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Genqiang Liu

Publications and source records attributed to Genqiang Liu.

27 records · Page 2Linked to original sources
arXiv2017

Irreducible Witt modules from Weyl modules and $\mathfrak{gl}_{n}$-modules

For an irreducible module $P$ over the Weyl algebra $\mathcal{K}_n^+$ (resp. $\mathcal{K}_n$) and an irreducible module $M$ over the general liner Lie algebra $\mathfrak{gl}_n$, using Shen's monomorphism, we make $P\otimes M$ into a module over the Witt algebra $W_n^+$ (resp. over $W_n$). We obtain the necessary and sufficient conditions for $P\otimes M$ to be an irreducible module over $W_n^+$ (resp. $W_n$), and determine all submodules of $P\otimes M$ when it is reducible. Thus we have constructed a large family of irreducible weight modules with many different weight supports and many irreducible non-weight modules over $W_n^+$ and $W_n$.

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arXiv2016

New families of irreducible weight modules over $\mathfrak{sl}_{3}$

Let $n>1$ be an integer, $α\in{\mathbb C}^n$, $b\in{\mathbb C}$, and $V$ a $\mathfrak{gl}_n$-module. We define a class of weight modules $F^α_{b}(V)$ over $\sl_{n+1}$ using the restriction of modules of tensor fields over the Lie algebra of vector fields on $n$-dimensional torus. In this paper we consider the case $n=2$ and prove the irreducibility of such 5-parameter $\mathfrak{sl}_{3}$-modules $F^α_{b}(V)$ generically. All such modules have infinite dimensional weight spaces and lie outside of the category of Gelfand-Tsetlin modules. Hence, this construction yields new families of irreducible $\mathfrak{sl}_{3}$-modules.

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arXiv2016

Jet modules for the centerless Virasoro-like algebra

In this paper, we studied the jet modules for the centerless Virasoro-like algebra which is the Lie algebra of the Lie group of the area-preserving diffeomorphisms of a $2$-torus. The jet modules are certain natural modules over the Lie algebra of semi-direct product of the centerless Virasoro-like algebra and the Laurent polynomial algebra in two variables. We reduce the irreducible jet modules to the finite-dimensional irreducible modules over some infinite-dimensional Lie algebra and then characterize the irreducible jet modules with irreducible finite dimensional modules over $\mathfrak{sl}_2$. To determine the indecomposable jet modules, we use the technique of polynomial modules in the sense of \cite{BB, BZ}. Consequently, indecomposable jet modules are described using modules over the algebra $\BB_+$, which is the "positive part" of a Block type algebra studied first by \cite{DZ} and recently by \cite{IM, I}).

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arXiv2016

Zero product determined Lie algebras

A Lie algebra $L$ over a field $\mathbb{F}$ is said to be zero product determined (zpd) if every bilinear map $f:L\times L\to \mathbb{F}$ with the property that $f(x,y)=0$ whenever $x$ and $y$ commute is a coboundary. The main goal of the paper is to determine whether or not some important Lie algebras are zpd. We show that the Galilei Lie algebra $\mathfrak{sl}_2\ltimes V$, where $V$ is a simple $\mathfrak{sl}_2$-module, is zpd if and only if $\dim V =2$ or $\dim V$ is odd. The class of zpd Lie algebras also includes the quantum torus Lie algebras $\mathcal{L}_q$ and $\mathcal{L}^+_q$, the untwisted affine Lie algebras, the Heisenberg Lie algebras, and all Lie algebras of dimension at most $3$, while the class of non-zpd Lie algebras includes the ($4$-dimensional) aging Lie algebra $\mathfrak {age}(1)$ and all Lie algebras of dimension more than $3$ in which only linearly dependent elements commute. We also give some evidence of the usefulness of the concept of a zpd Lie algebra by using it in the study of commutativity preserving linear maps.

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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 $F(M, Ω(λ,β))$.

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arXiv2014

Irreducible weight modules over Witt algebras with infinite dimensional weight spaces

Let $d>1$ be an integer. In 1986, Shen defined a class of weight modules $F^α_b(V)$ over the Witt algebra $\mathcal{W}_d$ for $\a\in\C^d$, $b\in\C$, and an irreducible module $ V$ over the special linear Lie algebra $\sl_d$. In 1996, Eswara Rao determined the necessary and sufficient conditions for these modules to be irreducible when $V$ is finite dimensional. In this note, we will determine the necessary and sufficient conditions for all these modules $F^α_b(V)$ to be irreducible where $V$ is not necessarily finite dimensional. Therefore we obtain a lot of irreducible $\mathcal{W}_d$-modules with infinite dimensional weight spaces.

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arXiv2012

A class of simple weight Virasoro modules

For a simple module $M$ over the positive part of the Virasoro algebra (actually for any simple module over some finite dimensional solvable Lie algebras $\mathfrak{a}_r$) and any $α\in\C$, a class of weight modules $\mathcal {N}(M, α)$ over the Virasoro algebra are constructed. The necessary and sufficient condition for $\mathcal {N}(M, \a)$ to be simple is obtained. We also determine the necessary and sufficient conditions for two such irreducible Virasoro modules to be isomorphic. Many examples for such irreducible Virasoro modules with different features are provided. In particular the irreducible weight Virasoro modules $Γ(α_1, α_2, λ_1, λ_2)$ are defined on the polynomial algebra $\C[x]\otimes \C[t, t^{-1}]$ for any $α_1, α_2, λ_1, λ_2\in\C$ with $λ_1$ or $λ_2$ nonzero. By twisting the weight modules $\mathcal {N}(M, α)$ we also obtain nonweight simple Virasoro modules $\mathcal {N}(M, β)$ for any $β\in\C[t,t^{-1}]$.

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