Johan RICHTER and Sergei SILVESTROV
Centre for Mathematical Sciences, Lund University, P.O. Box 118, SE-221 00 Lund, Sweden
Received date: May 21, 2009; Revised date: September 20, 2009;
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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 dierent 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 dening 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.
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 dierential 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 modications 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 dierential 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 , the key Burchnall-Chaundy-type theorem on algebraic dependence of commuting elements in q-deformed Heisenberg algebras (and thus as a corollary for q-dierence 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 . The proof in  is totally dierent 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 specic 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 , a step in that direction was taken by oering 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 , 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 conrming 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 dierential operators to the q-deformed case, after passing to a specic 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- dierence 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 . 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 , 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 , 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 . 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 dierent 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  but putting it into general context of the elements of the q-deformed Heisenberg algebra rather then operators of a specic 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 dening some algebraic curves. The commuting elements satisfy the equations of these algebraic curves . 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.
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 dene the q-deformed Heisenberg algebra over K as
The identity element will be denoted by I. For q = 1 we recover the classical Heisenberg algebra (called also Weyl algebra). One can dene 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 dened and does not depend on how the elements are written is proved in [4, Chapter 4]. We also dene the total degree function . In [4, Chapter 4] the following theorem is proved.
Theorem 2.1. Let for some Then
We dene the sets Rn for all integers i by
If the element belongs to some Rn, we say that it is homogeneous. We also dene a function
by dening to be the unique integer such that This 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 writewhere This decomposition is unique. We then dene the projection of The notation is intended to recall the notation for intersection. At this point we dene two new functions. They are dened for all nonzero elements of H(q).
These functions are known as the upper and lower chain functions, respectively.
We begin by noting that all elements of R0 commute with each other . Furthermore, the products of two elements are in is a commutative subalgebra. We want to show that it is in fact a maximal commutative subalgebra.
For an element we define 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 be two commuting elements in H(q). Then the following is true:
We now describe the centralizer of an element in R0
Theorem 3.2. Let q be of free type and Assume further that for all Then
Proof. As we noted above It remains to show the other inclusion. Let `be an arbitrary nonzero element of Theorem 3.1 we must have since Similarly we must have that So in the direct sum decomposition only elements in R0 occurs. Thus
Corollary 3.3. R0 is maximal commutative.
.Proof. Let 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
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.
Let be of free type. If commute, then there exists a nonzero such that
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
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
where are 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. will be placed as the element in row k+1 and column . 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 can be written as Every such will satisfy 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
be two commuting elements, the pj and rj being polynomials, and denote their eliminant by Then Furthermore has degree n seen as polynomial in s. If has leading coecientonce again seen as a polynomial in s. Symmetrically, will have degree m seen as a polynomial in t. The coecient of tm will be
Let We can write
Then at least one
In the general case the theorem does not rule out that one can get several nonzero in the expansion of the eliminant, This does not, however, occur when belong to R0
Theorem 5.1. Let Then, with the same notation as before, there will be only one nonzero when the eliminant is computed and this i will equal nm..
Proof. We begin by noting that will be of the form 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 (that is in the rst m rows) and otherwise (in the last n rows), where theare 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
where σ denotes a permutation. But looking at an arbitrary term of the sum we nd that it can be written as
for a polynomial But the two sums in the exponent cancel, since they have the same terms in dierent order, and we conclude that we get the exponent mn. Since we picked an arbitrary term, we are done.
We will include some examples here to give a feeling for the construction of the eliminant and our result. Let Then
On computing the determinant we nd that the annihilating polynomial is This is only a slight modication of the classical case when q = 1. (We note that it makes no dierence whether we set q = 1 at the beginning of the calculation or the end.) no dierence 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 Then we nd that
We get the annihilating polynomial
Once again no essential simplication occurs if we let q approach 1. Now set The determinant becomes
and we get the annihilating polynomial
In the classical case this polynomial becomes
As a nal example we can take The eliminant is
We then get the annihilating polynomial
The limit when q goes towards 1 is
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.
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.