On algebraic curves for commuting elements in $q$-Heisenberg algebras

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 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.