Search

Let A m , n be the tensor product of the Laurent polynomial algebra in m even variables and the exterior algebra in n odd variables over the complex field C , and the Witt superalgebra W m , n be the Lie superalgebra of superderivations of A m , n . In this paper, we classify the simple weight W m , n modules with finite-dimensional weight spaces with respect to the standard Cartan subalgebra of W m , 0 . Every such module is either a simple quotient of a tensor module or a module of highest weight type.

We classify Jet modules for the Lie (super)algebras L = W ⋉ ( g ⊗ C [ t , t − 1 ] ) , where W is the Witt algebra and g is a Lie superalgebra with an even diagonlizable derivation. Then we give a conceptional method to classify all simple Harish-Chandra modules for L and the map superalgebras, which are of the form L ⊗ R , where R is a Noetherian unital supercommutative associative superalgebra.

In this paper, we give a new approach to classify all simple Harish-Chandra modules for the N=1 Ramond algebra based on the so called A-cover theory developed in \cite{BF}, Comment: 11 pages

Let $d\ge1$ be an integer, $W_d$ and $\mathcal{K}_d$ be the Witt algebra and the weyl algebra over the Laurent polynomial algebra $A_d=\mathbb{C} [x_1^{\pm1}, x_2^{\pm1}, ..., x_d^{\pm1}]$, respectively. For any $\mathfrak{gl}_d$-module $M$ and any admissible module $P$ over the extended Witt algebra $\widetilde W_d$, we define a $W_d$-module structure on the tensor product $P\otimes M$. We prove in this paper that any simple $W_d$-module that is finitely generated over the cartan subalgebra is a quotient module of the $W_d$-module $P \otimes M$ for a finite dimensional simple $\mathfrak{gl}_d$-module $M$ and a simple $\mathcal{K}_d$-module $P$ that are finitely generated over the cartan subalgebra. We also characterize all simple $\mathcal{K}_d$-modules and all simple admissible $\widetilde W_d$-modules that are finitely generated over the cartan subalgebra., 25 pages

Let g = sl n + 1 , W n + or W n , D = ⊕ i = 1 n C ∂ ∂ t i . In this paper, we classify the full subcategory D ( g ) of g -Mod consisting of modules which are free of rank 1 when restricted to U ( D ) . The irreducibility of modules in D ( g ) is determined. Using modules in D ( sl n + 1 ) , we construct classes of modules which are free of finite rank and classes of infinitely generated modules when restricted to the Cartan subalgebra.

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., 28 pages

In this paper, we classify all simple jet modules for the Neveu-Schwarz algebra $\widehat{\mathfrak{k}}$ and its contact subalgebra $\mathfrak{k}^+$. Based on these results, we give a classification of simple Harish-Chandra modules for $\widehat{\mathfrak{k}}$ and $\mathfrak{k}^+$., Comment: arXiv admin note: text overlap with arXiv:2007.04483

For an irreducible module P over the Weyl algebra K n + (resp. K n ) and an irreducible module M over the general linear Lie algebra gl n , using Shen's monomorphism, we make P ⊗ M into a module over the Witt algebra W n + (resp. over W n ). We obtain the necessary and sufficient conditions for P ⊗ M to be an irreducible module over W n + (resp. W n ), and determine all submodules of P ⊗ 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 , including some known modules and many new modules.

Let n > 1 be an integer, α ∈ C n , b ∈ C , and V a gl n -module. We define a class of weight modules F b α ( 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 sl 3 -modules F b α ( 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 sl 3 -modules.

A Lie algebra L over a field $$\mathbb {F}$$ is said to be zero product determined (zpd) if every bilinear map 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 , 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 $$\mathscr {L}_q$$ and $$\mathscr {L}^+_q$$ , 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 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 zpd Lie algebra by using it in the study of commutativity preserving linear maps.

Using simple modules over the derivation Lie algebra C [ t ] d d t for the polynomial algebra C [ t ] , we construct new weight modules over the Witt algebra where all the weight spaces are infinite dimensional. We determine the necessary and sufficient conditions for these new modules being simple, as well as determining the necessary and sufficient conditions for two Witt modules being isomorphic. If a module is not simple, we obtain all its submodules. As a consequence, we fully determine the simplicity and isomorphism classes for the Witt modules defined by [3] , which are a small proportion of the modules constructed in this study.

In this paper we address the problem of classification of simple weight modules over weak generalized Weyl algebras of rank one. The principal difference between weak generalized Weyl algebras and generalized Weyl algebras is that weak generalized Weyl algebras are defined using an endomorphism rather than an automorphism of a commutative ring R. We reduce classification of simple weight modules over weak generalized Weyl algebras to description of the dynamics of the action of the above mentioned endomorphism on the set of maximal ideals. We also describe applications of our results to the study of generalized Heisenberg algebras.

Generalized Heisenberg algebras H ( f ) for any polynomial f ( h ) ∈ C [ h ] have been used to explain various physical systems and many physical phenomena for the last 20 years. In this paper, we first obtain the center of H ( f ) , and the necessary and sufficient conditions on f for two H ( f ) to be isomorphic. Then we determine all finite dimensional simple modules over H ( f ) for any polynomial f ( h ) ∈ C [ h ] . More precisely, there are three classes of them, A H ( f ) ′ ( λ , z ˙ , a ) , B H ( f ) ′ ( λ , z ˙ , a ) , and C H ( f ) ( z ˙ , n ) . If f = w h + c for any c ∈ C and n-th ( n > 1 ) primitive root w of unity we actually obtain a complete classification of all irreducible modules over H ( f ) .

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 a r ) and any α ∈ C , a class of weight modules N ( M , α ) over the Virasoro algebra is constructed. These weight modules have infinite dimensional weight spaces if dim ⁡ M > 1 . The necessary and sufficient conditions for N ( M , α ) to be simple are 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 ] ⊗ C [ t , t − 1 ] for any α 1 , α 2 , λ 1 , λ 2 ∈ C with λ 1 or λ 2 nonzero. By twisting the weight modules N ( M , α ) we also obtain nonweight simple Virasoro modules N ( M , β ) for any nonconstant β ∈ C [ t , t − 1 ] .

We study category O for the (centrally extended) Schrodinger algebra. We determine the quivers for all blocks and relations for blocks of nonzero central charge. We also describe the quiver and rel ...

We study irreducible representations of two classes of conformal Galilei algebras in 1-spatial dimension. We construct a functor which transforms simple modules with nonzero central charge over the Heisenberg subalgebra into simple modules over the conformal Galilei algebras. This can be viewed as an analogue of oscillator representations. We use oscillator representations to describe the structure of simple highest weight modules over conformal Galilei algebras. We classify simple weight modules with finite dimensional weight spaces over finite dimensional Heisenberg algebras and use this classification and properties of oscillator representations to classify simple weight modules with finite dimensional weight spaces over conformal Galilei algebras.

We use Block's results to classify irreducible modules over the differential operator algebra C [ t , t − 1 , d d t ] . From modules A over C [ t , t − 1 , d d t ] and using the “twisting technique” we construct a class of modules A b over the Virasoro algebra for any b ∈ C . These new Virasoro modules are generally not weight modules. The necessary and sufficient conditions for A b to be irreducible are obtained. Then we determine the necessary and sufficient conditions for two such irreducible Virasoro modules to be isomorphic. Many interesting examples for such irreducible Virasoro modules with different features are provided at the end of the paper. In particular the class of irreducible Virasoro modules Ω ( λ , b ) for any λ ∈ C ⁎ and any b ∈ C are defined on the polynomial algebra C [ x ] .

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 subalgebra $Vir[G_0]$ of $Vir[G]$ for a direct summand $G_0$ of $G$ with corank 1., Comment: 14 pages

Representations of various one-dimensional central extensions of quantum tori (called quantum torus Lie algebras) were studied by several authors. Now we define a central extension of quantum tori so that all known representations can be regarded as representations of the new quantum torus Lie algebras . The center of now is generally infinite dimensional.In this paper, Z-graded Verma modules over and their corresponding irreducible highest weight modules are defined for some linear functions . Necessary and sufficient conditions for to have all finite dimensional weight spaces are given. Also necessary and sufficient conditions for Verma modules e to be irreducible are obtained.

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 nonzero weight spaces are all 1-dimensional; the other class consists of the irreducible highest weight modules and lowest weight modules.

In this paper, the Harish-Chandra modules and Verma modules over Block algebra \( \mathfrak{L} \)[G] are investigated. More precisely, the irreducibility of the Verma modules over \( \mathfrak{L} \)[G] is completely determined, and the Harish-Chandra modules over \( \mathfrak{L} \)[ℤ] are classified.

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., Comment: 24 pages

Let [Formula: see text], [Formula: see text]. Then we have the algebraic curve [Formula: see text], and its coordinate algebras (the Riemann surfaces) [Formula: see text] and [Formula: see text] The Lie algebras [Formula: see text] and [Formula: see text] are called the [Formula: see text]th superelliptic Lie algebras associated to [Formula: see text]. In this paper, we determine the necessary and sufficient conditions for such Lie algebras to be simple, and determine their universal central extensions and their derivation algebras. We also study the isomorphism and automorphism problem for these Lie algebras, which will help to understand the birational equivalence of some algebraic curves of the form [Formula: see text].

In this paper we use Block's classification of simple modules over the first Weyl algebra to obtain a complete classification of simple weight modules, in particular, of Harish-Chandra modules, over the 1-spatial ageing algebra age(1). Most of these modules have infinite dimensional weight spaces and so far the algebra age(1) is the only Lie algebra having simple weight modules with infinite dimensional weight spaces for which such a classification exists. As an application we classify all simple weight modules over the (1+1)-dimensional space-time Schrodinger algebra S that have a simple age(1)-submodule thus constructing many new simple weight S-modules., Comment: 18 pages

In this paper we introduce and study n-point Virasoro algebras, [Formula: see text], 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 Cn, Dn, A4, S4 and A5. We also construct a large class of modules which we call modules of densities, and determine necessary and sufficient conditions for them to be irreducible.

In this paper, we introduce pre-exp-polynomial Lie algebras which include loop algebras, Virasoro-like algebras and some quantum torus Lie algebras. We study “highest weight” representations of these Z n + 1 -graded Lie algebras. More precisely, we show that non-graded and graded irreducible highest weight modules with the same highest weight simultaneously have all finite-dimensional weight spaces or not, and they have all finite-dimensional weight spaces if and only if the highest weight is an exp-polynomial “highest weight”. We also show that non-graded and graded highest weight Verma modules with the same highest weight are simultaneously irreducible or not, and we determine necessary and sufficient conditions for a Verma module to be irreducible.