arXiv subjects

Yueqiang Zhao

Publications and source records attributed to Yueqiang Zhao.

6 recordsLinked to original sources

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 $2$.

math.RT↗

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 $V$.

math.RT↗

arXiv2020

Local and $2$-local automorphisms of simple generalized Witt algebras

In this paper, we prove that every invertible $2$-local or local automorphism of a simple generalized Witt algebra over any field of characteristic $0$ is an automorphism. In particular, every $2$-local or local automorphism of Witt algebras $W_n$ is an automorphism for all $n\in \mathbb{N}$. But some simple generalized Witt algebras indeed have $2$-local (and local) automorphisms that are not automorphisms.

math.RA↗

arXiv2020

2-local derivations on Witt algebras

In this paper, we prove that every 2-local derivation on Witt algebras $W_n, W_n^+$ or $W_n^{++} $ is a derivation for all $n=1,2,\cdots,\infty$. As a consequence we obtain that every 2-local derivation on any centerless generalized Virasoro algebra of higher rank is a derivation.

math.RA↗

arXiv2019

local derivations on Witt algebras

In this paper, we prove that every local derivation on Witt algebras $W_n, W_n^+$ or $W_n^{++} $ is a derivation for any $n\in\mathbb{N}$. As a consequence, we obtain that every local derivation on a centerless generalized Virasoro algebra of higher rank is a derivation.

math.RA↗

arXiv2016

Generalized Polynomial modules over the Virasoro algebra

Let $\mathcal{B}_r$ be the $(r+1)$-dimensional quotient Lie algebra of the positive part of the Virasoro algebra $\mathcal{V}$. Irreducible $\mathcal{B}_r$-modules were used to construct irreducible Whittaker modules in [MZ2] and irreducible weight modules with infinite dimensional weight spaces over $\mathcal{V}$ in [LLZ].In the present paper, we construct non-weight Virasoro modules $F(M, Ω(λ,β))$ from irreducible $\mathcal{B}_r$-modules $M$ and $(\mathcal{A},\mathcal{V})$-modules $Ω(λ,β)$. We give necessary and sufficient conditions for the Virasoro module $F(M, Ω(λ,β))$ to be irreducible. Using the weighting functor introduced by J. Nilsson, we also we also give the isomorphism criterion for two $F(M, Ω(λ,β))$.

math.RT↗