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Journal of Generalized Lie Theory and Applications
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On algebraic curves for commuting elements in q-Heisenberg algebras

Johan RICHTER and Sergei SILVESTROV

Centre for Mathematical Sciences, Lund University, P.O. Box 118, SE-221 00 Lund, Sweden

E-mails: [email protected], [email protected]

Received date: May 21, 2009; Revised date: September 20, 2009;

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Abstract

In the present article we continue investigating the algebraic dependence of commuting elements in q-deformed Heisenberg algebras. We provide a simple proof that the 0-chain subalgebra is a maximal commutative subalgebra when q is of free type and that it coincides with the centralizer (commutant) of any one of its elements di erent from the scalar multiples of the unity. We review the Burchnall-Chaundy-type construction for proving algebraic dependence and obtaining corresponding algebraic curves for commuting elements in the q-deformed Heisenberg algebra by computing a certain determinant with entries depending on two commuting variables and one of the generators. The coe cients in front of the powers of the generator in the expansion of the determinant are polynomials in the two variables de ning some algebraic curves and annihilating the two commuting elements. We show that for the elements from the 0-chain subalgebra exactly one algebraic curve arises in the expansion of the determinant. Finally, we present several examples of computation of such algebraic curves and also make some observations on the properties of these curves.

Introduction

In 1994, one of the authors of the present paper, S. Silvestrov, based on consideration of the previous literature and a series of trial computations, made the following three-part conjecture.

• The rst part of the conjecture stated that the Burchnall-Chaundy-type result on algebraic dependence of commuting elements can be proved in greater generality, that is, for much more general classes of noncommutative algebras and rings than the Heisenberg algebra and related algebras of di erential operators treated by Burchnall and Chaundy and in subsequent literature [1, 2, 3, 7, 8, 11].

• The second part stated that the Burchnall-Chaundy eliminant construction of annihilating algebraic curves formulated in determinant (resultant) form works after some appropriate modi cations for most or possibly all classes of algebras where the Burchnall- Chaundy-type result on algebraic dependence of commuting elements can be proved.

• Finally, the third part of the conjecture stated that the proof of the vanishing of the corresponding determinant algebraic curves on the commuting elements can be performed in a purely algebraic way for all classes of algebras or rings where this fact is true, that is, using only the internal structure and calculations with the elements in the corresponding algebras or rings and the algebraic combinatorial expansion formulas for the corresponding determinants without any need of passing to operator representations and use of analytic methods as in the Burchnall-Chaundy-type proofs.

This third part of the conjecture remains widely open with no general such proofs available for any classes of algebras and rings, even in the case of the usual Heisenberg algebra and di erential operators, and with only a series of examples calculated for the Heisenberg algebra, q-Heisenberg algebra, and some more general algebras, all supporting the conjecture. In the rst and second parts of the conjecture progress has been made. In [4], the key Burchnall-Chaundy-type theorem on algebraic dependence of commuting elements in q-deformed Heisenberg algebras (and thus as a corollary for q-di erence operators as operators representing q-deformed Heisenberg algebras) was obtained. The result and the methods have been extended to more general algebras and rings generalizing q-deformed Heisenberg algebras (generalized Weyl structures and graded rings) in [5]. The proof in [4] is totally di erent from the Burchnall-Chaundy-type proof. It is an existence argument based only on the intrinsic properties of the elements and internal structure of q-deformed Heisenberg algebras, thus supporting the rst part of the conjecture. It can be used successfully for an algorithmic implementation for computing the corresponding algebraic curves for given commuting elements. However, it does not give any speci c information on the structure or properties of such algebraic curves or any general formulae. It is thus important to have a way of describing such algebraic curves by some explicit formulae, as, for example, those obtained using the Burchnall-Chaundy eliminant construction for the q = 1 case, that is, for the classical Heisenberg algebra. In [10], a step in that direction was taken by o ering a number of examples, all supporting the claim that the eliminant determinant method should work in the general case. However, no general proof for this was provided. The complete proof following the Burchnall-Chaundy approach in the case of q not a root of unity has been recently obtained [6], by showing that the determinant eliminant construction, properly adjusted for the q-deformed Heisenberg algebras, gives annihilating curves for commuting elements in the q-deformed Heisenberg algebra when q is not a root of unity, thus con rming the second part of the conjecture for these algebras. That proof was obtained by adapting the Burchnall-Chaundy eliminant determinant method of the case q = 1 of di erential operators to the q-deformed case, after passing to a speci c faithful representation of the q-deformed Heisenberg algebra on Laurent series and then performing a detailed analysis of the kernels of arbitrary operators in the image of this representation. While exploring the determinant eliminant construction of the annihilating curves, we also obtained some further information on such curves and some other results on dimensions and bases in the eigenspaces of the q- di erence operators in the image of the chosen representation of the q-deformed Heisenberg algebra. Recently, a further extension of Burchnall-Chaundy eliminant determinant method to the context of σ derivations and Ore extension rings has been considered in [9]. In the case of q being a root of unity the algebraic dependence of commuting elements holds only over the center of the q-deformed Heisenberg algebra [4], and it is still unknown how to modify the eliminant determinant construction to yield annihilating curves for this case.

In the present article we continue investigation of the algebraic dependence of commuting elements in q-deformed Heisenberg algebras within the context of [4, 6, 10]. In Section 2, following [4], we recall some preliminaries on q-deformed Heisenberg algebra, including degree function, decomposition into the direct sum of the \chain" subspaces indexed by the integers and corresponding to this decomposition the upper and lower chain functions. In Section 3, we consider in more detail the 0-chain subspace (indexed by zero). This subspace is a commutative subalgebra in the q-deformed Heisenberg algebra playing a pivotal role for the structure of this algebra [4]. We provide a simple proof that this subalgebra is a maximal commutative subalgebra when q is of free type, and that it coincides with the centralizer (commutant) of any one of its elements di erent from the scalar multiples of the unity. In Section 4, we review the Burchnall-Chaundy-type construction for proving algebraic dependence and obtaining corresponding algebraic curves for commuting elements in the q- deformed Heisenberg algebra following [6] but putting it into general context of the elements of the q-deformed Heisenberg algebra rather then operators of a speci c representation. The construction is based on computing a certain determinant of a matrix with entries depending on two commuting variables and containing one of the generators of the q-deformed Heisenberg algebra. This matrix is constructed from commuting elements. The coecients in front of the powers of the generator in the expansion of the determinant are polynomials in the two variables de ning some algebraic curves. The commuting elements satisfy the equations of these algebraic curves [6]. In Section 5, we show that for the elements from the 0-chain subalgebra exactly one algebraic curve arises via this construction in the expansion of the determinant and then present several examples of computations of such algebraic curves and also make some observations on the properties of these curves based on these examples and further computer experiments.

Preliminaries

Let K be a eld of characteristic 0, and q a nonzero element of K. We say that q is of free type if it is 1 or not a root of unity. If q is a root of unity, we say it is of torsion type. We de ne the q-deformed Heisenberg algebra over K as

IMAGE

The identity element will be denoted by I. For q = 1 we recover the classical Heisenberg algebra (called also Weyl algebra). One can de ne degree functions degA and degB with respect to A and B on i just as on the commutative algebra of polynomials. One computes these functions by inspection just as one would in a commutative algebra. That the functions are well de ned and does not depend on how the elements are written is proved in [4, Chapter 4]. We also de ne the total degree function image . In [4, Chapter 4] the following theorem is proved.

Theorem 2.1. Let imagefor some imageThen

image

We de ne the sets Rn for all integers i by

image

If the element image belongs to some Rn, we say that it is homogeneous. We also de ne a function

image

by de ning imageto be the unique integer such that iamgeThis function is called the chain function.

All Ri are vector spaces over K. Further H(q) is the direct sum of all the Ri. We can use this to define a projection operation. Let α be an element of H(q). We can writeIMAGEwhere IMAGE This decomposition is unique. We then de ne the projection ofIMAGEIMAGE The notation is intended to recall the notation for intersection. At this point we de ne two new functions. They are de ned for all nonzero elements of H(q).

IMAGE

These functions are known as the upper and lower chain functions, respectively.

R0 is maximal commutative

We begin by noting that all elements of R0 commute with each other [4]. Furthermore, the products of two elements image are inimage is a commutative subalgebra. We want to show that it is in fact a maximal commutative subalgebra.

For an element image we defineimage In [4, Chapter 6] the following theorem is proved (as a part of Theorem 6.10).

Theorem 3.1. Let q be of free type. Let imagebe two commuting elements in H(q). Then the following is true:

image

We now describe the centralizer of an element in R0

Theorem 3.2. Let q be of free type and image Assume further thatiamge for allimge Thenimage

Proof. As we noted above imageIt remains to show the other inclusion. Let image`be an arbitrary nonzero element of imageTheorem 3.1 we must have image sinceimage Similarly we must have thatimage So in the direct sum decomposition only elements in R0 occurs. Thus image

Corollary 3.3. R0 is maximal commutative.

.Proof. Let image be an element that commutes with everything in R0. Then in particular it must commute with BA. But Cen(BA) = R0 by the preceding theorem. Thus image

Annihilating polynomials

As mentioned in the introduction any two commuting elements in H(q) must be algebraically dependent when q is of free type. More formally, we have the following.

Theorem 4.1.

Let image be of free type. Ifimage commute, then there exists a nonzero image such thatimage

We now describe an explicit construction of this polynomial. We let s and t be variables that take values in the base eld K. We write the commuting elements image

imaeg

where the pi and ri are polynomials. We will form an n + m determinant that will give us the annihilating polynomial.

Consider the expressions obtained by reordering all A to the right of B in

image

where imageare functions of B; s; t arising after reordering. The coecients of the powers of A will be the elements in the determinant that we compute. IMAGE will be placed as the element in row k+1 and column image. will be placed in row k+m+1 and column i. The determinant will thus be a polynomial in s; t and B. This polynomial, which we will call the eliminant of imagecan be written as image Every suchimage will satisfyimage and at least one of them will not be identically zero.

A more precise formulation with additional information about the construction can be found in the following.

Theorem 4.2. Let

image

be two commuting elements, the pj and rj being polynomials, and denote their eliminant by image Thenimage Furthermoreimage has degree n seen as polynomial in s. Ifimage has leading coecientimageonce again seen as a polynomial in s. Symmetrically, imagewill have degree m seen as a polynomial in t. The coecient of tm will be image

Let image We can writeIMAGE

Then at least one IMAGE

The eliminant when the elements belong to R0

In the general case the theorem does not rule out that one can get several nonzero imagein the expansion of the eliminant, imageThis does not, however, occur when image belong to R0

Theorem 5.1. Let imageThen, with the same notation as before, there will be only one nonzero image when the eliminant is computed and this i will equal nm..

Proof. We begin by noting that image will be of the form image where the ai belong to K.

We use this result to describe the structure of the eliminant. Denote the element in row u and column v by eu;v. Then we will have image (that is in the rst m rows) and image otherwise (in the last n rows), where theimageare polynomials over K. Many of them will of course be zero, in particular those where B would otherwise occur with a negative exponent.

We know, from ordinary linear algebra, that image

where σ denotes a permutation. But looking at an arbitrary term of the sum we nd that it can be written as

image

for a polynomial imageBut the two sums in the exponent cancel, since they have the same terms in di erent order, and we conclude that we get the exponent mn. Since we picked an arbitrary term, we are done.

Examples

We will include some examples here to give a feeling for the construction of the eliminant and our result. Let imageThen IMAGE

On computing the determinant we nd that the annihilating polynomial is IMAGEThis is only a slight modi cation of the classical case when q = 1. (We note that it makes no di erence whether we set q = 1 at the beginning of the calculation or the end.) no di erence whether we set q = 1 at the beginning of the calculation or the end.) For our next example let α be as before and let imageThen we nd that

image

We get the annihilating polynomial iamge

Once again no essential simpli cation occurs if we let q approach 1. Now setimage The determinant becomes

image

and we get the annihilating polynomial

image

In the classical case this polynomial becomes

image

As a nal example we can take imageThe eliminant is image

We then get the annihilating polynomialIMAGE

The limit when q goes towards 1 is IMAGE

This is a simpler expression but only because the coecients are simpler. No coecient has become zero.

This illustrates that the complexity of the resulting polynomial grows pretty fast. Computer experiments indicate that Theorem 5.1 can be generalized substantially. We would be also interested to know whether the annihilating polynomials always have genus 0, a conjecture we have been unable to nd any counterexamples to.

Acknowledgements

This work was supported by the Swedish Research Council, the Swedish Foundation for International Cooperation in Research and Higher Education (STINT), the Crafoord Foundation, the Royal Physiographic Society in Lund, and the Royal Swedish Academy of Sciences.

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