Field Generators for the Quantum Plane

Abstract

Let k be a commutative field and q a (nonzero and not root of one) quantization parameter in k. Manin’s quantum plane P = k q[x,y] is the k-algebra of noncommutative polynomials in two variables with commutation law xy = qyx. The quantum torus R = k q[x ±1, y ±1 ] is the simple localization of P consisting of quantum Laurent polynomials. We denote by k q(x,y) = Frac R = Frac P the skew field of quantum rational functions over k. For any nonzero polynomials A,B ∈R such that AB = q BA, the (skew) subfield k q(A, B) of k q(x, y) generated by A and B is isomorphic to k q(x,y); the main question discussed in the paper is then: do we have k q(x,y) = k q(A,B)? We prove that this equality holds if at least one of the generators A or B is a monomial in R, or if the support of at least one of them is based on a line.