Let $G$ be a rank $n$ additive subgroup of $\bC$ and $\Vir[G]$ the corresponding Virasoro algebra of rank $n$. In the present paper, irreducible weight modules with finite dimensional weight spaces over $\Vir[G]$ are completely determined. There are two different classes of them. One class consists of simple modules of intermediate series whose weight spaces are all 1-dimensional. The other is constructed by using intermediate series modules over a Virasoro subalgebra of rank $n-1$. The classification of such modules over the classical Virasoro algebra was obtained by O. Mathieu in 1992 using a completely different approach.
In this paper, we obtain a class of Virasoro modules by taking tensor products of the irreducible Virasoro modules $Ω(λ,α,h)$ defined in \cite{CG}, with irreducible highest weight modules $V(θ,h)$ or with irreducible Virasoro modules Ind$_θ(N)$ defined in \cite{MZ2}. We obtain the necessary and sufficient conditions for such tensor product modules to be irreducible, and determine the necessary and sufficient conditions for two of them to be isomorphic. These modules are not isomorphic to any other known irreducible Virasoro modules.
In this paper, we obtain a class of Virasoro modules by taking tensor products of the irreducible Virasoro modules $Ω(λ,α,h)$ and $Ω(μ, b)$ with irreducible highest weight modules $V(θ,h)$ or with irreducible Virasoro modules Ind$_θ(N)$ defined in [MZ2]. We obtain the necessary and sufficient conditions for such tensor product modules to be irreducible, and determine the necessary and sufficient conditions for two of them to be isomorphic. We also compare these modules with other known non-weight Virasoro modules.
In this paper, we present a class of non-weight Virasoro modules $\mathcal{M}\big(V,Ω(λ_0,α_0)\big)\otimes\bigotimes_{i=1}^mΩ(λ_i,α_i)$ where $Ω(λ_i,α_i)$ and $\mathcal{M}\big(V,Ω(λ_0,α_0)\big)$ are irreducible Virasoro modules defined in \cite{LZ2} and \cite{LZ} respectively. The necessary and sufficient conditions for $\mathcal{M}\big(V,Ω(λ_0,α_0)\big)\otimes\bigotimes_{i=1}^mΩ(λ_i,α_i)$ to be irreducible are obtained. Then we determine the necessary and sufficient conditions for two such irreducible Virasoro modules to be isomorphic. At last, we show that the irreducible modules in this cla…
In this paper, all irreducible weight modules with finite dimensional weight spaces over the twisted Heisenberg-Virasoro algebra are determined. There are two different classes of them. One class is formed by simple modules of intermediate series, whose weight spaces are all 1-dimensional; the other class consists of the irreducible highest weight modules and lowest weight modules.
In this paper, we construct a class of non-weight modules over the affine-Virasoro algebra of type $A_1$ by taking tensor products of irreducibles defined in [Q. Chen, J. Han, Non-weight modules over the affine-Virasoro algebra of type $A_1$, J. Math. Phys. 60, 071707 (2019)] with irreducible highest weight modules. The irreducibility and the isomorphism classes of these modules are determined. Moreover, we show that these tensor product modules are different from the known non-weight modules. Finally, we realize some tensor product modules as induced modules from modules over certain subalgeb…
In this paper, we realize polynomial $\H$-modules $Ω(λ,α,β)$ from irreducible twisted Heisenberg-Virasoro modules $\A_{α,β}$. It follows from $\H$-modules $Ω(λ,α,β)$ and $\mathrm{Ind}(M)$ that we obtain a class of natural non-weight tensor product modules $\big(\bigotimes_{i=1}^mΩ(λ_i,α_i,β_i)\big)\otimes \mathrm{Ind}(M)$. Then we give the necessary and sufficient conditions under which these modules are irreducible and isomorphic, and also give that the irreducible modules in this class are new.
In this paper, we construct a class of non-weight modules over the affine-Virasoro algebra of type $A_1$ by taking tensor products of a finite number of irreducible modules $M(λ, α, β, γ)$ with irreducible highest weight modules $V(η, ε, θ)$. We obtain the necessary and sufficient conditions for such tensor product modules to be irreducible, and determine the necessary and sufficient conditions for such two modules to be isomorphic. We also compare these modules with other known non-weight modules, showing that these irreducible modules are new.
Tensor product of highest weight modules and intermediate modules for Virasoro algebra have been studied around 1997. Since then the irreducibility problem for tensor product of modules is open. We consider the loop-Virasoro algebra $Vir \otimes B$, where $Vir$ is the Virasoro algebra and $B$ a commutative associative unital algebra over $\mathbb C$. In this paper we study the irreducibility problem for the tensor product of highest weight modules and intermediate modules for $Vir\otimes B$. Finally we find out a necessary and sufficient conditions for such modules to be isomorphic.
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. …
In this paper, we study irreducible weight modules with infinite dimensional weight spaces over the mirror-twisted Heisenberg-Virasoro algebra $\mathcal{D}$. More precisely, the necessary and sufficient conditions for the tensor products of irreducible highest weight modules and irreducible modules of intermediates series over $\mathcal{D}$ to be irreducible are determined by using "shifting technique". This leads to a family of new irreducible weight modules over $\mathcal{D}$. Then we obtain that any two such tensor products are isomorphic if and only if the corresponding highest weight modu…
We use Block's results to classify irreducible modules over the differential operator algebra $\mathbb{C}[t,t^{-1}, \frac d{dt}]$. From this classification and using "the twisting technique" we construct a lot of new irreducible modules over the Virasoro algebra. These new irreducible Virasoro modules are generally not weight modules. We 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 at the end of the paper. In particular the class of irreducib…
The loop-Virasoro algebra is the Lie algebra of the tensor product of the Virasoro algebra and the Laurent polynomial algebra. This paper classifies irreducible Harish-Chandra modules over the loop-Virasoro algebra, which turn out to be highest weight modules, lowest weight modules and evaluation modules of the intermediate series (all wight spaces are 1-dimensional). As a by-product, we obtain a classification of irreducible Harish-Chandra modules over truncated Virasoro algebras. We also determine the necessary and sufficient conditions for highest weigh irreducible modules over the loop-Vir…
In this paper we study the representations of loop Affine-Virasoro Algebras. As they have canonical triangular decomposition, we define Verma modules and its irreducible quotients. We give necessary and sufficient condition for an irreducible highest weight module to have finite dimensional weight spaces. We prove that an irreducible integrable module is either an highest weight module or a lowest weight module whenever the canonical central element acts non trivially. At the end we construct Affine central operators for each integer and they commute with the action of the Affine Lie 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 $\mathfrak{sl…
We show that the support of an irreducible weight module over the twisted Heisenberg-Virasoro algebra, which has an infinite dimensional weight space, coincides with the weight lattice and that all nontrivial weight spaces of such a module are infinite dimensional. As a corollary, we obtain that every irreducible weight module over the twisted Heisenberg-Virasoro algebra, having a nontrivial finite dimensional weight space, is a Harish-Chandra module (and hence is either an irreducible highest or lowest weight module or an irreducible module from the intermediate series
We prove that any irreducible Harish-Chandra modules for a class of Lie algebras, which we call gap-$p$ Virasoro algebras, must be a highest weight module, a lowest weight module, or a module of intermediate series.These algebras are closely related to the Heisenberg-Virasoro algebra and the algebra of derivations over a quantum torus. They also contain subalgebras which are isomorphic to the Virasoro algebra $Vir$, but graded by $p\mathbb Z$(unlike $Vir$ by $\mathbb Z$).
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 irr…
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 an…
Lie conformal algebras $\mathcal{W}(a,b)$ are the semi-direct sums of Virasoro Lie conformal algebra and its nontrivial conformal modules of rank one. In this paper, we give a complete classification of extensions of finite irreducible conformal modules of $\mathcal{W}(a,b)$. With a similar method, we characterize all extensions of finite irreducible conformal modules of Schrödinger-Virasoro type Lie conformal algebras $TSV(a,b)$ and $TSV(c)$.
It is shown that the support of an irreducible weight module over the Schrödinger-Virasoro Lie algebra with an infinite-dimensional weight space, coincides with the weight lattice and that all nontrivial weight spaces of such a module are infinite-dimensional. As a side-product, it is obtained that every simple weight module over the Schrödinger-Virasoro Lie algebra with a nontrivial finite-dimensional weight space, is a Harish-Chandra module.
Lie conformal algebras $\mathcal{W}(a,b)$ are the semi-direct sums of Virasoro Lie conformal algebra and its nontrivial conformal modules of rank one. In this paper, we first give a complete classification of all finite nontrivial irreducible conformal modules of $\mathcal{W}(a,b)$. It is shown that all such modules are of rank one. Moreover, with a similar method, all finite nontrivial irreducible conformal modules of Schrödinger-Virasoro type Lie conformal algebras $TSV(a,b)$ and $TSV(c)$ are characterized.
In the present paper, we construct two classes of non-weight modules $Ω(λ,α,β)\otimes\mathrm{Ind}(M)$ and $\mathcal{M}\big(V,Ω(λ,α,β)\big)$ over the twisted Heisenberg-Virasoro algebra, which are both associated with the modules $Ω(λ,α,β)$. We present the necessary and sufficient conditions under which modules in these two classes are irreducible and isomorphic, and also show that the irreducible modules in these two classes are new. Finally, we construct non-weight modules $\mathrm{Ind}_{\underline y,λ}(\C_{RS})$ and $\mathrm{Ind}_{\underline z,λ}(\C_{PQ})$ over the twisted Heisenberg-Virasor…
In this paper, it is proved that all irreducible Harish-Chandra modules over the $\Q$ Heisenberg-Virasoro algebra are of intermediate series (all weight spaces are 1-dimensional).
In this paper, we obtain a class of irreducible Virasoro modules by taking tensor products of the irreducible Virasoro modules $Ω(λ,b)$ defined in [LZ], with irreducible highest weight modules $V(θ,h)$ or with irreducible Virasoro modules Ind$_θ(N)$ defined in [MZ2]. We determine the necessary and sufficient conditions for two such irreducible tensor products to be isomorphic. Then we prove that the tensor product of $Ω(λ,b)$ with a classical Whittaker module is isomorphic to the module $\mathrm{Ind}_{θ,λ}(\mathbb{C_\mathbf{m}})$ defined in [MW]. As a by-product we obtain the necessary and suf…
Let $G$ be an arbitrary additive subgroup of $C$ and $Vir[G]$ the corresponding generalized Virasoro algebra. In the present paper, irreducible weight modules with finite dimensional weight spaces over $Vir[G]$ are completely determined. The classification strongly depends on the index group $G$. If $G$ does not have a direct summand $Z$, then such irreducible modules over $Vir[G]$ are only modules of intermediate series whose weight spaces are all 1-dimensional. Otherwise, there is one more class of modules which are constructed by using intermediate series modules over a generalized Virasoro…
Previously we gave a proof of the Feigin--Fuchs character formula for the irreducible unitary discrete series of the Virasoro algebra with 0<c<1. The proof showed directly that the mutliplicity space arising in the coset construction of Goddard, Kent and Olive was irreducible, using the elementary part of the unitarity criterion of Friedan, Qiu and Shenker, giving restrictions on h for c=1-6/m(m+1) with m>2. In this paper we consider the same problem in the limiting case of the coset construction c=1. Using primary fields, we directly establish that the Virasoro algebra acts irreducibly on the…
In this paper we classify the irreducible integrable modules for the loop affine-Virasoro algebra $(( \overset{\circ}{\mathfrak{g}} \otimes \mathbb{C}[t, t^{-1}] \oplus \mathbb{C} K) \rtimes \text{Vir}) \otimes A$, where $A$ is a finitely generated commutative associative algebra with unity.
Let $\preceq$ be a compatible total order on the additive group $\mathbb{Z}^2$, and $L$ be the rank two Heisenberg-Virasoro algebra. For any $\mathbf{c}=(c_1,c_2,c_3,c_4) \in \mathbb{C}^4$, we define $\mathbb{Z}^2$-graded Verma module $M(\mathbf{c}, \preceq)$ for the Lie algebra $L$. A necessary and sufficient condition for the Verma module $M(\mathbf{c}, \preceq)$ to be irreducible is provided. Moreover, the maximal $\mathbb{Z}^2$-graded submodules of the Verma module $M(\mathbf{c}, \preceq)$ are characterized when $M(\mathbf{c}, \preceq)$ is reducible.
We give a complete classification of the irreducible quasifinite modules for algebras of the form Vir \otimes A, where Vir is the Virasoro algebra and A is a Noetherian commutative associative unital algebra over the complex numbers. It is shown that all such modules are tensor products of generalized evaluation modules. We also give an explicit sufficient condition for a Verma module of Vir \otimes A to be reducible. In the case that A is an infinite-dimensional integral domain, this condition is also necessary.
In this paper, we construct a family of non-weight modules over the super-Virasoro algebras. Those modules when regarded as modules of the Ramond algebra and further restricted as modules over the Cartan subalgebra $\mathfrak{h}$ are free of rank $1$, while when regarded as modules of the Neveu-Schwarz algebra and further restricted as modules over the Cartan subalgebra $\mathfrak{H}$ are free of rank $2$. We obtain a sufficient and necessary condition for such modules to be simple. Moreover, we determine the isomorphism classes of these modules. Finally, we show that these modules constitute …
We construct and classify the free $U(\mathbb{C}L_0\oplus \mathbb{C}M_0\oplus\mathbb{C}Y_0)$-modules of rank $1$ over the Schrödinger-Virasoro algebra $\mathfrak{sv}(s)$ for $s=0$.Moreover, we show that the class of free $U(\mathbb{C}L_0\oplus\mathbb{C}M_0)$-modules of rank $1$ over the Schrödinger-Virasoro algebra $\mathfrak{sv}(s)$ for $s=\frac{1}{2}$ is nonexistent.
In this paper, we introduce a class of infinite Lie conformal algebras $\mathfrak{B}(α,β,p)$, which are the semi-direct sums of Block type Lie conformal algebra $\mathfrak{B}(p)$ and its non-trivial conformal modules of $\Z$-graded free intermediate series. The annihilation algebras are a class of infinite-dimensional Lie algebras, which include a lot of interesting subalgebras: Virasoro algebra, Block type Lie algebra, twisted Heisenberg-Virasoro algebra and so on. We give a complete classification of all finite non-trivial irreducible conformal modules of $\mathfrak{B}(α,β,p)$ for $α,β\in\C,…
In this paper, we consider the classification of irreducible ${\bf Z}$- and ${\bf Z}^2$-graded modules with finite dimensional homogeneous subspaces over the Virasoro-like algebra. We first prove that such a module is a uniformly bounded module or a generalized highest weight module. Then we determine all generalized highest weight irreducible modules. As a consequence, we also determine all the modules with nonzero center. Finally, we prove that there does not exist any nontrivial ${\bf Z}$-graded modules of intermediate series.
It is obtained that an irreducible weight module with finite weight multiplicities over a higher rank Virasoro or super-Virasoro algebra is either a module of the intermediate series, or a so-called finitely-dense module.
We classify all graded compatible left-symmetric algebraic structures on high rank Witt algebras, and classify all non-graded ones satisfying a minor condition. Furthermore, graded compatible left-symmetric algebraic structures on high rank Virasoro algebras are also classified.
We study a family of fermionic oscillator representations of the Virasoro algebra via 2-point-local Virasoro fields on the Fock space $\mathit{F^{\otimes \frac{1}{2}}}$ of a neutral (real) fermion. We obtain the decomposition of $\mathit{F^{\otimes \frac{1}{2}}}$ as a direct sum of irreducible highest weight Virasoro modules with central charge $c=1$. As a corollary we obtain the decomposition of the irreducible highest weight Virasoro modules with central charge $c=\frac{1}{2}$ into irreducible highest weight Virasoro modules with central charge $c=1$. As an application we show how positive s…
In this paper, we study a class of non-weight modules over two kinds of algebras related to the Virasoro algebra, i.e., the loop-Virasoro algebras $\mathfrak{L}$ and a class of Block type Lie algebras $\mathfrak{B(q)}$, where $q$ is a nonzero complex number. We determine those modules whose restriction to the Cartan subalgebra (modulo center) are free of rank one. We also provide a sufficient and necessary condition for such modules to be simple, and determine their isomorphism classes. Moreover, we obtain the simplicity of modules over loop-Virasoro algebras by taking tensor products of some …
We present the list of irreducible (generalized) highest weight modules over the Virasoro algebra and N=1 super-Virasoro algebras obtained as factor-modules of (generalized) Verma modules. We present also the character formulae of all these modules and single out the unitary irreducible ones. Most formulas are valid for the three algebras under consideration, the different cases being distinguished by two parameters. This is a slightly extended version of an Encyclopedia entry.
It is proved that an irreducible module over the non-graded Virasoro-like algebra, which satisfies a natural condition, is a GHW module or uniformly bounded. Furthermore, the classification of some uniformly bounded modules is given.
Intrigued by a well-known theorem of Mathieu's on Harish-Chandra modules over the Virasoro algebra, we give an analogous result for a class of Block type Lie algebras $\BB$, where the parameter $q$ is a nonzero complex number. We also classify quasifinite irreducible highest weight $\BB$-modules and irreducible $\BB$-modules of the intermediate series. In particular, we obtain that an irreducible $\BB$-module of the intermediate series may be a nontrivial extension of a $\Vir$-module of the intermediate series if $q$ is half of a negative integer, where $\Vir$ is a subalgebra of $\BB$ isomorph…
It is proved that uniformly bounded simple modules over higher rank super-Virasoro algebras are modules of the intermediate series, and that simple modules with finite dimensional weight spaces are either modules of the intermediate series or generalized highest weight modules.
In this paper we introduce and study $n$-point Virasoro algebras, $\tilde{\W_a}$, which are natural generalizations of the classical Virasoro algebra and have as quotients multipoint genus zero Krichever-Novikov type algebras. We determine necessary and sufficient conditions for the latter two such Lie algebras to be isomorphic. Moreover we determine their automorphisms, their derivation algebras, their universal central extensions, and some other properties. The list of automorphism groups that occur is $C_n$, $D_n$, $A_4$, $S_4$ and $A_5$. We also construct a large class of modules which we …
Using simple modules over the derivation Lie algebra $C[t]\frac{d}{d t}$ of the associative polynomial algebra $C[t]$, we construct new weight Virasoro modules with all weight spaces infinite dimensional. We determine necessary and sufficient conditions for these new weight Virasoro modules to be simple, and determine necessary and sufficient conditions for two such weight Virasoro modules to be isomorphic. If such a weight Virasoro module is not simple, we obtain all its submodules. In particular, we completely determine the simplicity and the isomorphism classes of the weight modules defined…
A notion of generalized highest weight modules over the high rank Virasoro algebras is introduced, and a theorem, which was originally given as a conjecture by Kac over the Virasoro algebra, is generalized. Mainly, we prove that a simple Harish-Chandra module over a high rank Virasoro algebra is either a generalized highest weight module, or a module of the intermediate series.
In this paper, we give a complete classification of all free $U(\mathbb{C}L_0 \oplus \mathbb{C}Y_0\oplus \mathbb{C}M_0)$-modules of rank 1 over a Schr{ö}dinger-Virasoro type algebra $\mathfrak{tsv}$.
In the present paper, we prove that any finite non-trivial irreducible module over a rank two Lie conformal algebra $\mathcal{H}$ is of rank one. We also describe the actions of $\mathcal{H}$ on its finite irreducible modules explicitly. Moreover, we show that all finite non-trivial irreducible modules of finite Lie conformal algebras whose semisimple quotient is the Virasoro Lie conformal algebra are of rank one.
We describe the structure of the irreducible highest weight modules for the twisted Heisenberg-Virasoro Lie algebra at level zero. We prove that such a module is either isomorphic to a Verma module or to a quotient of two Verma modules.
Irreducible Virasoro representations in Liouville conformal field theory
Aix–Marseille Université（艾克斯-马赛大学）；Institut de Mathématiques de Marseille（马赛数学研究所）；Beijing International Center for Mathematical Research (BICMR)（北京国际数学研究中心）
In this paper, we study a class of non-weight modules over the affine-Virasoro algebra of type $A_1$, which are free modules of rank one when restricted to the Cartan subalgebra (modulo center). We give the classification of such modules. Moreover, the simplicity and the isomorphism classes of these modules are determined.
