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On irreducible weight representations of a new deformation Uq(sl2) of U(sl2) | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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On irreducible weight representations of a new deformation Uq(sl2) of U(sl2)

Xin TANG*

 Department of Mathematics and Computer Science, Fayetteville State University, Fayetteville NC 28304, U.S.A.

*Corresponding Author:
Xin TANG
Department of Mathematics and Computer Science
Fayetteville State University
Fayetteville NC 28304, U.S.A.
E-mail: [email protected]

 

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Abstract

Starting from a Hecke R-matrix, Jing and Zhang constructed a new deformation Uq(sl2) of U(sl2) and studied its finite dimensional representations in [Pacific J. Math., 171 (1995), 437-454]. In this note, more irreducible representations for this algebra are constructed. At first, by using methods in noncommutative algebraic geometry the points of the spectrum of the category of representations over this new deformation are studied. The construction recovers all finite dimensional irreducible representations classified by Jing and Zhang, and yields new families of infinite dimensional irreducible weight representations.

1 Introduction

Spectral theory of abelian categories was first initiated by Gabriel in [5]. In particular, Gabriel defined the injective spectrum of any noetherian Grothendieck category. The injective spectrum consists of isomorphism classes of indecomposable injective objects in the category endowed with the Zariski topology. If R is a commutative noetherian ring, then the injective spectrum of the category of all R-modules is homeomorphic to the prime spectrum of R. This homeomorphism is a part (and the main step in the argument) of the Gabriel’s reconstruction Theorem [5], according to which any noetherian commutative scheme can be uniquely reconstructed up to isomorphism from the category of quasi-coherent sheaves on it.

The general spectrum of arbitrary abelian category was defined by Rosenberg [15]. Using this spectrum, one can reconstruct any quasi-separated and quasi-compact scheme from the category of quasi-coherent sheaves on the scheme.

Isomorphism classes of simple objects of any abelian category correspond to closed points of its spectrum, and, under some mild finiteness conditions, this correspondence is bijective. For instance, the correspondence is bijective for the category of modules over an associative ring, or, more generally, for the category of quasi-coherent sheaves on a noncommutative (that is not necessarily commutative) scheme.

Thus, in order to study irreducible representations, one can first study the spectrum of the category of all representations, then single out its closed points.

As a specific application of spectral theory to representation theory, points of the spectrum of the category of modules have been constructed for a large family of algebras, which are called Hyperbolic algebras in [15]. And it is a pure luck that a lot of important “small” algebras, including U(sl2) and its quantized versions, are Hyperbolic algebras.

Starting from a Hecke R-matrix, Jing and Zhang constructed a new deformation Uq(sl2) of U(sl2) (which is denoted by Uq(sl2) in [7]). This algebra shares quite a few properties with U(sl2) , and all its finite dimensional irreducible representations are constructed explicitly in [7]. On the other hand, Uq(sl2) has a natural bialgebra structure, but, it is not a Hopf algebra. Besides, an example constructed in [7] shows that not all of its finite dimensional representations are completely reducible. So the representation theory of this new deformation differs from the representation theory of the standard quantized enveloping algebra of sl2. Therefore, it seems to be an interesting problem to further study the irreducible representations of Uq(sl2).

Note that Uq(sl2) also belongs to a general class of algebras studied in [2], where the irreducible weight modules for these algebras are classified based on the study of certain dynamical system. However, this note studies representations from the perspective of noncommutative algebraic geometry, and serves the purpose of providing a more transparent construction of irreducible weight representations for Uq(sl2). Indeed, based on sufficient knowledge about the structure of Uq(sl2), we are able to carry out explicit calculations.

To solve the problem, we first construct families of points for the spectrum of the category of representations for this deformation. Applied to the study of representations, our construction recovers all finite dimensional irreducible representations of Uq(sl2) constructed in [7], and produces new families of infinite dimensional irreducible representations as well. This work can also be regarded as one more nice application of the methods in noncommutative algebraic geometry to representation theory. For more details about spectral theory, we refer the reader to [15].

The paper is organized as follows. In Section 2, we give a very brief review on the spectrum of an abelian category. In Section 3, we review the concept of Hyperbolic algebras. In Section 4, we review some basic facts about the new deformation Uq(sl2) introduced by Jing and Zhang, and prove some supplementary useful Lemmas. In Section 5, we construct families of points of the spectrum for Uq(sl2). Then we use them to construct irreducible representations for this new deformation Uq(sl2). We follow the notations in [7], but we will always denote the new deformation by Uq(sl2). The base field will be fixed to be the complex field C, and q is not a root of unity.

2 Basic facts about the spectrum of any abelian category

In this section, we are going to review some basic notions and facts about the spectrum of any abelian category for the purpose of understanding the rest of this work. First, we review the definition of the spectrum of any abelian category, then we explain its applications in representation theory.

Let image be an abelian category. Let image be any two objects in image; We say that image if and only if N is a subquotient of the direct sum of finite copies of M. It is easy to verify that is a preorder. We say image if and only if image and image. It is obvious that image is an equivalence. Let Spec(X) be the family of all nonzero objects imagesuch for all nonzero subobject N of M, N M. The spectrum of the abelian category image is defined [15] by

image

It is endowed with a natural analogue of the Zariski topology.

The spectrum of an abelian category is one of the fundamental notions of noncommutative algebraic geometry.

The spectrum also has important applications in representation theory. This is due to the fact that there is a natural embedding of the set of isomorphism classes of simple objects of the category image into the set of closed points of Spec(X). If every nonzero object of the category image has a simple subquotient, then the embedding is a bijection. In particular, if A is an algebra and image is the category of left A-modules, then the closed points of Spec(X) are in a bijective correspondence with isomorphism classes of irreducible A-modules. The spectrum Spec(X) has much better functorial properties than the set of its closed points, like in the case of commutative algebraic geometry. So one can study the spectrum via the methods in noncommutative algebraic geometry, then apply to representation theory ([15]).

2.1 The left spectrum of a ring

If CX is the category A−mod of left modules over a ring A, then it is sometimes convenient to express the points of Spec(X) in terms of left ideals of the ring A. In order to do it, the left spectrum Specl(A) was defined in [15], which is by definition the set of all left ideals p of A such that A/p is an object of Spec(X). The relation on A − mod induces a specialization relation among left ideals, in particular, the specialization relation on Specl(A). Namely, A/m A/n iff there exists a finite subset x of elements of A such that the ideal image is contained in m. Following [15], we denote this by imageNote that the relation ≤ is just the inclusion if n is a two-sided ideal. In particular, it is the inclusion if the ring A is commutative. The map which assigns to an element of Specl(A) induces a bijection of the quotient image of Specl(A) by the equivalence relation associated with ≤ onto Spec(X). From now on, we will not distinguishimage and will express results in terms of the left spectrum.

The rest of this paper is a typical application of spectral theory to representation theory of “small” algebras.

3 Hyperbolic algebras imageand points of the spectrum

Hyperbolic algebras are studied by Rosenberg in [15] and by Bavula under the name of Generalized Weyl algebras in [1]. Hyperbolic algebra structure is very convenient for the construction of points of the spectrum. And a lot of interesting algebras such as U(sl2) and its quantized versions have a Hyperbolic algebra structure. Points of the spectrum of the category of modules over these algebras have been constructed in [15]. In this section, we review some basic facts about Hyperbolic algebras and two important construction theorems due to Rosenberg ([15]). Let  be an automorphism of a commutative algebra R; and let image be an element of R. Then we have the following definition from [15].

Definition 3.1. We denote by image the corresponding R−algebra generated by x, y subject to the following relations:

image

for all image

First, we look at some basic examples of Hyperbolic algebras.

Example 3.1. The first Weyl algebra A1 is a Hyperbolic algebra over imageimage and its quantized versions are Hyperbolic algebras too. And the reader to referred to [15] for more details about Hyperbolic algebras.

Let image mod be the category of left modules over image The Hyperbolic algebra structure is very convenient for the construction of points of the spectrum Spec(X). We replace the study of Spec(X) by the study of the left spectrum Specl(image) of the hyperbolic algebra image (see 2.1 above).

For the left spectrum of the Hyperbolic algebra, we have the following two crucial construction theorems due to Rosenberg from [15].

Theorem 3.1 ([15], Theorem 3.2.2.). 1. Letimage the orbit of P under the action of the automorphism θ is infinite.

image image

2. If the ideal P in (b), (c) or (d) is maximal, then the corresponding left ideal of Specl(image) is maximal.

3. Every left ideal image such thatimage is equivalent to one left ideal as defined above uniquely from a prime ideal image The latter means that if imageare two prime ideals of R and (α,β) and (v, μ) take values image or (1, n), then image , is equivalent toimage if and only if image

Theorem 3.2 ([15], Proposition 3.2.3.). 1. Let image be a prime ideal of such that image

2. Moreover, if P is a left ideal of image such thatimage In particular, if P is a maximal ideal, then image is a maximal left ideal.

3. If a prime ideal image is such thatimagethen image for some integer n. Conversely, image

4 A new deformationimage

Starting from an R−matrix, Jing and Zhang constructed a new deformation image (which is still denoted byimage in [7]). This new deformation is a bialgebra deformation of U(sl2) [7]. In this section, we first recall the definition of this new deformation image Then we verify thatimagehas a Hyperbolic algebra structure over a polynomial ring in two variables. Finally, we will state and verify some supplementary useful formulas, which will be used in the next section.

Let image be the field of complex numbers and image be an element of image. Let image be the image−algebra generated by e, f, h subject to following relations:

image

It is easy to see that this new deformation image shares a lot of properties with U(sl2) . However, this new deformation image is just a bialgebra deformation of U(sl2) without having a Hopf algebra structure. The finite dimensional irreducible representations of this algebra were constructed in [7], and an example was constructed to show that not every finite dimensional representation is completely reducible.

In addition, it has a Casimir element which is defined as follows:

image

We have the following basic lemma about this Casmir element C from [7].

Lemma 4.1 ([7], Lemma 3.4). The Casimir element C q−commutes with generators of image in the following sense:

image

We have the following corollary of Lemma 4.1.

Corollary 4.1. The element h commutes with ef.

Proof. This follows directly from the definition of C and Lemma 4.1.

Let us denote ef by image respectively. Letimage be the subalgebra of image generated by image is a polynomial ring in two variables image which is thus commutative. We will verify that image is a Hyperbolic algebra over R.

First of all, let us define an endomorphism θ of R by

image

It is obvious that θ is an algebra automorphism. In addition, we have the following basic

Lemma 4.2. The following identities hold:

image

Proof. The verification of the above lemma is straightforward, and is left to the reader.

From Lemma 4.2, we obtain the following

Proposition 4.1. image is a Hyperbolic algebra over R.

Proof. This follows directly from the definition of Hyperbolic algebras and Lemma 4.2.

Corollary 4.2. The Gelfand-Kirillov dimension of image is 3.

Proof. This follows from the fact that image is a Hyperbolic algebra over a polynomial algebra in two variables. Since the Gelfand-Kirillov dimension of the latter is 2, the Gelfand-Kirillov dimension of image is 3.

Before we finish this section, we would like to state another useful lemma, which will be needed in the next section.

Lemma 4.3. One has

image

and

image

Proof. First of all, we prove the statement is true for image When n = 0, the statement is obviously true. And we have image Suppose that the statement is true for n − 1. Note that we have

image

So we have proved the first statement for image by using induction. Similar argument shows that the statement is true for all image

Now we are going to prove the second statement. Since C is in R, then we have image and image by Lemma 4.1. So we have image.Thus

image

Therefore, we have

imagefor all image

5 Construction of points of the spectrum for image

In this section, we construct families of points of the spectrum of the category of representations of image using the construction theorems quoted in Section 2 from [15]. As a result, we will obtain families of irreducible weight representations of image

First of all, we have the following basic

Proposition 5.1. Let image be a closed point of Spec(R). Thenimage is a finite set if and only ifimage

Proof. If

image

then

image

In addition,

image

So the orbit of P is finite if and only if

image

and

image

hence if and only if

imagethat completes the proof.

 

For the rest of this section, we may assume that image We have the following

Theorem 5.1. Suppose that q is not a root of unity andimage is a half non-negative integer.

Let

image

be a maximal ideal of R, then the corresponding point

image

of the left spectrum image is a closed point. Hence the representationimage corresponding to this point is a finite dimensional irreducible representation of image.

Proof. If

image

then we have image andimageThus the statement follows from (b) of part (1) of the Theorem 3.1.

Remark 5.1. The representations constructed above recover all finite dimensional irreducible representations as constructed in [7].

Now we are going to construct some new families of infinite dimensional irreducible weight representations. Suppose image is a maximal ideal of R, then we have the following

Theorem 5.2. 1. If

image

for all non-negative half integer s, then the corresponding point

image

of the spectrum is closed. And the corresponding representation image is an infinite dimensional irreducible highest weight representation.

2. If

image

for all half positive integers s, then the corresponding point

image

of the spectrum is closed, and the corresponding representation image is an infinite dimensional irreducible lowest weight representation.

Proof. We will only verify the first part of this statement, and the rest is similar. According to Lemma 4.4, we have

image

If

image

then we have image a closed point of the left spectrumimage by Theorem 3.1, hence the corresponding representation image is an infinite dimensional highest weight irreducible representation.

Theorem 5.3. Let image be a maximal ideal of R such that

image

for allimage then the pointimage is a closed point of the left spectrum, and the corresponding representation image is an infinite dimensional irreducible weight representation.

Proof. The proof is a direct verification of the conditions in Theorem 3.2, and we will omit it here.

Remark 5.2. It is tempting to construct some nonweight irreducible representations for image [17, 18]. Unfortunately, the Whittaker model does not work here. The difficulty lies in that the algebra image has a trivial center. So it would be an interesting problem to find a way of constructing nonweight irreducible representations for this new deformation image.

Acknowledgements

The author is very grateful to the referees for their comments and suggestions, which have significantly improved the paper.

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