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Genqiang Liu

Publications and source records attributed to Genqiang Liu.

At least 19 recordsLinked to original sources
arXiv2026

Classification of Simple Cuspidal Modules over Nongraded Witt Lie Algebras

For a positive integer $n$, let $A_n=\mathbb{C}[t_1^{\pm1},\ldots,t_n^{\pm1},x_1,\ldots,x_n]$ and $\mathfrak{g}_n=\bigoplus_{i=1}^n A_nd_i$, where $d_i=t_i\frac{\partial}{\partial t_i} +\frac{\partial}{\partial x_i}$. We first determine when the tensor module $T(P,V)=P\otimes V$ is simple, where $P$ is a simple module over the Weyl type algebra $D_n$ and $V$ is a simple $\mathfrak{gl}_n$-module. We then prove a canonical algebra isomorphism $A_n\#U(\mathfrak{g}_n)\cong D_n\otimes U(\mathfrak{m}_{\mathbf{1},\mathbf{0}}\Delta)$, and use it to show that every simple cuspidal $\mathfrak{g}_n$-module is isomorphic to a simple quotient of some $T(A_n(\lambda),V)$, where $V$ is a finite-dimensional simple $\mathfrak{gl}_n$-module.

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arXiv2026

The category of Whittaker modules over the Cartan Type Lie algebra $\bar{S}_2$

The Lie algebra $\bar{S}_2$ of polynomial vector fields on $\mathbb{C}^2$ with constant divergence is an important Cartan type Lie algebra. In this paper, we study Whittaker $\bar{S}_2$-modules that are locally finite over $\text{span}\{\frac{\partial}{\partial t_1}, \frac{\partial}{\partial t_2}\}$. We first show that each block $\Omega^{\widetilde{S}_2}_{\mathbf{a}}$ of the category of $(A_2, \bar{S}_2)$-Whittaker modules with finite-dimensional Whittaker vector spaces is equivalent to the category of finite-dimensional modules over the parabolic subalgebra $\bar{S}_2^{\geq 0}$. Then we classify all simple Whittaker $\bar{S}_2$-modules in every block $\Omega^{\bar{S}_2}_{\mathbf{a}}$ . Finally, we establish an equivalence between $\Omega^{\bar{S}_2}_{\mathbf{1}}$ and the category $H_{\mathbf{1}}$-fmod of finite-dimensional modules over an associative algebra $H_{\mathbf{1}}$, whose generators are also determined.

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arXiv2025

Irreducible cuspidal $\mathfrak{sl}_{n+1}$-modules from finite-dimensional modules over the minimal nilpotent finite $W$-algebra

A weight $\mathfrak{sl}_{n+1}$-module with finite-dimensional weight spaces is called a cuspidal module, if every root vector of $\mathfrak{sl}_{n+1}$ acts injectively on it. In \cite{LL}, it has been shown that any block with a generalized central character of the cuspidal $\mathfrak{sl}_{n+1}$-module category is equivalent to a block of the category of finite-dimensional modules over the minimal nilpotent finite $W$-algebra $W(e)$ for $\mathfrak{sl}_{n+1}$. In this paper, using a centralizer realization of $W(e)$ and an explicit embedding $W(e)\rightarrow U(\mathfrak{gl}_n)$, we show that every finite-dimensional irreducible $W(e)$-module is isomorphic to an irreducible $W(e)$-quotient module of some finite-dimensional irreducible $\mathfrak{gl}_n$-module. As an application, we can give very explicit realizations of all irreducible cuspidal $\mathfrak{sl}_{n+1}$-modules using finite-dimensional irreducible $\mathfrak{gl}_n$-modules, avoiding using the twisted localization method and the coherent family introduced in [M].

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arXiv2024

Minimal nilpotent finite $W$-algebra and cuspidal module category of $\mathfrak{sp}_{2n}$

Let $U_S$ be the localization of $U(\mathfrak{sp}_{2n})$ with respect to the Ore subset $S$ generated by the root vectors $X_{\epsilon_1-\epsilon_2},\dots,X_{\epsilon_1-\epsilon_n}, X_{2\epsilon_1}$. We show that the minimal nilpotent finite $W$-algebra $W(\mathfrak{sp}_{2n}, e)$ is isomorphic to the centralizer $C_{U_S}(B)$ of some subalgebra $B$ in $U_S$, and it can be identified with a tensor product factor of $U_S$. As an application, we show that the category of weight $\mathfrak{sp}_{2n}$-modules with injective actions of all root vectors and finite-dimensional weight spaces is equivalent to the category of finite-dimensional modules over $W(\mathfrak{sp}_{2n}, e)$, explaining the coincidence that both of them are semi-simple.

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arXiv2023

A category equivalence on the Lie algebra of polynomial vector fields

For any positive integer $n$, let $W_n=\text{Der}(\mathbb{C}[t_1,\dots,t_n])$. The subspaces $\mathfrak{h}_n=\text{Span}\{t_1\frac{\partial}{\partial{t_1}},\dots,t_n\frac{\partial}{\partial{t_n}}\}$ and $Δ_n=\text{Span}\{\frac{\partial}{\partial{t_1}},\dots,\frac{\partial}{\partial{t_n}}\}$ are two abelian subalgebras of $W_n$. We show that a full subcategory $Ω_{\mathbf{1}}$ of the category of $W_n$-modules $M$ which are locally finite over $Δ_n$ is equivalent to some full subcategory of weight $W_n$-modules $M$ which are cuspidal modules when restricted to the subalgebra $\mathfrak{sl}_{n+1}$ of $W_n$.

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arXiv2023

Whittaker categories and the minimal nilpotent finite $W$-algebra for $\mathfrak{sl}_{n+1}$

For any $\mathbf{a}=(a_1,\dots,a_n)\in \mathbb{C}^n$, we introduce a Whittaker category $\mathcal{H}_{\mathbf{a}}$ whose objects are $\mathfrak{sl}_{n+1}$-modules $M$ such that $e_{0i}-a_i$ acts locally nilpotently on $M$ for all $i \in \{1,\dots,n\}$, and the subspace $\mathrm{wh}_{\mathbf{a}}(M)=\{v\in M \mid e_{0i} v=a_iv, \ i=1,\dots,n\}$ is finite dimensional. In this paper, we first give a tensor product decomposition $U_S=W\otimes B$ of the localization $U_S$ of $U(\mathfrak{sl}_{n+1})$ with respect to the Ore subset $S$ generated by $e_{01},\dots, e_{0n}$. We show that the associative algebra $W$ is isomorphic to the type $A_n$ finite $W$-algebra $W(e)$ defined by a minimal nilpotent element $e$ in $\mathfrak{sl}_{n+1}$. Then using $W$-modules as a bridge, we show that each block with a generalized central character of $\mathcal{H}_{\mathbf{1}}$ is equivalent to the corresponding block of the cuspidal category $\mathcal{C}$, which is completely characterized by Grantcharov and Serganova. As a consequence, each regular integral block of $\mathcal{H}_{\mathbf{1}}$ and the category of finite dimensional modules over $W(e) can be described by a well-studied quiver with certain quadratic relations.

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arXiv2023

Category $\mathcal{O}$ for the Lie algebra of vector fields on the line

Let $\mathfrak{W}$ be the Lie algebra of vector fields on the line. Via computing extensions between all simple modules in the category $\mathcal{O}$, we give the block decomposition of $\mathcal{O}$, and show that the representation type of each block of $\mathcal{O}$ is wild using the Ext-quiver. Each block of $\mathcal{O}$ has infinite simple objects. This result is very different from that of $\mathcal{O}$ for complex semisimple Lie algebras. To find a connection between $\mathcal{O}$ and the module category over some associative algebra, we define a subalgebra $H_1$ of $U(\mathfrak{b})$. We give an exact functor from $\mathcal{O}$ to the category $Ω$ of finite dimensional modules over $H_1$. We also construct new simple $\mathfrak{W}$-modules from Weyl modules and modules over the Borel subalgebra $\mathfrak{b}$ of $\mathfrak{W}$.

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arXiv2022

A Whittaker category for the Symplectic Lie algebra

For any $n\in \mathbb{Z}_{\geq 2}$, let $\mathfrak{m}_n$ be the subalgebra of $\mathfrak{sp}_{2n}$ spanned by all long negative root vectors $X_{-2ε_i}$, $i=1,\dots,n$. An $\mathfrak{sp}_{2n}$-module $M$ is called a Whittaker module with respect to the Whittaker pair $(\mathfrak{sp}_{2n},\mathfrak{m}_n)$ if the action of $\mathfrak{m}_n$ on $M$ is locally finite, according to a definition of Batra and Mazorchuk. This kind of modules are more general than the classical Whittaker modules defined by Kostant. In this paper, we show that each non-singular block $\mathcal{WH}_{\mathbf{a}}^μ$ with finite dimensional Whittaker vector subspaces is equivalent to a module category $\mathcal{W}^{\mathbf{a}}$ of the even Weyl algebra $\mathcal{D}_n^{ev}$ which is semi-simple. As a corollary, any simple module in the block $\mathcal{WH}_{\mathbf{i}}^{-\frac{1}{2}ω_n}$ for the fundamental weight $ω_n$ is equivalent to the Nilsson's module $N_{\mathbf{i}}$ up to an automorphism of $\mathfrak{sp}_{2n}$. We also characterize all possible algebra homomorphisms from $U(\mathfrak{sp}_{2n})$ to the Weyl algebra $\mathcal{D}_n$ under a natural condition.

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arXiv2021

Whittaker category for the Lie algebra of polynomial vector fields

For any positive integer $n$, let $A_n=\mathbb{C}[t_1,\dots,t_n]$, $W_n=\text{Der}(A_n)$ and $Δ_n=\text{Span}\{\frac{\partial}{\partial{t_1}},\dots,\frac{\partial}{\partial{t_n}}\}$. Then $(W_n, Δ_n)$ is a Whittaker pair. A $W_n$-module $M$ on which $Δ_n$ operates locally finite is called a Whittaker module. We show that each block $Ω_{\mathbf{a}}^{\widetilde{W}}$ of the category of $(A_n,W_n)$-Whittaker modules with finite dimensional Whittaker vector spaces is equivalent to the category of finite dimensional modules over $L_n$, where $L_n$ is the Lie subalgebra of $W_n$ consisting of vector fields vanishing at the origin. As a corollary, we classify all simple non-singular Whittaker $W_n$-modules with finite dimensional Whittaker vector spaces using $\mathfrak{gl}_n$-modules. We also obtain an analogue of Skryabin's equivalence for the non-singular block $Ω_{\mathbf{a}}^W$.

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arXiv2021

An algebra isomorphism on $U(\mathfrak{gl}_n)$

For each positive integer $n$, let $\mathfrak{s}_n=\mathfrak{gl}_n\ltimes \mathbb{C}^n$. We show that $U(\mathfrak{s}_{n})_{X_{n}}\cong \mathcal{D}_{n}\otimes U(\mathfrak{s}_{n-1})$ for any $n\in\mathbb{Z}_{\geq 2}$, where $U(\mathfrak{s}_{n})_{X_{n}}$ is the localization of $U(\mathfrak{s}_{n})$ with respect to the subset $X_n:=\{e_1^{i_1}\cdots e_{n}^{i_{n}}\mid i_1,\dots,i_{n}\in \mathbb{Z}_+\}$, and $\mathcal{D}_{n}$ is the Weyl algebra $\mathbb{C}[x_1^{\pm 1}, \cdots, x_{n}^{\pm 1}, \frac{\partial}{\partial x_1},\cdots, \frac{\partial}{\partial x_{n}}]$. As an application, we give a new proof of the Gelfand-Kirillov conjecture for $\mathfrak{s}_n$ and $\mathfrak{gl}_n$. Moreover we show that the category of Harish-Chandra $U(\mathfrak{s}_{n})_{X_n}$-modules with a fixed weight support is equivalent to the category of finite dimensional $\mathfrak{s}_{n-1}$-modules whose representation type is wild, for any $n\in \mathbb{Z}_{\geq 2}$.

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arXiv2020

Irreducible Jet modules for the vector field Lie algebra on $\mathbb{S}^1\times \mathbb{C}$

For a commutative algebra $A$ over $\mathbb{C}$,denote $\mathfrak{g}=\text{Der}(A)$. A module over the smash product $A\# U(\mathfrak{g})$ is called a jet $\mathfrak{g}$-module, where $U(\mathfrak{g})$ is the universal enveloping algebra of $\mathfrak{g}$.In the present paper, we study jet modules in the case of $A=\mathbb{C}[t_1^{\pm 1},t_2]$.We show that $A\#U(\mathfrak{g})\cong\mathcal{D}\otimes U(L)$, where $\mathcal{D}$ is the Weyl algebra $\mathbb{C}[t_1^{\pm 1},t_2, \frac{\partial}{\partial t_1},\frac{\partial}{\partial t_2}]$, and $L$ is a Lie subalgebra of $A\# U(\mathfrak{g})$ called the jet Lie algebra corresponding to $\mathfrak{g}$.Using a Lie algebra isomorphism $θ:L \rightarrow \mathfrak{m}_{1,0}Δ$, where $\mathfrak{m}_{1,0}Δ$ is the subalgebra of vector fields vanishing at the point $(1,0)$, we show that any irreducible finite dimensional $L$-module is isomorphic to an irreducible $\mathfrak{gl}_2$-module. As an application, we give tensor product realizations of irreducible jet modules over $\mathfrak{g}$ with uniformly bounded weight spaces.

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arXiv2020

Irreducible weight modules over the Schr{ö}dinger Lie algebra in $(n+1)$ dimensional space-time

In this paper, we study weight representations over the Schr{ö}dinger Lie algebra $\mathfrak{s}_n$ for any positive integer $n$. It turns out that the algebra $\mathfrak{s}_n$ can be realized by polynomial differential operators. Using this realization, we give a complete classification of irreducible weight $\mathfrak{s}_n$-modules with finite dimensional weight spaces for any $n$. All such modules can be clearly characterized by the tensor product of $\mathfrak{so}_n$-modules, $\mathfrak{sl}_2$-modules and modules over the Weyl algebra.

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arXiv2019

The category of weight modules for symplectic oscillator Lie algebras

The rank $n$ symplectic oscillator Lie algebra $\mathfrak{g}_n$ is the semidirect product of the symplectic Lie algebra $\mathfrak{sp}_{2n}$ and the Heisenberg Lie algebra $H_n$. In this paper, we study weight modules with finite dimensional weight spaces over $\mathfrak{g}_n$. When $\dot z\neq 0$, it is shown that there is an equivalence between the full subcategory $\mathcal{O}_{\mathfrak{g}_n}[\dot z]$ of the BGG category $\mathcal{O}_{\mathfrak{g}_n}$ for $\mathfrak{g}_n$ and the BGG category $\mathcal{O}_{\mathfrak{sp}_{2n}}$ for $\mathfrak{sp}_{2n}$. Then using the technique of localization and the structure of generalized highest weight modules, we also give the classification of simple weight modules over $\mathfrak{g}_n$ with finite-dimensional weight spaces.

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arXiv2019

BGG category for the quantum Schrödinger algebra

In this paper, we study the BGG category $\mathcal{O}$ for the quantum Schr{ö}dinger algebra $U_q(\mathfrak{s})$, where $q$ is a nonzero complex number which is not a root of unity. If the central charge $\dot z\neq 0$, using the module $B_{\dot z}$ over the quantum Weyl algebra $H_q$, we show that there is an equivalence between the full subcategory $\mathcal{O}[\dot z]$ consisting of modules with the central charge $\dot z$ and the BGG category $\mathcal{O}^{(\mathfrak{sl}_2)}$ for the quantum group $U_q(\mathfrak{sl}_2)$. In the case that $\dot z=0$, we study the subcategory $\mathcal{A}$ consisting of finite dimensional $U_q(\mathfrak{s})$-modules of type $1$ with zero action of $Z$. Motivated by the ideas in \cite{DLMZ, Mak}, we directly construct an equivalent functor from $\mathcal{A}$ to the category of finite dimensional representations of an infinite quiver with some quadratic relations. As a corollary, we show that the category of finite dimensional $U_q(\mathfrak{s})$-modules is wild.

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arXiv2017

Localization of highest weight modules of a class of Extended Affine Lie Algebras

In 2006, Gao and Zeng \cite{GZ} gave the free field realizations of highest weight modules over a class of extended affine Lie algebras. In the present paper, applying the technique of localization to those free field realizations, we construct a class of new weight modules over the extended affine Lie algebras. We give necessary and sufficient conditions for these modules to be irreducible. In this way, we construct free field realizations for a class of simple weight modules with infinite weight multiplicities over the extended affine Lie algebras.

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arXiv2017

Simple Witt modules that are finitely generated over the cartan subalgebra

Let $d\ge1$ be an integer, $W_d$ and $\mathcal{K}_d$ be the Witt algebra and the weyl algebra over the Laurent polynomial algebra $A_d=\mathbb{C} [x_1^{\pm1}, x_2^{\pm1}, ..., x_d^{\pm1}]$, respectively. For any $\mathfrak{gl}_d$-module $M$ and any admissible module $P$ over the extended Witt algebra $\widetilde W_d$, we define a $W_d$-module structure on the tensor product $P\otimes M$. We prove in this paper that any simple $W_d$-module that is finitely generated over the cartan subalgebra is a quotient module of the $W_d$-module $P \otimes M$ for a finite dimensional simple $\mathfrak{gl}_d$-module $M$ and a simple $\mathcal{K}_d$-module $P$ that are finitely generated over the cartan subalgebra. We also characterize all simple $\mathcal{K}_d$-modules and all simple admissible $\widetilde W_d$-modules that are finitely generated over the cartan subalgebra.

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arXiv2017

Simple weight modules over the quantum Schrödinger algebra

In the present paper, using the technique of localization, we determine the center of the quantum Schrödinger algebra $§_q$ and classify simple modules with finite-dimensional weight spaces over $§_q$, when $q$ is not a root of unity. It turns out that there are four classes of such modules: dense $U_q(\mathfrak{sl}_2)$-modules, highest weight modules, lowest weight modules, and twisted modules of highest weight modules.

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