# Non-weight modules over algebras related to the Virasoro algebra

###### Abstract.

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 and a class of Block type Lie algebras , where 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 irreducible modules mentioned above with irreducible highest weight modules or Whittaker modules.

###### Key words and phrases:

Virasoro algebra, loop-Virasoro algebra, Block type Lie algebra, non-weight module, simple module###### 2010 Mathematics Subject Classification:

17B10, 17B35, 17B65, 17B68## 1. Introduction

Throughout the paper, we denote by

The Virasoro algebra, denoted by , is an infinite dimensional Lie algebra over with basis and defining relations

which is the universal central extension of the so-called infinite dimensional Witt algebra of rank one. The Virasoro algebra is one of the most important Lie algebras in both mathematics and mathematical physics (cf. [13]). The representation theory of the Virasoro algebra has been extensively studied. In [20], O. Mathieu classified all simple Harish-Chandra modules, which was conjectured by Kac [12]. In recent years, many authors constructed various simple non-Harish-Chandra modules and simple non-weight modules(cf. [1, 17, 21, 22, 28, 14, 18, 27, 7, 19, 4]). In particular, the authors in [27] constructed a class of -modules that are free of rank one when restricted to the Cartan subalgebra. Explicitly, the -module structure on is given by

(1.1) | ||||

Actually, this class of modules were first introduced and studied in [18] as quotient modules of fraction -module. Recently, this kind of non-weight modules, which many authors call -free modules, have been extensively studied.

The notation of -free modules was first introduced by J. Nilsson [23] for the simple Lie algebra . The idea originated in the attempt to understand whether the general setup for study of Whittaker modules proposed in [1] can be used to construct some explicit families of simple -modules. In this paper and a subsequent paper [24], Nilsson showed that a finite dimensional simple Lie algebra has nontrivial -free modules if and only if it is of type or . Furthermore, the -free modules of rank one for the Kac-Moody Lie algebras were determined in [2]. They proved that there are no nontrivial -free modules of rank one for affine type and indefinite type. And the idea was exploited and generalized to consider modules over infinite dimensional Lie algebras, such as the Witt algebras of all ranks [27], Heisenberg-Virasoro algebra and algebra [5], simple finite-dimensional Lie superalgebras [3], the algebras [11], the Lie algebras related to the Virasoro algebra [8] and so on. The aim of this paper is to classify such modules for the loop-Virasoro algebras and a class of Block type Lie algebras.

This paper is organized as follows. In Section 2, we construct a class of non-weight modules over the loop-Virasoro algebra. The simplicity and isomorphism classes of these modules are explicitly determined. Moreover, we obtain the simplicity of modules over the loop-Virasoro algebra by taking tensor products of some irreducible modules mentioned above with irreducible highest weight modules or Whittaker modules. Section 3 is devoted to classifying the modules whose restriction to the Cartan subalgebra are free of rank one over a class of Block type Lie algebras. We also provide a necessary and sufficient condition for such modules to be simple.

## 2. Modules over the loop-Virasoro algebra

The loop-Virasoro algebra is the Lie algebra that is the tensor product of the Virasoro algebra and the Laurent polynomial algebra , i.e., with -basis and defining relations

We see that has a copy of Virasoro algebra which is . For convenience, we simply write and . It is obvious that has the natural -grading

Note that is an infinite dimensional abelian subalgebra of , and that is the center of . The Cartan subalgebra (modulo center) of is spanned by . Various classes of representation of the loop-Virasoro algebra were studied and classified in [10, 15]. In this section, we will determine the -modules which are free of rank when regarded as -modules.

###### Definition 2.1.

For , define the action of on as follows:

(2.1) | ||||

where .

###### Proposition 2.2.

is an -module under the action given in Definition 2.1.

###### Proof.

The following result determines the isomorphism classes of the -modules .

###### Proposition 2.3.

Let . Then the -modules and are isomorphic if and only if

###### Proof.

It suffices to show the necessary part. Suppose is an isomorphism of -modules with the inverse . Regard and as -modules, we get by [18]. To complete the proof, we only need to show . For that, let , we have

Similarly,

Hence,

which implies . Combining this with and

we konw that . ∎

We are now in the position to present the following main result of this section.

###### Theorem 2.4.

Let be an -module such that it is free of rank one as a -module. Then for some and . Moreover, is simple if and only if for some .

###### Proof.

Viewed as the Virasoro-module, we have defined in (1.1), that is,

where and . For , now we consider the actions of (resp. ) on . They are completely determined by the actions of and on . Explicitly, for any polynomial , we have

where . From , we know that

(2.2) |

This shows that divides . These entail us to assume that for some . Inserting this into (2.2) gives , which in turn requires for some with . Thus,

Using

we obtain

It follows from the above recurrence relations that () for all . Hence,

By , we obtain

It follows that

(2.3) |

Combining this with gives

From

we obtain . Thus, (2.3) is simply written as for any . Consequently,

Hence,

Moreover, from the following equality

we immediately get for any . The above discussion implies that .

Furthermore, if is a simple -module, then it is obvious simple as a -module. It follows directly from [18] that . On the other hand, if , then it is a routine to check that is an -submodule of . ∎

###### Remark 2.5.

The following result asserts the simplicity of -modules by taking tensor products of some irreducible modules in Theorem 2.4 with irreducible highest weight modules or Whittaker modules.

###### Theorem 2.6.

Let for with the pairwise distinct. Let be a highest weight module or Whittaker module over . Then the tensor product is an irreducible -module. Especially, is an irreducible -module.

###### Proof.

The proof is similar to that of [28, Theorem 1]. We omit the details. ∎

## 3. Modules over Block type Lie algebras

For , we denote if , and otherwise. For any positive integer , we use to denote the positive integer greater than or equal to . For any nonzero complex number , the Block type Lie algebra has a basis over subject to the following relations

The Lie algebra is in fact a subalgebra of some special case of generalized Block algebras studied in [9]. It is also a half part of the Block type algebras studied in [16] and is the Block type Lie algebra considered in [29]. One sees that the center of is . In this section, we concentrate on the algebra

with basis

Here and below, when , we abuse the notation by writing rather than (we will make sure that this abuse of notation will not create any confusion). The Lie algebra is interesting in the sense that it contains the following subalgebra

which is isomorphic to the well-known Virasoro algebra. The Lie algebra is interesting to us in another aspect that it also contains many important subquotient algebras for , where

For instance, is the (centerless) Virasoro algebra (thus the Virasoro algebra is both a subalgebra and a quotient algebra of ), is the (centerless) twisted Heisenberg-Virasoro algebra. Note that the Cartan subalgebra of is spanned by . The representation theory of has been extensively studied by many authors. For example, the authors in [29, 25, 26] presented a classification of the irreducible Harish-Chandra modules over . In [6] and [30], the authors classified the unitary Harish-Chandra modules over . In this section, we will determine the -modules which are free of rank one when regarded as -modules.

If , the -module (i.e.,-module) becomes a -module by setting

The resulting -module will be also denoted by for brevity. Similarly, the twisted Heisenberg-Virasoro-module (i.e., -module) constructed in [5] can be extended to a -module by setting

The resulting -module will be also denoted by for brevity. In a uniform way, the -module structure on and is given by

(3.1) | ||||

where .

###### Theorem 3.1.

Let be a -module such that it is free of rank one as a -module. Then there exist some and such that

(3.2) |

Moreover, is simple if and only if or in (3.2).

###### Proof.

Regarded as the Virasoro-module, we have , that is,

For , now we consider the actions of on , which are completely determined by the actions of on . Explicitly, for any polynomial , we have

where .

###### Case 1.

.

In this case, from , we obtain

which implies for any . Combining this with , we see that for any . Moreover, since , it follows that for any . Now using

we get for any . Putting our observation together, we know that .

###### Case 2.

.

In this case, the subalgebra generated by is isomorphic to the twisted Heisenberg-Virasoro algebra. Following from [5, Theorem 2], we have

for some . By , we get

from which we see that

(3.3) |

From , we know that

(3.4) |

If , then the above formula together with (3.3) gives that for any . If , it follows from (3.3) and (3.4) that for any . Using , we get for any . Consequently, the subalgebra vanishes on . Then is isomorphic to .

###### Case 3.

.

Before the proof of this case, we present some formulae here, which will be used to do calculations in this following. For any , we have

which gives rise to

(3.5) |

Using this, we further have

which shows that for any ,

(3.6) | |||||

Assume that

(3.7) |

Substituting this expression into (3.6) (for the case ) and comparing the coefficients of of both sides, we obtain

(3.8) |

###### Subcase 1.

.

It follows from (3.8) that , i.e., . Since , it follows that for any . While the identity implies that for . That is to say that the subalgebra vanishes on . Then .

###### Subcase 2.

.

We claim that in this subcase. Suppose on the contrary that . One sees from (3.8) that

First we assert that for in (3.7). The proof is given by induction on . We can write

Substituting this into (3.6) (for the case ) and comparing the coefficient of , we obtain . Assume that the conclusion holds for , that is,

Also, by inserting this expression into (3.6) (for the case ) and comparing the coefficient of , we obtain

(3.9) | |||||

If is an odd number, we can simplify (3.9) as

This together with . If is an even number, then rewrite (3.9) as indicates

Since can be an arbitrary integer, it follows from the above formula that , completing the induction step. Hence, we have

(3.10) |

Similarly, one has

(3.11) |

where

(3.11) in the case together with (3.5) gives rise to

(3.12) |

Now (3.10) along with (3.5) forces

(3.13) |

Combining (3.10), (3.13) with , we know that

(3.14) |

(3.15) |