Modules Of Intermediate Series Research Articles

A family of simple non-weight modules over the twisted N = 2 superconformal algebra

We construct a class of non-weight modules over the twisted N = 2 superconformal algebra L . Let h = C L 0 ⊕ C G 0 be the Cartan subalgebra of L , and let t = C L 0 be the Cartan subalgebra of even part L 0 ¯ . These modules over L when restricted to the h are free of rank 1 or when restricted to the t are free of rank 2. We provide the necessary and sufficient conditions for these modules to be simple, and give the necessary and sufficient conditions for two such L -modules to be isomorphic. We also compute the action of an automorphism on them. Moreover, based on the weighting functor introduced in [15] , a class of intermediate series modules A t ( σ ) are obtained.

Tensor product weight modules for the mirror Heisenberg-Virasoro algebra

In this paper, we study irreducible weight modules with infinite dimensional weight spaces over the mirror Heisenberg-Virasoro algebra D . More precisely, the necessary and sufficient conditions for the tensor products of irreducible highest weight modules and irreducible modules of intermediate series over D to be irreducible are determined by using “shifting technique”. This leads to a family of new irreducible weight modules over D . Then we obtain that any two such tensor products are isomorphic if and only if the corresponding highest weight modules and modules of intermediate series are isomorphic respectively. Also we discuss submodules of the tensor product module when it is not irreducible.

Lie structure on the Hochschild cohomology of a family of subalgebras of the Weyl algebra

For each nonzero $h\in\mathbb{F}\[x]$, where $\mathbb{F}$ is a field, let $\mathsf{A}\_h$ be the unital associative algebra generated by elements $x,y$, satisfying the relation $yx-xy=h$. This gives a parametric family of subalgebras of the Weyl algebra $\mathsf{A}\_1$, containing many well-known algebras which have previously been studied independently. In this paper, we give a full description of the Hochschild cohomology $\operatorname{\mathsf{HH}}^{\bullet}(\mathsf{A}h)$ over a field of an arbitrary characteristic. In case $\mathbb{F}$ has a positive characteristic, the center $\mathsf{Z}(\mathsf{A}{h})$ of $\mathsf{A}\_h$ is nontrivial and we describe $\operatorname{\mathsf{HH}}^\bullet(\mathsf{A}h)$ as a module over $\mathsf{Z}(\mathsf{A}{h})$. The most interesting results occur when $\mathbb{F}$ has a characteristic $0$. In this case, we describe $\operatorname{\mathsf{HH}}^\bullet(\mathsf{A}\_h)$ as a module over the Lie algebra $\operatorname{\mathsf{HH}}^1(\mathsf{A}\_h)$ and find that this action is closely related to the intermediate series modules over the Virasoro algebra. We also determine when $\operatorname{\mathsf{HH}}^\bullet(\mathsf{A}\_h)$ is a semisimple $\operatorname{\mathsf{HH}}^1(\mathsf{A}\_h)$-module.

On a Class of Infinite Simple Lie Conformal Algebras

In this paper, we study a class of infinite simple Lie conformal algebras associated to a class of generalized Block type Lie algebras. The central extensions by a one-dimensional center \(\mathbb {C}\mathfrak {c}\), conformal derivations and free intermediate series modules of this class of Lie conformal algebras are determined. Moreover, we also show that these Lie conformal algebras do not have any non-trivial finite conformal modules. Consequently, these Lie conformal algebras cannot be embedded into gcN for any positive integer N.

On a family of non-weight modules over Witt-type Lie algebras and superalgebras

We classify modules over a Witt-type Lie algebra and superalgebra such that when considered as modules of U(h) they are free of rank 1. We provide sufficient conditions for simplicity, and compute the action of an automorphism on the module. Lastly, we apply the weighting functor and show that the resulting modules are intermediate series modules.

Lie conformal algebras related to Galilean conformal algebras

Let [Formula: see text] be a Lie conformal algebra related to Galilean conformal algebras, where [Formula: see text] are complex numbers. All the conformal derivations of [Formula: see text] are shown to be inner. The rank one conformal modules and [Formula: see text]-graded free intermediate series modules over [Formula: see text] are completely classified. The corresponding results of the finite conformal subalgebra of [Formula: see text] are also obtained as byproducts.

Classification of irreducible Harish-Chandra modules over gap-p Virasoro algebras

We prove that any irreducible Harish-Chandra module 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 pZ (unlike Vir by Z).

Generalized Oscillator Representations of the Twisted Heisenberg-Virasoro Algebra

In this paper, we first obtain a general result on sufficient conditions for tensor product modules to be simple over an arbitrary Lie algebra. We classify simple modules with a nice property over the infinite-dimensional Heisenberg algebra ${\H}$, and then obtain a lot of simple modules over the twisted Heisenberg-Virasoro algebra $\LL$ from generalized oscillator representations of $\LL$ by extending these $\H$-modules. We give the necessary and sufficient conditions for Whittaker modules over $\LL$ (in the more general setting) to be simple. We use the "shifting technique" to determine the necessary and sufficient conditions for the tensor products of highest weight modules and modules of intermediate series over $\LL$ to be simple. At last we establish the "embedding trick" to obtain a lot more simple $\LL$-modules.

Tensor product weight modules of Schrödinger-Virasoro algebras

It is known that the Schrodinger-Virasoro algebras, including the original Schrodinger-Virasoro algebra and the twisted Schrodinger-Virasoro algebra, are playing important roles in mathematics and statistical physics. In this paper, we study the tensor products of weight modules over the Schrodinger-Virasoro algebras. The irreducibility criterion for the tensor products of highest weight modules with intermediate series modules over the Schrodinger-Virasoro algebra is obtained.

Lie Conformal Algebras of Planar Galilean Type

Motivated by the Lie structure of the planar Galilean conformal algebra, we construct a class of infinite rank Lie conformal algebras C P G ( a , b ) , where a, b are complex numbers. All their conformal derivations are shown to be inner. The rank-one conformal modules and ℤ-graded free intermediate series modules over C P G ( a , b ) are completely classified. The parallel results of the finite Lie conformal subalgebra C P G ( a , b ) of C P G ( a , b ) are also presented.

Generalised Witt algebras and idealizers

Let k be an algebraically closed field of characteristic zero, and let Γ be an additive subgroup of k. Results of Kaplansky–Santharoubane and Su classify intermediate series representations of the generalised Witt algebraWΓ in terms of three families, one parameterised by A2 and two by P1. In this note, we use the first family to construct a homomorphism Φ from the enveloping algebra U(WΓ) to a skew extension k[A2]⋊Γ of the coordinate ring of A2. We show that the image of Φ is contained in a (double) idealizer subring of this skew extension and that the representation theory of idealizers explains the three families. We further show that the image of U(WΓ) under Φ is not left or right noetherian, giving a new proof that U(WΓ) is not noetherian.We construct Φ as an application of a general technique to create ring homomorphisms from shift-invariant families of modules. Let G be an arbitrary group and let A be a G-graded ring. A graded A-module M is an intermediate series module if Mg is one-dimensional for all g∈G. Given a shift-invariant family of intermediate series A-modules parametrised by a scheme X, we construct a homomorphism Φ from A to a skew extension of k[X]. The kernel of Φ consists of those elements which annihilate all modules in X.

Infinite-dimensional 3-Lie algebras and their connections to Harish-Chandra modules

We construct two kinds of infinite-dimensional 3-Lie algebras from a given commutative associative algebra, and show that they are all canonical Nambu 3-Lie algebras. We relate their inner derivation algebras to Witt algebras, and then study the regular representations of these 3-Lie algebras and the natural representations of the inner derivation algebras. In particular, for the second kind of 3-Lie algebras, we find that their regular representations are Harish-Chandra modules, and the inner derivation algebras give rise to intermediate series modules of the Witt algebras and contain the smallest full toroidal Lie algebras without center.

A class of simple weight modules over the twisted Heisenberg-Virasoro algebra

A class of weight modules M(V,a) over the twisted Heisenberg-Virasoro H are constructed, which includes modules of intermediate series, where V is an H̄r,d-module and a is a complex number. We give the necessary and sufficient conditions under which these modules are simple and also determine all the equivalent simple modules in this class. Moreover, we show that simple modules in this class are new. Finally, we construct a class of new simple non-weight H-modules.

Infinite rank Schrödinger-Virasoro type Lie conformal algebras

Motivated by the structure of certain modules over the loop Virasoro Lie conformal algebra and the Lie structures of Schrödinger-Virasoro algebras, we construct a class of infinite rank Lie conformal algebras CSV(a, b), where a, b are complex numbers. The conformal derivations of CSV(a, b) are uniformly determined. The rank one conformal modules and ℤ-graded free intermediate series modules over CSV(a, b) are classified. Corresponding results of the conformal subalgebra CHV(a, b) of CSV(a, b) are also presented.

Loop Schrödinger–Virasoro Lie conformal algebra

In this paper, we introduce two kinds of Lie conformal algebras, associated with the loop Schrödinger–Virasoro Lie algebra and the extended loop Schrödinger–Virasoro Lie algebra, respectively. The conformal derivations, the second cohomology groups of these two conformal algebras are completely determined. And nontrivial free conformal modules of rank one and [Formula: see text]-graded free intermediate series modules over these two conformal algebras are also classified in the present paper.

Classification of ℤ2-graded modules of intermediate series over a Block-type Lie algebra

Let [Formula: see text] be the Lie algebra of Block type with basis {Li,j | i,j ∈ ℤ} and relations [Li,j, Lk,l] = ((j + 1)k - (l + 1)i)Li+k,j+l. Since [Formula: see text] is ℤ2-graded, it is natural to study ℤ2-graded modules over [Formula: see text]. In this paper, ℤ2-graded [Formula: see text]-modules of the intermediate series are classified.

Tensor Product Weight Representations of the Neveu–Schwarz Algebra

We study the tensor product of a highest weight module with an intermediate series module over the Neveu–Schwarz algebra. If the highest weight module is nontrivial, the weight spaces of such a tensor product are infinite dimensional. We show that such a tensor product is indecomposable. Using a “shifting technique” developed by H. Chen, X. Guo, and K. Zhao for the Virasoro algebra case, we give necessary and sufficient conditions for such a tensor product to be irreducible. Furthermore, we give necessary and sufficient conditions for two such tensor products to be isomorphic.

Loop Heisenberg-Virasoro Lie conformal algebra

Let HV be the loop Heisenberg-Virasoro Lie algebra over ℂ with basis {Lα,i, Hβ,j∣α, β, i, j ∈ ℤ} and brackets [Lα,i, Lβ,j] = (α − β) Lα+β,i+j, [Lα,i, Hβ,j] = − βHα+β,i+j, [Hα,i, Hβ,j] = 0. In this paper, a formal distribution Lie algebra of HV is constructed. Then, the associated conformal algebra CHV is studied, where CHV has a ℂ[∂]-basis {Li, Hi∣i ∈ ℤ} with λ-brackets [Li λ Lj] = (∂ + 2λ) Li+j, [Li λ Hj] = (∂ + λ) Hi+j, [Hi λ Lj] = λLi+j, and [Hi λ Hj] = 0. In particular, the conformal derivations of CHV are determined. Finally, rank one conformal modules and ℤ-graded free intermediate series modules over CHV are classified.

Irreducible quasifinite modules over a class of Lie algebras of block type

For any nonzero complex number $q$, there is a Lie algebra of Block type, denoted by $\mathcal{B}(q)$. In this paper, a complete classification of irreducible quasifinite modules is given. More precisely, an irreducible quasifinite module is a highest weight or lowest weight module, or a module of intermediate series. As a consequence, a classification for uniformly bounded modules over another class of Lie algebras, the semi-direct product of the Virasoro algebra and a module of intermediate series, is also obtained. Our method is conceptional, instead of computational.

Loop Virasoro Lie conformal algebra

The Lie conformal algebra of loop Virasoro algebra, denoted by \documentclass[12pt]{minimal}\begin{document}$\mathscr {CW}$\end{document}CW, is introduced in this paper. Explicitly, \documentclass[12pt]{minimal}\begin{document}$\mathscr {CW}$\end{document}CW is a Lie conformal algebra with \documentclass[12pt]{minimal}\begin{document}$\mathbb {C}[\partial ]$\end{document}C[∂]-basis \documentclass[12pt]{minimal}\begin{document}$\lbrace L_i\,|\,i\in \mathbb {Z}\rbrace$\end{document}{Li|i∈Z} and λ-brackets [Li λ Lj] = (−∂−2λ)Li+j. Then conformal derivations of \documentclass[12pt]{minimal}\begin{document}$\mathscr {CW}$\end{document}CW are determined. Finally, rank one conformal modules and \documentclass[12pt]{minimal}\begin{document}$\mathbb {Z}$\end{document}Z-graded free intermediate series modules over \documentclass[12pt]{minimal}\begin{document}$\mathscr {CW}$\end{document}CW are classified.