symmetric biderivation radicals and characteristic subalgebras of Lie algebras, and study their properties. Based on these results, we precisely determine biderivations of some Lie algebras including finite-dimensional simple Lie algebras over arbitrary fields of characteristic not $2$ or $3$, and the Witt algebras $\mathcal{W}^+_n$ over fields of characteristic $0$. As an application, commutative post-Lie algebra structure on aforementioned Lie algebras is shown to be trivial.

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arXiv2023

Smooth modules over the N=1 Bondi-Metzner-Sachs superalgebra

In this paper, we present a determinant formula for the contravariant form on Verma modules over the N=1 Bondi-Metzner-Sachs (BMS) superalgebra. This formula establishes a necessary and sufficient condition for the irreducibility of the Verma modules. We then introduce and characterize a class of simple smooth modules that generalize both Verma and Whittaker modules over the N=1 BMS superalgebra. We also utilize the Heisenberg-Clifford vertex superalgebra to construct a free field realization for the N=1 BMS superalgebra. This free field realization allows us to obtain a family of natural smooth modules over the N=1 BMS superalgebra, which includes Fock modules and certain Whittaker modules.

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arXiv2023

Representations of the Fermion-Virasoro algebras

Let $δ=0$ or $\frac{1}{2}$. In this paper, we introduce the Fermion algebra $F(δ)$ and the Fermion-Virasoro algebra $\mathcal S(δ)$. They are infinite-dimensional Lie superalgebras. All simple smooth $F(δ)$-modules, all simple weight $F(δ)$-modules, all simple smooth $\mathcal S(δ)$-modules of nonzero level, and all simple Harish-Chandra $\mathcal S(δ)$-modules are determined. Surprisingly, we have found four different $\mathcal S(δ)$-module structures on free $\mathbb C[L_0]$-modules of rank $2$.

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arXiv2023

Simple smooth modules over the superconformal current algebra

In this paper, we classify simple smooth modules over the superconformal current algebra $\frak g$. More precisely, we first classify simple smooth modules over the Heisenberg-Clifford algebra, and then prove that any simple smooth $\frak g$-module is a tensor product of such modules for the super Virasoro algebra and the Heisenberg-Clifford algebra, or an induced module from a simple module over some finite-dimensional solvable Lie superalgebras. As a byproduct, we provide characterizations for both simple highest weight $\frak g$-modules and simple Whittaker $\frak g$-modules. Additionally, we present several examples of simple smooth $\frak g$-modules that are not tensor product of modules over the super Virasoro algebra and the Heisenberg-Clifford algebra.

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arXiv2023

Stretching maps for tensors

We consider an algebra of even-order square tensors and introduce a stretching map which allows us to represent tensors as matrices. The stretching map could be understood as a generalized matricization. It conserves algebraic properties of the tensors. In the same time, we don't necessarily assume injectivity of the stretching map. Dropping the injectivity condition allows us to construct examples of stretching maps with additional symmetry properties. Furthermore, the noninjectivity leads to the averaging of the tensor and possibly could be used to compress the data.

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arXiv2022

Actions of monoidal categories and representations of Cartan type Lie algebras

Using crossed homomorphisms, we show that the category of weak representations (resp. admissible representations) of Lie-Rinehart algebras (resp. Leibniz pairs) is a left module category over the monoidal category of representations of Lie algebras. In particular, the corresponding bifunctor of monoidal categories is established to give new weak representations (resp. admissible representations) of Lie-Rinehart algebras (resp. Leibniz pairs). This generalizes and unifies various existing constructions of representations of many Lie algebras by using this new bifunctor. We construct some crossed homomorphisms in different situations and use our actions of monoidal categories to recover some known constructions of representations of various Lie algebras, also to obtain new representations for generalized Witt algebras and their Lie subalgebras. The cohomology theory of crossed homomorphisms between Lie algebras is introduced and used to study linear deformations of crossed homomorphisms.

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arXiv2021

Non-weight modules over the mirror Heisenberg-Virasoro algebra

In this paper, we study irreducible non-weight modules over the mirror Heisenberg-Virasoro algebra $\mathcal{D}$, including Whittaker modules, $\mathcal{U}(\mathbb{C} d_0)$-free modules, and their tensor products. More precisely, we give the necessary and sufficient conditions for the Whittaker modules to be irreducible. We determine all $\mathcal{D}$-module structures on $\mathcal{U}(\mathbb{C} d_0)$, and find the necessary and sufficient conditions for these modules to be irreducible. At last we determine the necessary and sufficient conditions for the tensor products of Whittaker modules and $\mathcal{U}(\mathbb{C} d_0)$-free modules to be irreducible, and obtain that any two such tensor products are isomorphic if and only if the corresponding Whittaker modules and $\mathcal{U}(\mathbb{C} d_0)$-free modules are isomorphic. These lead to many new irreducible non-weight modules over $\mathcal{D}$.

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arXiv2021

Simple restricted modules over the Heisenberg-Virasoro algebra as VOA modules

In this paper, we determine all simple restricted modules over the mirror Heisenberg-Virasoro algebra ${\mathfrak{D}}$, and the twisted Heisenberg-Virasoro algebra $\bar\mathfrak{D}$ with nonzero level. As applications, we characterize simple Whittaker modules and simple highest weight modules over ${\mathfrak{D}}$. A vertex-algebraic interpretation of our result is the classification of simple weak twisted and untwisted modules over the Heisenberg-Virasoro vertex operator algebras $\mathcal V^{c} \cong V_{Vir}^{c}\otimes M(1)$. We also present a few examples of simple restricted ${\mathfrak{D}}$-modules and $\bar\mathfrak{D}$-modules induced from simple modules over finite dimensional solvable Lie algebras, that are not tensor product modules of Virasoro modules and Heisenberg modules. This is very different from the case of simple highest weight modules over $\mathfrak{D}$ and $\bar\mathfrak{D}$ which are always tensor products of simple Virasoro modules and simple Heisenberg modules.

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