Simple weight modules of the Drinfeld double of the Jordan plane

Classification of simple locally finite modules over the affine-Virasoro algebras

In this paper, we classify simple smooth modules over the mirror Heisenberg–Virasoro algebra ${\mathfrak {D}}$, and simple smooth modules over the twisted Heisenberg–Virasoro algebra $\bar {\mathfrak {D}}$ with non-zero level. To this end we generalize Sugawara operators to smooth modules over the Heisenberg algebra, and develop new techniques. As applications, we characterize simple Whittaker modules and simple highest weight modules over ${\mathfrak {D}}$. A vertex-algebraic interpretation of our result is the classification of simple weak twisted and untwisted modules over the Heisenberg–Virasoro vertex algebras. We also present a few examples of simple smooth ${\mathfrak {D}}$-modules and $\bar {\mathfrak {D}}$-modules induced from simple modules over finite dimensional solvable Lie algebras, that are not tensor product modules of Virasoro modules and Heisenberg modules. This is very different from the case of simple highest weight modules over $\mathfrak {D}$ and $\bar {\mathfrak {D}}$ which are always tensor products of simple Virasoro modules and simple Heisenberg modules.

Let 𝔤 be a classical Lie superalgebra of type I or a Cartan-type Lie superalgebra W(n). We study weight 𝔤-modules using a method inspired by Mathieu's classification of the simple weight modules with finite weight multiplicities over reductive Lie algebras, [Mathieu O.: Classification of irreducible weight modules. Ann. Inst. Fourier 50 (2000), 537–592]. Our approach is based on the fact that every simple weight 𝔤-module with finite weight multiplicities is obtained via a composition of a twist and localization from a highest weight module. This allows us to transfer many results for category 𝒪 modules to the category of weight modules with finite weight multiplicities. As a main application of the method we reduce the problems of finding a 𝔤0-composition series and a character formula for all simple weight modules to the same problems for simple highest weight modules. In this way, using results of Serganova we obtain a character formula for all simple weight W(n)-modules and all simple atypical nonsingular -modules. Some of our results are new already in the case of a classical reductive Lie algebra 𝔤.

On simple modules over conformal Galilei algebras

Simple Schrödinger modules which are locally finite over the positive part

Classification of simple weight Virasoro modules with a finite-dimensional weight space

Generalized down-up algebras were first introduced in Cassidy and Shelton (2004). Their simple weight modules were classified in Cassidy and Shelton (2004) in the noetherian case, and in Praton (2007) in the non-noetherian case. Here we concentrate on non-noetherian down-up algebras. We show that almost all simple modules are weight modules. We also classify the corresponding primitive ideals.

Given any simple Lie superalgebra g, we investigate the structure of an arbitrary simple weight 0-module. We introduce two invariants of simple weight modules: the shadow and the small Weyl group. Generalizing results of Fernando and Futorny we show that any simple module is obtained by parabolic induction from a cuspidal module of a Levi subsuperalgebra. Then we classify the cuspidal Levi subsuperalgebras of all simple classical Lie superalgebras and of the Lie superalgebra W(n). Most of them are simply Levi subalgebras of go, in which case the classification of all finite cuspidal representations has recently been carried out by one of us (Mathieu). Our results reduce the classification of the finite simple weight modules over all classical simple Lie superalgebras to classifying the finite cuspidal modules over certain Lie superalgebras which we list explicitly.

Locally finite simple weight modules over twisted generalized Weyl algebras

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 $\mathfrak {age(1)}$ . Most of these modules have infinite dimensional weight spaces and so far the algebra $\mathfrak {age(1)}$ 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 Schrödinger algebra $\mathcal {S}$ that have a simple $\mathfrak {age(1)}$ -submodule thus constructing many new simple weight $\mathcal {S}$ -modules.

We propose a very general construction of simple Virasoro modules generalizing and including both highest weight and Whittaker modules. This construction enables us to classify all simple Virasoro modules that are locally finite over a positive part. To obtain those irreducible Virasoro modules, we use simple modules over a family of finite dimensional solvable Lie algebras. For one of these algebras, all simple modules are classified by R. Block and we extend this classification to the next member of the family. As a result, we recover many known but also construct a lot of new simple Virasoro modules. We also propose a revision of the setup for study of Whittaker modules.

BGG algebras and the BGG reciprocity principle

Let $\mathfrak{g}$ be a reductive Lie algebra over $\mathbb{C}$. For any simple weight module of $\mathfrak{g}$ with finite-dimensional weight spaces, we show that its Dirac cohomology is vanished unless it is a highest weight module. This completes the calculation of Dirac cohomology for simple weight modules since the Dirac cohomology of simple highest weight modules was carried out in our previous work. We also show that the Dirac index pairing of two weight modules which have infinitesimal characters agrees with their Euler-Poincar\'{e} pairing. The analogue of this result for Harish-Chandra modules is a consequence of the Kazhdan's orthogonality conjecture which was settled by the first named author and Binyong Sun.

In this paper, we study the representation theory of Hopf-Ore extensions of group algebras and pointed Hopf algebras of rank one over an arbitrary field $k$. Let $H=kG(\chi, a,\d)$ be a Hopf-Ore extension of $kG$ and $H'$ a rank one quotient Hopf algebra of $H$, where $k$ is a field, $G$ is a group, $a$ is a central element of $G$ and $\chi$ is a $k$-valued character for $G$ with $\chi(a)\neq 1$. We first show that the simple weight modules over $H$ and $H'$ are finite dimensional. Then we describe the structures of all simple weight modules over $H$ and $H'$, and classify them. We also consider the decomposition of the tensor product of two simple weight modules over $H'$ into the direct sum of indecomposable modules. Furthermore, we describe the structures of finite dimensional indecomposable weight modules over $H$ and $H'$, and classify them. Finally, when $\chi(a)$ is a primitive $n$-th root of unity for some $n>2$, we determine all finite dimensional indecomposable projective objects in the category of weight modules over $H'$.

Simple restricted modules for Neveu-Schwarz algebra