arXiv subjects

Yufang Zhao

Publications and source records attributed to Yufang Zhao.

6 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.

math.RT↗

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.

math.RT↗

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 $W_n$.

math.RT↗

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 $Ω_{\mathbf{a}}^W$.

math.RT↗

arXiv2021

2-local derivations on the twisted Heisenberg-Virasoro algebra

2-local derivation is a generalized derivation for a Lie algebra, which plays an important role to the study of local properties of the structure of the Lie algebra. In this paper, we prove that every 2-local derivation on the twisted Heisenberg-Virasoro algebra is a derivation.

math.RA↗

arXiv2021

2-local derivation on the conformal Galilei algebra

math.RA↗